url
stringlengths 15
1.13k
| text
stringlengths 100
1.04M
| metadata
stringlengths 1.06k
1.1k
|
|---|---|---|
https://www.physicsforums.com/threads/mean-energy-and-preassure-inolving-partition-function.430629/ | # Mean energy and preassure inolving partition function
1. Sep 20, 2010
### rayman123
1. The problem statement, all variables and given/known data
Can someone explain to me how this work out this formula
2. Relevant equations
we are supposed to find the mean energy and preasure of a gas with given partition function
3. The attempt at a solution
mean energy is given $$\overline{-}U=\sum_{r}E_{r}p_{r}$$
I know also that Boltzman's probability distribution is described by
$$p_{r}= \frac{e^{-\beta E_{r}}}{\sum_{r}e^{-\beta E_{r}}}$$
because the partition function is definied as $$z=\sum_{r}e^{-\beta E_{r}^}$$
so rewriting now the Boltzman's probablility distribution I get
$$p_{r}= \frac{e^{-\beta E_{r}}}{z}$$
1. The problem statement, all variables and given/known data
now going back to the mean energy I can write
$$\overline{-}U=\frac{1}{z}\sum_{r}E_{r}e^{-\beta E_{r}$$
These are operations I do not understand. Could someone explain them step by step ?
$$\sum_{r}E_{r}e^{-\beta E_{r}}= -\frac{\partial}{\partial \beta}\sum_{r}e^{-\beta E_{r}}= -\frac{\partial}{\partial \beta}z$$
and the final one
$$U= -\frac{1}{z}\frac{\partial z}{\partial \beta}=-\frac{\partial lnz}{\partial \beta}$$
2. Sep 20, 2010
### zhermes
The first part of the equation is how you find the (ensemble) average energy---you find the weighted average over all ensemble states. The second part is showing that you can rewrite this as the partial derivative with respect to beta; if you take the derivative of the boltzmann factor WRT beta, the energy E gets pulled down
$$\frac{\partial}{\partial \beta}e^{-\beta E_{r}}= E_{r} e^{-\beta E_{r}}$$
In the expression using the partial derivative wrt beta, the sum over boltzmann factors is the definition of the partition function. So you end up with the partial derivative of the partition function = the average energy.
Again, this is just rewriting things with derivatives. Using the chain rule, if you take the derivative of lnz, you get a 1/z term out front.
'z' is some function of beta $$z = z(\beta)$$, so using the chain rule: $$\frac{\partial ln z(\beta)}{\partial \beta} = \frac{1}{z} \frac{\partial z(\beta)}{\partial \beta}$$
Does that make sense?
Last edited: Sep 20, 2010
3. Sep 21, 2010
### rayman123
yes it does!:) thank you
Similar Discussions: Mean energy and preassure inolving partition function | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9837461113929749, "perplexity": 894.207972972425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948609934.85/warc/CC-MAIN-20171218063927-20171218085927-00617.warc.gz"} |
http://math.stackexchange.com/questions/310513/from-weak-and-weak-star-to-norm-convergence | # From weak and weak star to norm convergence
I haven't found this yet and I'm somehow not sure if my idea is correct.
The Problem: Let $X$ be a separable Banach-Space, let $x_k\to x$ weakly and such that for every $\lambda_k \to \lambda$ weakly-* there holds $\lambda_k(x_k)\to \lambda(x)$. Then $x_k\to x$ strongly (in the norm of $X$).
My idea was to give a proof with contradiction. Hence assume there holds for some $\epsilon >0$ and a subsequence of $x_k$ denoted again by $k$: $$\epsilon <||x_k-x||=|\lambda^*_k(x_k-x)|$$ for a functional provided by Hahn-Banach theorem with norm 1. From that and the separabilty we conclude that there is a further subsequence such that $\lambda_{k_l}^*\to \lambda^*$ weakly-* . Since each subsequence of $x_k$ also converges weakly to $x$ we use the "weakly-*" assumption to receive a contradiction since for the previous subsequence $$|\lambda^*_{k_l}(x_{k_l}-x)|=|\lambda^*_{k_l}(x_{k_l})-\lambda^*_{k_l}(x)|\to 0$$
Somehow this seems to easy and I feel like I'm not using especially the weak convergence in the right way.
-
I made an edit fixing what must surely have been a typo in the post: I inserted an $\epsilon$ in what seems the intended place. – Harald Hanche-Olsen Feb 21 '13 at 20:40
I think the reason for your confusion is the condition made on the convergence $x_k \rightarrow x$. The condition that for any $\lambda_k \rightarrow \lambda$ weak* we have $|\lambda_k(x_k) - \lambda(x)| \rightarrow 0$ implies $x_k \rightarrow x$ weakly, by taking $\lambda_k$ to be the constant sequence. You certainly use this condition, so there's nothing to worry about.
Thanks. You're right. I forgot to mention the first subsequence. Somehow I was doubting since this looked somehow like only exploiting $||x_k||\to ||x||$ (I considered $x=0$ first and then this is of course norm conergence :)). But this instead of the weakly-* assumption from above does not lead to strong convergence in general separable spaces. Additionally the weak convergence of $x_k$ is somehow only implicitly used in the weakly-* assumption. – Quickbeam2k1 Feb 21 '13 at 20:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9851915240287781, "perplexity": 227.81965602185778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049278385.1/warc/CC-MAIN-20160524002118-00067-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://mathinsight.org/historical_theoretical_comments_mean_value_theorem_refresher | # Math Insight
### Historical and theoretical comments: Mean Value Theorem
For several reasons, the traditional way that Taylor polynomials are taught gives the impression that the ideas are inextricably linked with issues about infinite series. This is not so, but every calculus book I know takes that approach. The reasons for this systematic mistake are complicated. Anyway, we will not make that mistake here, although we may talk about infinite series later.
Instead of following the tradition, we will immediately talk about Taylor polynomials, without first tiring ourselves over infinite series, and without fooling anyone into thinking that Taylor polynomials have the infinite series stuff as prerequisite!
The theoretical underpinning for these facts about Taylor polynomials is The Mean Value Theorem, which itself depends upon some fairly subtle properties of the real numbers. It asserts that, for a function $f$ differentiable on an interval $[a,b]$, there is a point $c$ in the interior $(a,b)$ of this interval so that $$f'(c)={f(b)-f(a)\over b-a}$$
Note that the latter expression is the formula for the slope of the ‘chord’ or ‘secant’ line connecting the two points $(a,f(a))$ and $(b,f(b))$ on the graph of $f$. And the $f'(c)$ can be interpreted as the slope of the tangent line to the curve at the point $(c,f(c))$.
In many traditional scenarios a person is expected to commit the statement of the Mean Value Theorem to memory. And be able to respond to issues like ‘Find a point $c$ in the interval $[0,1]$ satisfying the conclusion of the Mean Value Theorem for the function $f(x)=x^2$.’ This is pointless and we won't do it. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9612252712249756, "perplexity": 173.80191956504285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917120461.7/warc/CC-MAIN-20170423031200-00227-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/394000/borceux-handbook-of-categorical-algebra-i-proposition-3-4-2 | # Borceux. Handbook of Categorical Algebra I. Proposition 3.4.2.
I'm trying to understand proposition 3.' HoCA (vol. I).
Proposition 3.4.2 Consider a functor $F\colon \mathcal A \to \mathcal B$ with both a left adjoint functor $G$ and a right adjoint functor $H$. If functor is full and faithful, si is the other adjoint functor.
There are some minor typos in the proof, but they're not a big deal.
Borceux calls $\epsilon \colon GF \Rightarrow 1_{\mathcal A}$, $\eta\colon 1_{\mathcal B} \Rightarrow FG$ and $\alpha \colon FH \Rightarrow 1_{\mathcal B}$, $\beta\colon 1_{\mathcal A} \Rightarrow HF$ the counits and units of the adjuctions.
By duality, it is enough to prove, wlog, that if H is fully faithful then the same holds for G. Further, this statement is equivalent (by proposition 3.4.1) to say that $\eta$ is iso whenever $\alpha$ is.
The claim is that
$\alpha \circ (1_F *\epsilon * 1_H) \circ (1_{FG}*\alpha^{-1}) = \eta^{-1}$.
By the naturality of $eta$,
$\alpha \circ (1_F *\epsilon * 1_H) \circ (1_{FG}*\alpha^{-1}) \circ \eta = \alpha \circ (1_F *\epsilon * 1_H) \circ (\eta*1_{FH})\circ \alpha^{-1}.$
The trangular equality, i.e., $(1_F *\epsilon) \circ (\eta*1_{F}) = 1_{\mathcal F}$ (thus, in particular $(1_F *\epsilon *1_H) \circ (\eta*1_{FH}) = 1_{\mathcal FH}$), gives us
$alpha \circ \alpha^{-1} = 1_{\mathcal B}$.
On the other hand, first note that the triangular equality
$(\alpha *1_F)\circ (1_F * \beta) = 1_F$
implies that $1_F* \beta = (\alpha *1_F)^{-1}\quad(\heartsuit)$.
Now consider
$\star = \eta\circ \alpha \circ (1_F *\epsilon * 1_H) \circ (1_{FG}*\alpha^{-1})$
$\star = (\text{by naturality of }\alpha) = (\alpha*1_{FG})\circ (1_{FH}*\eta) \circ (1_F *\epsilon * 1_H) \circ (1_{FG}*\alpha^{-1})$
$\star = (\text{by naturality of }1_F *\epsilon * 1_H) = (\alpha*1_{FG})\circ (1_F *\epsilon * 1_{HFG})\circ (1_{FGFH}*\eta) \circ (1_{FG}*\alpha^{-1})$
$\star = (\text{by naturality of }1_{FG}*\alpha^{-1}) = (\alpha*1_{FG})\circ (1_F *\epsilon * 1_{HFG})\circ (1_{FG}*\alpha^{-1}*1_{FG}) \circ (1_{FG} *\eta)$
Using $(\heartsuit)$, $\star$ is thus equal to
$(\alpha *1_{FG}) \circ (1_F*\epsilon *1_{HFG}) \circ (1_{FGF}*\beta *1_G) \circ (1_{FG}*\eta)$.
(Borceux writes
$(1_{FG}*\alpha) \circ (1_F*\epsilon *1_{HFG}) \circ (1_{FGF}*\beta *1_G) \circ (1_{FG}*\eta)$,
but I think there's a small typo).
The next (and last) equality is
$(\alpha *1_{FG}) \circ (1_F*\epsilon *1_{HFG}) \circ (1_{FGF}*\beta *1_G) \circ (1_{FG}*\eta) = (1_F*1_G)\circ (1_F*1_G)$.
Where does this equality come from?
Aside question. Proving all these last equalities I've realized that, if $A$ is a category, $F, G\colon A \to A$ are endofunctors, $\alpha\colon F\to 1_A$, $\beta\colon 1_A \to G$ are natural transformations, then
$\beta \circ \alpha = (\alpha *1_G)\circ (1_F *\beta)$.
I suppose that, maybe using the interchange law for 2-cells and doing some checks, lemma general enough to prove immediately the equalities in the last part of the proof.
-
It's a good idea to at least explain what the proposition is (as well as an outline of the argument up to this point), for the benefit of everyone who does not have that book at hand. – Zev Chonoles May 16 '13 at 23:01
Presumably $\eta$ and $\epsilon$ are adjunction unit and counit and $\alpha$ and $\beta$ are mutually inverse, or something like that? Then just push $\epsilon$ past $\beta$ using naturality. – Zhen Lin May 16 '13 at 23:04
I have improved the question with some more background. @ZhenLin, do you mean $(1_F*\epsilon *1_{HFG}) \circ (1_{FGF}*\beta *1_G) = (1_F * \beta * 1_G)\circ (1_F * \epsilon 1_G)$? – Andrea Gagna May 16 '13 at 23:51
I will use the same notation. By naturality, $$(1_F * \epsilon * 1_{H F G}) \circ (1_{F G F} * \beta * 1_G) = (1_F * \beta * 1_G) \circ (1_F * \epsilon * 1_G)$$ and so $$(\alpha * 1_{F G}) \circ (1_F * \epsilon * 1_{H F G}) \circ (1_{F G F} * \beta * 1_G) \circ (1_{F G} * \eta)$$ is equal to $$(\alpha * 1_{F G}) \circ (1_F * \beta * 1_G) \circ (1_F * \epsilon * 1_G) \circ (1_{F G} * \eta)$$ which by the triangle identities is just $1_{F G}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9611241221427917, "perplexity": 628.9074180068321}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644060413.1/warc/CC-MAIN-20150827025420-00185-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/148722/beamer-notes-with-color-overlay | # Beamer notes with color overlay
I am having trouble with the overlay system of beamer when using side notes. I have a slide with an overlay specification and I have some notes for each stage of this slide. For example, from the beamer docs:
\documentclass[compress,mathserif]{beamer}
\usepackage{pgfpages}
\usetheme{Frankfurt}
\setbeameroption{show notes}
\setbeameroption{show notes on second screen=right}
\begin{document}
\begin{frame}
\frametitle{There Is No Largest Prime Number}
\framesubtitle{The proof uses \textit{reductio ad absurdum}.}
\begin{theorem}
There is no largest prime number.
\end{theorem}
\begin{proof}
\begin{enumerate}
\item<1-| alert@1> Suppose $p$ were the largest prime number.
\item<2-> Let $q$ be the product of the first $p$ numbers.
\item<3-> Then $q+1$ is not divisible by any of them.
\item<1-> Thus $q+1$ is also prime and greater than $p$.\qedhere
\end{enumerate}
\end{proof}
\note<1->{ note one\\ }
\note<2->{ note 2\\ }
\note<3->{ note 3\\ }
\note<4->{ note 4\\ }
\end{frame}
\end{document}\endinput
With this example, notes appear as the overlay advances. What I would like is for the notes to be always visible, but somehow highlighted (e.g. in red) in its corresponding slide. I tried with \note<1-| alert@1>, but this didn't work so it seems that I am misunderstanding something about the overlays.
• I thought this can be achieved by defining a macro \notealert as: \newcommand<>\notealert[1]{\textcolor#2{red}{#1}}, and use \note[item]{\notealert<1>{note 1}}. But this gives a wrong output: the overlay effect has a one-slide lag. Apparently this has something to do with the pgfpages package. A related post: tex.stackexchange.com/questions/71206/… – Herr K. Dec 8 '13 at 0:48
You can use \alt{}{} to color the notes:
\documentclass[compress,mathserif]{beamer}
\usepackage{pgfpages}
\usetheme{Frankfurt}
\setbeameroption{show notes}
\setbeameroption{show notes on second screen=right}
\newcommand<>{\mynote}[1]{%
\alt#2{%
\note{\textcolor{red}{#1}}
}
{%
\note{#1}
}
}
\begin{document}
\begin{frame}
\frametitle{There Is No Largest Prime Number}
\framesubtitle{The proof uses \textit{reductio ad absurdum}.}
\begin{theorem}
There is no largest prime number.
\end{theorem}
\begin{proof}
\begin{enumerate}
\item<1-| alert@1> Suppose $p$ were the largest prime number.
\item<2-> Let $q$ be the product of the first $p$ numbers.
\item<3-> Then $q+1$ is not divisible by any of them.
\item<1-> Thus $q+1$ is also prime and greater than $p$.\qedhere
\end{enumerate}
\end{proof}
\mynote<1>{note one\\}
\mynote<2>{ note 2\\ }
\mynote<3>{ note 3\\ }
\mynote<1>{ note 4\\ }
\end{frame}
\end{document} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9719341397285461, "perplexity": 2115.8089847189144}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195529406.97/warc/CC-MAIN-20190723130306-20190723152306-00329.warc.gz"} |
http://mathoverflow.net/questions/58148/polynomial-group-laws-on-mathbbr2 | # Polynomial group Laws on $\mathbb{R}^2$
When students are first learning about groups, a classic example of a group that is not defined as a set of functions is the group whose underlying set is $\mathbb{R}\setminus-1$, and whose operation is $x*y=x+y+xy$. This naturally leads one to wonder about what other polynomials in two variables give rise to a group law on $\mathbb{R}$. Is there any nice criteria for such polynomials, or, in the case that there is not, are there any nice classes of polynomials that are group laws?
-
Your operation only defines a group on $\mathbb R\smallsetminus\{-1\}$. It can be generalized to $x∗y=L^{−1}(L(x)L(y))$ on $\mathbb R\smallsetminus\{L^{-1}(0)\}$, where $L$ is any nonconstant linear function. This also shows that the example is a cheat, the group is just ordinary multiplication, except that the domain has been shifted. – Emil Jeřábek Mar 11 '11 at 13:05
I don't follow. By Cayley's theorem, every group is a set of functions. – Qiaochu Yuan Mar 11 '11 at 13:55
@Qiaochu: Although every group is isomorphic to a group of functions, the context of the original question ("When students are first learning ...") significantly reduces the appropriateness of working up to isomorphism. In particular, if I'm teaching an introductory algebra class, I'm likely to use examples like the one in the question without immediately mentioning an isomorphic copy consisting of functions. (The isomorphic copy and Cayley's theorem would come up somewhat later.) – Andreas Blass Mar 11 '11 at 14:42
Emil's basic observation can be extended. Polynomials are smooth (i.e., infinitely differentiable) functions. What can you say about smooth group laws on the set of real numbers?
In different language: a Lie group is a manifold which comes equipped with group operations which are smooth with respect to the manifold structure. What can you say about Lie groups whose underlying manifold is the set of real numbers with its standard manifold structure?
Answer: they are all isomorphic. In particular, they are all isomorphic to the standard example. That is, there is always a smooth bijection $L$ (with smooth inverse) such that the group operation is given by $x \ast y = L^{-1}(L(x) + L(y))$. This is proven in any text which treats Lie groups. So: all examples are "cheats".
Of course, your example is slightly different: it is isomorphic to $(\mathbb{R} - \{0\}, \cdot)$. In fact, all Lie group structures on the punctured line are isomorphic to this one, which is
$$\mathbb{Z}_2 \times \mathbb{R}$$
where the Lie group structure on $\mathbb{R}$ is, of course, given by addition.
-
I wrote up a handout, for the case of the real line, at a level suitable for someone who just knows calculus: math.uconn.edu/~kconrad/blurbs/grouptheory/relativity.pdf – KConrad Mar 11 '11 at 17:21
The only polynomial group law on $\mathbf{R}$ such that the identity element is $0$ is given by the polynomial $P(X,Y)=X+Y$.
Proof. Let $P \in \mathbf{R}[X,Y]$ be the group law, with identity element $0$. For any $y \in \mathbf{R}$, let $P_y$ be the polynomial $P(X,y)$. If $y' \in \mathbf{R}$ is the inverse of $y$ with respect to the group law, then $P_y \circ P_{y'} = X$ so that $\operatorname{deg}(P_y) \cdot \operatorname{deg}(P_{y'}) = 1$. Thus the degree of $P$ with respect to $X$ is 1. Similarly the degree of $P$ with respect to $Y$ is 1. So $P=\alpha X+\beta Y + \gamma XY$. Since $P(X,0)=X$ and $P(0,Y)=Y$ then $\alpha=\beta=1$. Moreover for any $x \in \mathbf{R}$, the function $y \mapsto x+y+\gamma xy$ is a continuous bijection of $\mathbf{R}$, which forces $\gamma=0$.
Note : if the identity element of $P$ is $a \in \mathbf{R}$ instead, then by considering $(X,Y) \mapsto P(X+a,Y+a)-a$ we see that necessarily $P(X,Y)=X+Y-a$.
-
Apart from the final $\gamma=0$ step, the same argument works for any infinite subset of $\mathbb R$, which covers the motivating example $x+y+xy$. That is, if $p(x,y)$ defines a group operation on any infinite set of reals, then $p(x,y)=x+y-a$ or $p(x,y)=\gamma xy+bx+by+(b^2-b)/\gamma$, i.e., a linear shift of $x+y$ or $xy$. – Emil Jeřábek Mar 11 '11 at 15:55
It also seems that the same proof works with $\mathbf{R}$ replaced by any infinite field. – François Brunault Mar 11 '11 at 17:04
jstor.org/pss/2042946 seems to prove the generalisation to infinite fields and something more. – Gian Maria Dall'Ara May 28 '11 at 15:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9648104906082153, "perplexity": 152.32327884248875}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207931085.38/warc/CC-MAIN-20150521113211-00075-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://link.springer.com/article/10.1007/s10994-018-5723-3?error=cookies_not_supported&code=5c14b5d6-53bb-416f-ae60-dcb3844a09aa | # Deep Gaussian Process autoencoders for novelty detection
## Abstract
Novelty detection is one of the classic problems in machine learning that has applications across several domains. This paper proposes a novel autoencoder based on Deep Gaussian Processes for novelty detection tasks. Learning the proposed model is made tractable and scalable through the use of random feature approximations and stochastic variational inference. The result is a flexible model that is easy to implement and train, and can be applied to general novelty detection tasks, including large-scale problems and data with mixed-type features. The experiments indicate that the proposed model achieves competitive results with state-of-the-art novelty detection methods.
## Introduction
Novelty detection is a fundamental task across numerous domains, with applications in data cleaning (Liu et al. 2004), fault detection and damage control (Dereszynski and Dietterich 2011; Worden et al. 2000), fraud detection related to credit cards (Hodge and Austin 2004) and network security (Pokrajac et al. 2007), along with several medical applications such as brain tumor (Prastawa et al. 2004) and breast cancer (Greensmith et al. 2006) detection. Novelty detection targets the recognition of anomalies in test data which differ significantly from the training set (Pimentel et al. 2014), so this problem is also known as “anomaly detection”. Challenges in performing novelty detection stem from the fact that labelled data identifying anomalies in the training set is usually scarce and expensive to obtain, and that very little is usually known about the distribution of such novelties. Meanwhile, the training set itself might be corrupted by outliers and this might impact the ability of novelty detection methods to accurately characterize the distribution of samples associated with a nominal behavior of the system under study. Furthermore, there are many applications, such as the ones that we study in this work, where the volume and heterogeneity of data might pose serious computational challenges to react to novelties in a timely manner and to develop flexible novelty detection algorithms. As an example, the Airline IT company Amadeus provides booking platforms handling millions of transactions per second, resulting in more than 3 million bookings per day and Petabytes of stored data. This company manages almost half of the flight bookings worldwide and is targeted by fraud attempts leading to revenue losses and indemnifications. Detecting novelties in such large volumes of data is a daunting task for a human operator; thus, an automated and scalable approach is truly desirable.
Because of the difficulty in obtaining labelled data and since the scarcity of anomalies is challenging for supervised methods (Japkowicz and Stephen 2002), novelty detection is normally approached as an unsupervised machine learning problem (Pimentel et al. 2014). The considerations above suggest some desirable scalability and generalization properties that novelty detection algorithms should have.
We have recently witnessed the rise of deep learning techniques as the preferred choice for supervised learning problems, due to their large representational power and the possibility to train these models at scale (LeCun et al. 2015); examples of deep learning techniques achieving state-of-the-art performance on a wide variety of tasks include computer vision (Krizhevsky et al. 2012), speech recognition (Hinton et al. 2012), and natural language processing (Collobert and Weston 2008). A natural question is whether such impressive results can extend beyond supervised learning to unsupervised learning and further to novelty detection. Deep learning techniques for unsupervised learning are currently actively researched on Kingma and Welling (2014) and Goodfellow et al. (2014), but it is still unclear whether these can compete with state-of-the-art novelty detection methods. We are not aware of recent surveys on neural networks for novelty detection, and the latest one we could find is almost 15 years old (Markou and Singh 2003) and misses the recent developments in this domain.
Key challenges with the use of deep learning methods in general learning tasks are (1) the necessity to specify a suitable architecture for the problem at hand and (2) the necessity to control their generalization. While various forms of regularization have been proposed to mitigate the overfitting problem and improve generalization, e.g., through the use of dropout (Srivastava et al. 2014; Gal and Ghahramani 2016), there are still open questions on how to devise principled ways of applying deep learning methods to general learning tasks. Deep Gaussian Processes (dgps) are ideal candidate to simultaneously tackle issues (1) and (2) above. dgps are deep nonparametric probabilistic models implementing a composition of probabilistic processes that implicitly allows for the use of an infinite number of neurons at each layer (Damianou and Lawrence 2013; Duvenaud et al. 2014). Also, their probabilistic nature induces a form of regularization that prevents overfitting, and allows for a principled way of carrying out model selection (Neal 1996). While dgps are particularly appealing to tackle general deep learning problems, their training is computationally intractable. Recently, there have been contributions in the direction of making the training of these models tractable (Bui et al. 2016; Cutajar et al. 2017; Bradshaw et al. 2017), and these are currently in the position to compete with Deep Neural Networks (dnns) in terms of scalability, accuracy, while providing superior quantification of uncertainty (Gal and Ghahramani 2016; Cutajar et al. 2017; Gal et al. 2017).
In this paper, we introduce an unsupervised model for novelty detection based on dgps in autoencoder configuration. We train the proposed dgp autoencoder (dgp-ae) by approximating the dgp layers using random feature expansions, and by performing stochastic variational inference on the resulting approximate model. The key features of the proposed approach are as follows: (1) dgp-aes are unsupervised probabilistic models that can model highly complex data distribution and offer a scoring method for novelty detection; (2) dgp-aes can model any type of data including cases with mixed-type features, such as continuous, discrete, and count data; (3) dgp-aes training does not require any expensive and potentially numerically troublesome matrix factorizations, but only tensor products; (4) dgp-aes can be trained using mini-batch learning, and could therefore exploit distributed and GPU computing; (5) dgp-aes training using stochastic variational inference can be easily implemented taking advantage of automatic differentiation tools, making for a very practical and scalable methods for novelty detection. Even though we leave this for future work, it is worth mentioning that dgp-aes can easily include the use of special representations based, e.g., on convolutional filters for applications involving images, and allow for end-to-end training of the model and the filters.
We compare dgp-aes with a number of competitors that have been proposed in the literature of deep learning to tackle large-scale unsupervised learning problems, such as Variational Autoencoders (vae) (Kingma and Welling 2014), Variational Auto-Encoded Deep Gaussian Process (vae-dgp) (Dai et al. 2016) and Neural Autoregressive Distribution Estimator (nade) (Uria et al. 2016). Through a series of experiments, where we also compare against state-of-the-art novelty detection methods such as Isolation Forest (Liu et al. 2008) and Robust Kernel Density Estimation (Kim and Scott 2012), we demonstrate that dgp-aes offer flexible modeling capabilities with a practical learning algorithm, while achieving state-of-the-art performance.
The paper is organized as follows: Sect. 2 introduces the problem of novelty detection and reviews the related work on the state-of-the-art. Section 3 presents the proposed dgp-ae for novelty detection, while Sects. 4 and 5 report the experiments and conclusions.
## Novelty detection
Consider an unsupervised learning problem where we are given a set of input vectors $$X = [\mathbf {x}_1, \ldots , \mathbf {x}_n]^{\top }$$. Novelty detection is the task of classifying new test points $$\mathbf {x}_*$$, based on the criterion that they significantly differ from the input vectors X, that is the data available at training time. Such data is assumed to be generated by a different generative process and are called anomalies. Novelty detection is thus a one-class classification problem, which aims at constructing a model describing the distribution of nominal samples in a dataset. Unsupervised learning methods allow for the prediction on test data $$\mathbf {x}_*$$; given a model with parameters $${\varvec{\theta }}$$, define predictions as $$h(\mathbf {x}_* | X, {\varvec{\theta }})$$. Assuming $$h(\mathbf {x}_* | X, {\varvec{\theta }})$$ to be continuous, it is possible to interpret it as a means of scoring test points as novelties. The resulting scores allow for a ranking of test points $$\mathbf {x}_*$$ highlighting the patterns which differ the most from the training data X. In particular, it is possible to define a threshold $$\alpha$$ and flag a test point $$\mathbf {x}_*$$ as a novelty when $$h(\mathbf {x}_* | X, {\varvec{\theta }}) > \alpha$$.
After thresholding, it is possible to assess the quality of a novelty detection algorithm using scores proposed in the literature for binary classification. Based on a labelled testing dataset, where novelties and nominal cases are defined as positive and negative samples, respectively, we can compute the precision and recall metrics given in Eq. 1. True positives (TP) are examples correctly labelled as positives, false positives (FP) refer to negative samples incorrectly labelled as positives, while false negatives (FN) are positive samples incorrectly labelled as negatives.
\begin{aligned} { precision} = \frac{{ TP}}{{ TP}+{ FP}} \quad { recall} = \frac{{ TP}}{{ TP}+{ FN}}. \end{aligned}
(1)
In the remainder of this paper we are going to assess results of novelty detection methods by varying $$\alpha$$ over the range of values taken by $$h(\mathbf {x}_* | X, {\varvec{\theta }})$$ over a set of test points. When we vary $$\alpha$$, we obtain a set of precision and recall measurements resulting in a curve. We can then compute the area under the precision–recall curve called the mean average precision (map), which is a relevant metric to compare the performance of novelty detection methods (Davis and Goadrich 2006). In practical terms, $$\alpha$$ is chosen to strike an appropriate balance between accuracy in identifying novelties and a low level of false positives.
Novelty detection has been thoroughly investigated by theoretical studies (Pimentel et al. 2014; Hodge and Austin 2004). The evaluation of state-of-the-art methods was also reported in experimental papers (Emmott et al. 2016), including experiments on the methods scalability (Domingues et al. 2018) and resistance to the curse of dimensionality (Zimek et al. 2012). In one of the most recent surveys on novelty detection (Pimentel et al. 2014), methods have been classified into the following categories. (1) Probabilistic approaches estimate the probability density function of X defined by the model parameters $${\varvec{\theta }}$$. Novelties are scored by the likelihood function $$P(\mathbf {x}_* | {\varvec{\theta }})$$, which computes the probability for a test point to be generated by the trained distribution. These approaches are generative, and provide a simple understanding of the underlying data through parameterized distributions. (2) Distance-based methods compute the pairwise distance between samples using various similarity metrics. Patterns with a small number of neighbors within a specified radius, or distant from the center of dense clusters of points, receive a high novelty score. (3) Domain-based methods learn the domain of the nominal class as a decision boundary. The label assigned to test points is then based on their location with respect to the boundary. (4) Information theoretic approaches measure the increase of entropy induced by including a test point in the nominal class. As an alternative, (5) isolation methods target the isolation of outliers from the remaining samples. As such, these techniques focus on isolating anomalies instead of profiling nominal patterns. (6) Most suitable unsupervised neural networks for novelty detection are autoencoders, i.e., networks learning a compressed representation of the training data by minimizing the error between the input data and the reconstructed output. Test points showing a high reconstruction error are labelled as novelties. Our model belongs to this last category, and extends it by proposing a nonparametric and probabilistic approach to alleviate issues related to the choice of a suitable architecture while accounting for the uncertainty in the autoencoder mappings; crucially, we show that this can be achieved while learning the model at scale.
## Deep Gaussian Process autoencoders for novelty detection
In this section, we introduce the proposed dgp-ae model and describe the approximation that we use to make inference tractable and scalable. Each iteration of the algorithm is linear in the dimensionality of the input, batch size, dimensionality of the latent representation and number of Monte Carlo samples used in the approximation of the objective function, which highlights the tractability of the model. We also discuss the inference scheme based on stochastic variational inference, and show how predictions can be made. Finally, we present ways in which we can make the proposed dgp-ae model handle various types of data, e.g., mixing continuous and categorical features. We refer the reader to Cutajar et al. (2017) for a detailed derivation of the random feature approximation of dgps and variational inference of the resulting model. In this work, we extend this dgp formulation to autoencoders.
### Deep Gaussian Process autoencoders
An autoencoder is a model combining an encoder and a decoder. The encoder part takes each input $$\mathbf {x}$$ and maps it into a set of latent variables $$\mathbf {z}$$, whereas the decoder part maps latent variables $$\mathbf {z}$$ into the inputs $$\mathbf {x}$$. Because of their structure, autoencoders are able to jointly learn latent representations for a given dataset and a model to produce $$\mathbf {x}$$ given latent variables $$\mathbf {z}$$. Typically this is achieved by minimizing a reconstruction error.
Autoencoders are not generative models, and variational autoencoders have recently been proposed to enable this feature (Dai et al. 2016; Kingma and Welling 2014). In the context of novelty detection, the possibility to learn a generative model might be desirable but not essential, so in this work we focus in particular on autoencoders. Having said that, we believe that extending variational autoencoders using the proposed framework is possible, as well as empowering the current model to enable generative modeling; we leave these avenues of research for future work. In this work, we propose to construct the encoder and the decoder functions of autoencoders using dgps. As a result, we aim at jointly learning a probabilistic nonlinear projection based on dgps (the encoder) and a dgp-based latent variable model (the decoder).
The building block of dgps are gps, which are priors over functions; formally, a gp is a set of random variables characterized by the property that any subset of them is jointly Gaussian (Rasmussen and Williams 2006). The gp covariance function models the covariance between the random variables at different inputs, and it is possible to specify a parametric function for their mean.
Stacking multiple gps into a dgp means feeding the output of gps at each layer as the input of the gps at the next; this construction gives rise to a composition of stochastic processes. Assume that we compose $$N_{\mathrm {L}}$$ possible functions modelled as multivariate gps, the resulting composition takes the form
\begin{aligned} \mathbf {f}(\mathbf {x}) = \left( \mathbf {f}^{(N_{\mathrm {L}})} \circ \cdots \circ \mathbf {f}^{(1)}\right) (\mathbf {x}). \end{aligned}
(2)
Without loss of generality, we are going to assume that the gp s at each layer have zero mean, and that gp covariances at layer (l) are parameterized through a set of parameters $${\varvec{\theta }}^{(l)}$$ shared across gps in the same layer.
Denote by $$F^{(i)}$$ the collection of the multivariate functions $$\mathbf {f}^{(i)}$$ evaluated at the inputs $$F^{(i-1)}$$, and define $$F^{(0)} := X$$. The encoder part of the proposed dgp-ae model maps the inputs X into a set of latent variables $$Z := F^{(j)}$$ through a dgp, whereas the decoder is another dgp mapping Z into X. The dgp controlling the decoding part of the model, assumes a likelihood function that allows one to express the likelihood of the observed data X as $$p\left( X | F^{(N_{\mathrm {L}})}, {\varvec{\theta }}^{(N_{\mathrm {L}})}\right)$$. The likelihood reflects the choice on the mappings between latent variables and the type of data being modelled, and it can include and mix various types and dimensionality; Sect. 3.5 discusses this in more detail.
By performing Bayesian inference on the proposed dgp-ae model we aim to integrate out latent variables at all layers, effectively integrating out the uncertainty in all the mappings in the encoder/decoder and the latent variables Z themselves. Learning and making predictions with dgp-aes, however, require being able to solve intractable integrals. To evaluate the marginal likelihood expressing the probability of observed data given model parameters, we need to solve the following
\begin{aligned} p(X | {\varvec{\theta }}) = \int p\left( X | F^{(N_{\mathrm {L}})}, {\varvec{\theta }}^{(N_{\mathrm {L}})}\right) \prod _{j = 1}^{N_L} p\left( F^{(j)} | F^{(j - 1)}, {\varvec{\theta }}^{(j - 1)}\right) \prod _{j = 1}^{N_L} dF^{(j)} \end{aligned}
(3)
A similar intricate integral can be derived to express the predictive probability $$p(\mathbf {x}_* | X, {\varvec{\theta }})$$. For any nonlinear covariance function, these integrals are intractable. In the next section, we show how random feature expansions of the gps at each layer expose an approximate model that can be conveniently learned using stochastic variational inference, as described in Cutajar et al. (2017).
### Random feature expansions for dgp-aes
To start with, consider a shallow multivariate gp and denote by F the latent variables associated with the inputs. For a number of gp covariance functions, it is possible to obtain a low-rank approximation of the processes through the use of a finite set of basis functions, and transform the multivariate gp into a Bayesian linear model. For example, in the case of an rbf covariance function of the form
\begin{aligned} k_{\mathrm {rbf}}(\mathbf {x}, \mathbf {x}^{\prime }) = \exp \left[ -\frac{1}{2} \left\| \mathbf {x}- \mathbf {x}^{\prime } \right\| ^{\top } \right] \end{aligned}
(4)
it is possible to employ standard Fourier analysis to show that $$k_{\mathrm {rbf}}$$ can be expressed as an expectation under a distribution over spectral frequencies, that is:
\begin{aligned} k_{\mathrm {rbf}}(\mathbf {x}, \mathbf {x}^{\prime }) = \int p(\varvec{\omega }) \exp \left[ i (\mathbf {x}- \mathbf {x}^{\prime })^{\top } \varvec{\omega }\right] d\varvec{\omega }. \end{aligned}
(5)
After standard manipulation, it is possible to obtain an unbiased estimate of the integral above by mean of a Monte Carlo average:
\begin{aligned} k_{\mathrm {rbf}}(\mathbf {x}, \mathbf {x}^{\prime }) \approx \frac{1}{N_{\mathrm {RF}}} \sum _{r=1}^{N_{\mathrm {RF}}} \mathbf {z}(\mathbf {x}| \tilde{\varvec{\omega }}_r)^{\top } \mathbf {z}(\mathbf {x}^{\prime } | \tilde{\varvec{\omega }}_r), \end{aligned}
(6)
where $$\mathbf {z}(\mathbf {x}| \varvec{\omega }) = [\cos (\mathbf {x}^{\top } \varvec{\omega }), \sin (\mathbf {x}^{\top } \varvec{\omega })]^{\top }$$ and $$\tilde{\varvec{\omega }}_{r} \sim p(\varvec{\omega })$$. It is possible to increase the flexibility of the rbf covariance above by scaling it by a marginal variance parameter $$\sigma ^2$$ and by scaling the features individually with length-scale parameters $$\varLambda = \mathrm {diag}(l_1^2,\ldots ,l_{DF}^2)$$; it is then possible to show that $$p(\varvec{\omega }) = \mathcal {N}\left( \varvec{\omega }| \mathbf {0}, \varLambda ^{-1} \right)$$ using Bochner’s theorem. By stacking the samples from $$p(\varvec{\omega })$$ by column into a matrix $$\varOmega$$, we can define
\begin{aligned} \varPhi _{\mathrm {rbf}} = \sqrt{\frac{(\sigma ^2)}{N_{\mathrm {RF}}}} \Big [ \cos \left( F \varOmega \right) , \sin \left( F \varOmega \right) \Big ], \end{aligned}
(7)
where the functions $$\cos ()$$ and $$\sin ()$$ are applied element-wise. We can now derive a low-rank approximation of K as follows:
\begin{aligned} K \approx \varPhi \varPhi ^{\top } \end{aligned}
(8)
It is straightforward to verify that the individual columns of F in the original gp can be approximated by the Bayesian linear model $$F_{\cdot j} = \varPhi W_{\cdot j}$$ with $$W_{\cdot j} \sim \mathcal {N}(\mathbf {0}, I)$$, as the covariance of $$F_{\cdot j}$$ is indeed $$\varPhi \varPhi ^{\top } \approx K$$.
The decomposition of the gp covariance in Eq. 4 suggests an expansion with an infinite number of basis functions, thus leading to a well-known connection with single-layered neural networks with infinite neurons (Neal 1996); the random feature expansion that we perform using Monte Carlo induces a truncation of the infinite expansion. Based on the expansion defined above, we can now build a cascade of approximate gps, where the output of layer l becomes the input of layer $$l+1$$. The layer $$\varPhi ^{(0)}$$ first expands the input features in a high-dimensional space, followed by a linear transformation parameterized by a weight matrix $$W^{(0)}$$ which results in the latent variables $$F^{(1)}$$ in the second layer. Considering a dgp with rbf covariances obtained by stacking the hidden layers previously described, we obtain Eqs. 9 and 10 derived from Eq. 6. These transformations are parameterized by prior parameters $$(\sigma ^2)^{(l)}$$ which determine the marginal variance of the gps and $$\varLambda ^{(l)} = \mathrm {diag}\left( \left( l_1^2\right) ^{(l)},\ldots ,\left( l_{DF^{(l)}}^2\right) ^{(l)}\right)$$ describing the length-scale parameters.
\begin{aligned} \varPhi _{\mathrm {rbf}}^{(l)}= & {} \sqrt{\frac{(\sigma ^2)^{(l)}}{N_{\mathrm {RF}}^{(l)}}} \left[ \cos \left( F^{(l)} \varOmega ^{(l)}\right) , \sin \left( F^{(l)} \varOmega ^{(l)}\right) \right] ,\end{aligned}
(9)
\begin{aligned} F^{(l+1)}= & {} \varPhi _{\mathrm {rbf}}^{(l)} W^{(l)} \end{aligned}
(10)
This leads to the proposed dgp-ae model’s topology given in Fig. 1. The resulting approximate dgp-ae model is effectively a Bayesian dnn where the priors for the spectral frequencies $$\varOmega ^{(l)}$$ are controlled by covariance parameters $${\varvec{\theta }}^{(l)}$$, and the priors for the weights $$W^{(l)}$$ are standard normal.
In our framework, the choice of the covariance function induces different basis functions. For example, a possible approximation of the arc-cosine kernel (Cho and Saul 2009) yields Rectified Linear Units (relu) basis functions (Cutajar et al. 2017) resulting in faster computations compared to the approximation of the rbf covariance, given that derivatives of relu basis functions are cheap to evaluate.
### Stochastic variational inference for dgp-aes
Let $${\varvec{\Theta }}$$ be the collection of all covariance parameters $${\varvec{\theta }}^{(l)}$$ at all layers; similarly, define $${\varvec{\Omega }}$$ and $$\mathbf {W}$$ to be the collection of the spectral frequencies $$\varOmega ^{(l)}$$ and weight matrices $$W^{(l)}$$ at all layers, respectively. We are going to apply stochastic variational inference techniques to infer $$\mathbf {W}$$ and optimize all covariance parameters $${\varvec{\Theta }}$$; we are going to consider the case where the spectral frequencies $${\varvec{\Omega }}$$ are fixed, but these can also be learned (Cutajar et al. 2017). The marginal likelihood $$p(X | {\varvec{\Omega }}, {\varvec{\Theta }})$$ can be bounded using standard variational inference techniques, following Kingma and Welling (2014) and Graves (2011), Defining $$\mathcal {L}= \log \left[ p(X | {\varvec{\Omega }}, {\varvec{\Theta }})\right]$$, we obtain
\begin{aligned} \mathcal {L}\ge \mathrm {E}_{q(\mathbf {W})} \left( \log \left[ p\left( X | \mathbf {W}, {\varvec{\Omega }}, {\varvec{\Theta }}\right) \right] \right) - \mathrm {DKL}\left[ q(\mathbf {W}) \Vert p\left( \mathbf {W}\right) \right] . \end{aligned}
(11)
Here the distribution $$q(\mathbf {W})$$ denotes an approximation to the intractable posterior $$p(\mathbf {W}| X, {\varvec{\Omega }}, {\varvec{\Theta }})$$, whereas the prior on $$\mathbf {W}$$ is the product of standard normal priors resulting from the approximation of the gp s at each layer $$p(\mathbf {W}) = \prod _{l=0}^{N_{\mathrm {L}} - 1} p(W^{(l)}).$$
We are going to assume an approximate Gaussian distribution that factorizes across layers and weights
\begin{aligned} q(\mathbf {W}) = \prod _{ijl} q\left( W^{(l)}_{ij}\right) = \prod _{ijl} \mathcal {N}\left( m^{(l)}_{ij}, (s^2)^{(l)}_{ij} \right) . \end{aligned}
(12)
We are interested in finding an optimal approximate distribution $$q(\mathbf {W})$$, so we are going to introduce the variational parameters $$m^{(l)}_{ij}, (s^2)^{(l)}_{ij}$$ to be the mean and the variance of each of the approximating factors. Therefore, we are going to optimize the lower bound above with respect to all variational parameters and covariance parameters $${\varvec{\Theta }}$$.
Because of the chosen Gaussian form of $$q(\mathbf {W})$$ and given that the prior $$p(\mathbf {W})$$ is also Gaussian, the DKL term in the lower bound to $$\mathcal {L}$$ can be computed analytically. The remaining term in the lower bound, instead, needs to be estimated. Assuming a likelihood that factorizes across observations, it is possible to perform a doubly-stochastic approximation of the expectation in the lower bound so as to enable scalable stochastic gradient-based optimization. The doubly-stochastic approximation amounts to replacing the sum over n input points with a sum over a mini-batch of m points selected randomly from the entire dataset:
\begin{aligned} \mathrm {E}_{q(\mathbf {W})} \left( \log \left[ p\left( X | \mathbf {W}, {\varvec{\Omega }}, {\varvec{\Theta }}\right) \right] \right) \approx \frac{n}{m} \sum _{k \in \mathcal {I}_m} \mathrm {E}_{q(\mathbf {W})} \left( \log \left[ p(\mathbf {x}_k | \mathbf {W}, {\varvec{\Omega }}, {\varvec{\Theta }})\right] \right) . \end{aligned}
(13)
Then, each element of the sum can itself be estimated unbiasedly using Monte Carlo sampling and averaging, with $$\tilde{\mathbf {W}}_r \sim q(\mathbf {W})$$:
\begin{aligned} \mathrm {E}_{q(\mathbf {W})} \left( \log \left[ p\left( X | \mathbf {W}, {\varvec{\Omega }}, {\varvec{\Theta }}\right) \right] \right) \approx \frac{n}{m} \sum _{k \in \mathcal {I}_m} \frac{1}{N_{\mathrm {MC}}} \sum _{r = 1}^{N_{\mathrm {MC}}} \log \left[ p\left( \mathbf {x}_k | \tilde{\mathbf {W}}_r, {\varvec{\Omega }}, {\varvec{\Theta }}\right) \right] . \end{aligned}
(14)
Because of the unbiasedness property of the last expression, computing its derivative with respect to the variational parameters and $${\varvec{\Theta }}$$ yields a so-called stochastic gradient that can be used for stochastic gradient-based optimization. The appeal of this optimization strategy is that it is characterized by theoretical guarantees to reach local optima of the objective function (Robbins and Monro 1951). Derivatives can be conveniently computed using automatic differentiation tools; we implemented our model in TensorFlow (Abadi et al. 2015) that has this feature built-in. In order to take derivatives with respect to the variational parameters we employ the so-called reparameterization trick (Kingma and Welling 2014)
\begin{aligned} \left( \tilde{W}^{(l)}_{r}\right) _{ij} = s^{(l)}_{ij} \epsilon ^{(l)}_{rij} + m^{(l)}_{ij}, \end{aligned}
(15)
to fix the randomness when updating the variational parameters, and $$\epsilon ^{(l)}_{rij}$$ are resampled after each iteration of the optimization.
### Predictions with dgp-aes
The predictive distribution for the proposed dgp-ae model requires solving the following integral
\begin{aligned} p(\mathbf {x}_* | X, {\varvec{\Omega }}, {\varvec{\Theta }}) = \int p(\mathbf {x}_* | \mathbf {W}, {\varvec{\Omega }}, {\varvec{\Theta }}) p(\mathbf {W}| X, {\varvec{\Omega }}, {\varvec{\Theta }}) d\mathbf {W}, \end{aligned}
(16)
which is intractable due to fact that the posterior distribution over $$\mathbf {W}$$ is unavailable. Stochastic variational inference yields an approximation $$q(\mathbf {W})$$ to the posterior $$p(\mathbf {W}| X, {\varvec{\Omega }}, {\varvec{\Theta }})$$, so we can use it to approximate the predictive distribution above:
\begin{aligned} p(\mathbf {x}_* | X, {\varvec{\Omega }}, {\varvec{\Theta }}) \approx \int p\left( \mathbf {x}_* | \mathbf {W}, {\varvec{\Omega }}, {\varvec{\Theta }}\right) q(\mathbf {W}) d\mathbf {W}\approx \frac{1}{N_{\mathrm {MC}}} \sum _{r = 1}^{N_{\mathrm {MC}}} p\left( \mathbf {x}_* | \tilde{\mathbf {W}}_r, {\varvec{\Omega }}, {\varvec{\Theta }}\right) , \end{aligned}
(17)
where we carried out a Monte Carlo approximation by drawing $$N_{\mathrm {MC}}$$ samples $$\tilde{\mathbf {W}}_r \sim q(\mathbf {W})$$. The overall complexity of each iteration is thus $$\mathcal {O}\left( m D_F^{(l-1)} N_{RF}^{(l)} N_{MC}\right)$$ to construct the random features at layer l and $$\mathcal {O}\left( mN_{RF}^{(l)}D_F^{(l)}N_{MC}\right)$$ to compute the value of the latent functions at layer l, where m is the batch size and $$D_F^{(l)}$$ is the dimensionality of $$F^{(l)}$$. Hence, by carrying out updates using mini-batches, the complexity of each iteration is independent of the dataset size.
For a given test set $$X_*$$ containing multiple test samples, it is possible to use the predictive distribution as a scoring function to identify novelties. In particular, we can rank the predictive probabilities $$p(\mathbf {x}_* | X, {\varvec{\Omega }}, {\varvec{\Theta }})$$ for all test points to identify the ones that have the lowest probability under the given dgp-ae model. In practice, for numerical stability, our implementation uses log-sum operations to compute $$\log [p(\mathbf {x}_* | X, {\varvec{\Omega }}, {\varvec{\Theta }})]$$, and we use this as the scoring function.
### Likelihood functions
One of the key features of the proposed model is the possibility to model data containing a mix of types of features. In order to do this, all we need to do is to specify a suitable likelihood for the observations given the latent variables at the last layer, that is $$p(\mathbf {x}| \mathbf {f}^{(N_{\mathrm {L}})})$$. Imagine that the vector $$\mathbf {x}$$ contains continuous and categorical features that we model using Gaussian and multinomial likelihoods; extensions to other combinations of features and distributions is straightforward. Consider a single continuous feature of $$\mathbf {x}$$, say $$x_{[G]}$$; the likelihood function for this feature is:
\begin{aligned} p(x_{[G]} | \mathbf {f}^{(N_{\mathrm {L}})}) = \mathcal {N}\left( x_{[G]} | f_{[G]}^{(N_{\mathrm {L}})}, \sigma _{[G]}^2\right) . \end{aligned}
(18)
For any given categorical feature, instead, assuming a one-hot encoding, say $$\mathbf {x}_{[C]}$$, we can use a multinomial likelihood with probabilities given by the softmax transformation of the corresponding latent variables:
\begin{aligned} p\left( (\mathbf {x}_{[C]})_j | \mathbf {f}^{(N_{\mathrm {L}})}\right) = \frac{\exp \left[ \left( f^{(N_{\mathrm {L}})}_{[C]}\right) _j\right] }{ \sum _i \exp \left[ \left( f^{(N_{\mathrm {L}})}_{[C]}\right) _i\right] }. \end{aligned}
(19)
It is now possible to combine any number of these into the following likelihood function:
\begin{aligned} p\left( \mathbf {x}| \mathbf {f}^{(N_{\mathrm {L}})}\right) = \prod _k p\left( \mathbf {x}_{[k]} | \mathbf {f}^{(N_{\mathrm {L}})}\right) \end{aligned}
(20)
Any extra parameters in the likelihood function, such as the variances in the Gaussian likelihoods, can be included in the set of all model parameters $${\varvec{\Theta }}$$ and learned jointly with the rest of parameters. For count data, it is possible to use the Binomial or Poisson likelihood, whereas for positive continuous variables we can use Exponential or Gamma. It is also possible to jointly model multiple continuous features and use a full covariance matrix for multivariate Gaussian likelihoods, multivariate Student-T, and the like. The nice feature of the proposed dgp-ae model is that the training procedure is the same regardless of the choice of the likelihood function, as long as the assumption of factorization across data points holds.
## Experiments
We evaluate the performance of our model by monitoring the convergence of the mean log-likelihood (mll) and by measuring the area under the Precision–Recall curve, namely the mean average precision (map). These metrics are taken on real-world datasets described in Sect. 4.2. In addition, we compare our model against state-of-the-art neural networks suitable for outlier detection and highlighted in Sect. 4.1. To demonstrate the value of our proposal as a competitive novelty detection method, we include top performance novelty detection methods from other domains, namely Isolation Forest (Liu et al. 2008) and Robust Kernel Density Estimation (rkde) (Kim and Scott 2012), which are recommended for outlier detection in Emmott et al. (2016).
### Selected methods
In order to retrieve a continuous score for the outliers and be able to compare the convergence of the likelihood for the selected models, our comparison focuses on probabilistic neural networks. Our dgp-ae is benchmarked against the Variational Autoencoder (vae) (Kingma and Welling 2014) and the Neural Autoregressive Distribution Estimator (nade) (Uria et al. 2016). We also include standard dnn autoencoders with sigmoid activation functions and dropout regularization to give a wider context to the reader. We initially intended to include Real nvp (Dinh et al. 2016) and Wasserstein gan (Arjovsky et al. 2017), but we found these networks and their implementations tightly tailored to images. The one-class classification with gps recently developed (Kemmler et al. 2013) is actually a supervised learning task where the authors regress on the labels and use heuristics to score novelties. Since this work is neither probabilistic nor a neural network, we did not include it. Parameter selection for the following methods was achieved by grid-search and maximized the average map over testing datasets labelled for novelty detection and described in Sect. 4.2. We append the depth of the networks as a suffix to the name, e.g., vae-2.
dgp -ae -g , dgp -ae -gs We train the proposed dgp-ae model for 100,000 iterations using 100 random features at each hidden layer. Due to the network topology, we use a number of multivariate gps equal to the number of input features when using a single-layer configuration, but use a multivariate gp of dimension 3 for the latent variables representation when using more than one layer. In the remainder of the paper, the term layer describes a hidden layer composed of two inner layers $$\varPhi ^{(i)}$$ and $$F^{(i+1)}$$. As observed in Duvenaud et al. (2014) and Neal (1996), deep architectures require to feed the input forward to the hidden layers in order to implement the modeling of meaningful functions. In the experiments involving more than 2 layers, we follow this advice by feed-forwarding the input to the encoding layers and feed-forward the latent variables to the decoding layers. The weights are optimized using a batch size of 200 and a learning rate of 0.01. The parameters $$q({\varvec{\Omega }})$$ and $${\varvec{\Theta }}$$ are fixed for 1000 and 7000 iterations respectively. $$N_{\mathrm {MC}}$$ is set to 1 during the training, while we use $$N_{\mathrm {MC}} = 100$$ at test time to score samples with higher accuracy. dgp-ae-g uses a Gaussian likelihood for continuous and one-hot encoded categorical variables. dgp-ae-gs is a modified dgp-ae-g where categorical features are modelled by a softmax likelihood as previously described. These networks use an rbf covariance function, except when the arc suffix is used, e.g., dgp-ae-g-1-arc.
vae-dgp-2Footnote 1 This network performs inducing points approximation to train a dgp model with variational inference. The network uses 2 hidden layers of dimensionality $$max(\frac{d}{2}, 5)$$ and $$max(\frac{d}{3}, 4)$$, and is trained for 1000 iterations over all training samples. All layers use a rbf kernel with 40 inducing points. The MLP in the recognition model has two layers of dimensionality 300 and 150.
vae-dgp,Footnote 2 vae-2 The variational autoencoder is a generative model which compresses the representation of the training data into a layer of latent variables, optimized with stochastic gradient descent. The sum of the reconstruction error and the latent loss, i.e., the negative of the Kullback–Leibler divergence between the approximate posterior over the latent variables and the prior, gives the loss term optimized during the training. The networks were trained for 4000 iterations using 50 hidden units and a batch size of 1000 samples. A learning rate of 0.001 was selected to optimize the weights. vae-1 is a shallow network using one layer for latent variables representation, while vae-2 uses a two-layer architecture with a first layer for encoding and a second one for decoding, each containing 100 hidden units. We use the reconstruction error to score novelties.
nade-2Footnote 3 This neural network is an autoencoder suitable for density estimation. The network uses mixtures of Gaussians to model p(x). The network yields an autoregressive model, which implies that the joint distribution is modelled such that the probability for a given feature depends on the previous features fed to the network, i.e., $$p(\varvec{x}) = p(x_{o_d}|\varvec{x}_{o_{<d}})$$, where $$x_{o_d}$$ is the feature of index d of $$\varvec{x}$$. We train a deep and orderless nade for 5000 iterations using batches of 200 samples, a learning rate of 0.005 and a weight decay of 0.02. Training the network for more iterations increases the risk of the training failing due to runtime errors. The network has a 2 layer-topology with 100 hidden units and a relu activation function. The number of components for the mixture of Gaussians was set to 20, and we use Bernoulli distributions instead of Gaussians to model datasets exclusively composed of categorical data. 15% of the training data was used for validation to select the final weights.
ae-1, ae-5 These two neural networks are feedforward autoencoders using sigmoid activation functions in the hidden layers and a dropout rate of 0.5 to provide regularization. The first network is a single layer autoencoder with a number of hidden units equal to the number of features, while the second one has a 5-layer topology with 80% of the number of input features on the second and fourth layer, and 60% on the third layer. The networks are trained for 100,000 iterations with a batch size of 200 samples and a learning rate of 0.01. The reconstruction error is used to detect outliers.
Isolation forestFootnote 4 is a random forest algorithm performing recursive random splits over the feature domain until each sample is isolated from the rest of the dataset. As a result, outliers are separated after few splits and are located in nodes close to the root of the trees. The average path length required to reach the node containing the specified point is used for scoring. A contamination rate of 5% was used for this experiment.
rkdeFootnote 5 is a probabilistic method which assigns a kernel function to each training sample, then sums the local contribution of each function to give an estimate of the density. The experiment uses the cross-validation bandwidth (lkcv) as a smoothing parameter on the shape of the density, and the Huber loss function to provide a robust estimation of the maximum likelihood.
### Datasets
Our evaluation is based on 11 datasets, including 7 datasets made publicly available by the UCI repository (Bache and Lichman 2013), while the 4 other datasets are proprietary datasets containing production data from the company Amadeus. This company provides online platforms to connect the travel industry and manages almost half of the flight bookings worldwide. Their business is targeted by fraud attempts reported as outliers in the corresponding datasets. The proprietary datasets are given thereafter; pnr describes the history of changes applied to booking records, transactions depicts user sessions performed on a Web application and targets the detection of bots and malicious users, shared-access was extracted from a backend application dedicated to shared rights management between customers, e.g., seat map display or cruise distribution, and payment-sub reports the booking records along with the user behavior through the booking process, e.g., searches and actions performed. Table 1 shows the datasets characteristics.
### Results
This section shows the outlier detection capabilities of the methods and monitors the mll to exhibit convergence. We also study the impact of depth and dimensionality on dgp-aes, and plot the latent representations learnt by the network.
#### Method comparison
Our experiment performs a fivefold Monte Carlo cross-validation, using 80% of the original dataset for the training and 20% for the testing. Training and testing datasets are normalized, and we use the characteristics of the training dataset to normalize the testing data. Both datasets contain the same proportion of anomalies. Since class distribution is by nature heavily imbalanced for novelty detection problems, we use the map as a performance metric instead of the average rocauc. Indeed, the precision metric strongly penalizes false positives, even if they only represent a small proportion of the negative class, while false positives have very little impact on the roc (Davis and Goadrich 2006). The detailed map are reported in Table 2. Bold results are similar to the best map achieved on the dataset with nonsignificant differences. We used a pairwise Friedman test (Garca et al. 2010) with a threshold of 0.05 to reject the null hypothesis. The experiments are performed on an Ubuntu 14.04 LTS powered by an Intel Xeon E5-4627 v4 CPU and 256 GB RAM. This amount of memory is not sufficient to train rkde on the airline dataset, resulting in missing data in Table 2.
Looking at the average performance, our dgps autoencoders achieve the best results for novelty detection. dgps performed well on all datasets, including high dimensional cases, and outperform the other methods on wine-quality, airline and pnr. By fitting a softmax likelihood instead of a Gaussian on one-hot encoded features, dgp-ae-gs-1 achieves better performance than dgp-ae-g-1 on 3 datasets containing categorical variables out of 4, e.g., mushroom-sub, german-sub and transactions, while showing similar results on the car dataset. This representation allows dgps to reach the best performance on half of the datasets and to outperform state-of-the-art algorithms for novelty detection, such as rkde and IForest. Despite the low-dimensional representation of the latent variables, dgp-ae-g-2 achieves performance comparable to dgp-ae-g-1, which suggests good dimensionality reduction abilities. The use of a softmax likelihood in dgp-ae-gs-2 resulted in better novelty detection capabilities than dgp-ae-g-2 on the 4 datasets containing categorical features. vae-dgp-2 achieves good results but is outperformed on most small datasets.
vae-1 also shows good outlier detection capabilities and handles binary features better than vae-2. However, the multilayer architecture outperforms its shallow counterpart on large datasets containing more than 10,000 samples. Both algorithms perform better than nade-2 which fails on high dimensional datasets such as mushroom-sub, pnr or transactions. We performed additional tests with an increased number of units for nade-2 to cope for the large dimensionality, but we obtained similar results.
While ae-1 shows unexpected detection capabilities for a very simple model, ae-5 reaches the lowest performance. Compressing the data to a feature space 40% smaller than the input space along with dropout layers may cause loss of information resulting in an inaccurate model.
#### Convergence monitoring
To assess the accuracy and the scalability of the selected neural networks, we measure the map and mean log-likelihood (mll) on test data during the training phase to monitor their convergence. The evolution of the two metrics for the dnns is reported in Fig. 2.
While the likelihood is the objective function of most networks, the monitoring of this metric reveals occasional decreases of the mll for all methods during the training process. If minor increases are part of the gradient optimization, the others indicate convergence issues for complex datasets. This is observed for vae-dgp-2 and vae-1 on mammography, or dgp-ae-g-1-arc and vae-1 on mushroom-sub.
Our dgps show the best likelihood on most datasets, in particular when using the arc kernel, with the exception of pnr and mushroom-sub where the rbf kernel is much more efficient. These results demonstrate the efficiency of regularization for dgps and their excellent ability to generalize while fitting complex models.
On the opposite, nade-2 barely reaches the likelihood of ae-1 and ae-5 at convergence. In addition, the network requires an extensive tuning of its parameters and has a computationally expensive prediction step. We tweaked the parameters to increase the model complexity, e.g., number of components and units, but it did not improve the optimized likelihood.
vae-dgp-2 does not reach a competitive likelihood, even with deeper architectures, and shows a computationally expensive prediction step.
Looking at the overall results of these networks, we observe that the model, depicted here by the likelihood, is refined during the entire training process, while the average precision quickly stabilizes. This behavior implies that the ordering of data points according to their outlier score converges much faster, even though small changes can still occur.
Additional convergence experiments have been performed in dgps and are reported in Fig. 3. The left part of the figure shows the ability of dgp-ae-g to generalize while increasing the number of layers. On the right, we compare the dimensionality reduction capabilities of dgp-ae-g-2 while increasing the number of gps on the latent variables layer.
The left part of the plot reports the convergence of dgp-ae-g for configurations ranging from one to ten layers. The plot highlights the correlation between a higher test likelihood and a higher average precision. Single-layer models show a good convergence of the mll on most datasets, though are outperformed by deeper models, especially 4-layer networks, on magic-gamma-sub, payment-sub and airline. Deep architectures result in models of higher capacity at the cost of needing larger datasets to be able to model complex representations, with a resulting slower convergence behavior. Using moderately deep networks can thus show better results on datasets where a single layer is not sufficient to capture the complexity of the data. Interestingly, the bound on the model evidence makes it possible to carry out model selection to decide on the best architecture for the model at hand (Cutajar et al. 2017).
In the right panel of Fig. 3, we increase the dimensionality of the latent representation fixing the architecture to a dgp-ae-g-2. Both the test likelihood and the average precision show that a univariate gp is not sufficient to model accurately the input data. The limitations of this configuration is observed on mammography, payment-sub and airline where more complex representations achieve better performance. Increasing the number of gps results in a higher number of weights for the model, thus in a slower convergence. While configurations using 5 GPs already perform a significant dimensionality reduction, they achieve good performance and are suitable for efficient novelty detection.
#### Latent representation
In this section we illustrate the capabilities of the proposed dgp-ae model to construct meaningful latent probabilistic representations of the data. We select a two-layer dgp-ae architecture with a two-dimensional latent representation $$Z := F^{(1)}$$. Since the mapping of the dgp-ae model is probabilistic, each input point is mapped into a cloud of latent variables. In order to obtain a generative model, we could then train a density estimation algorithm on the latent variables to construct a density $$q(\mathbf {z})$$ used together with the probabilistic decoder part of the dgp-ae to generate new observations.
In Fig. 4, we draw 300 Monte Carlo samples from the approximate posterior over the weights $$\mathbf {W}$$ to construct a latent representation of the old faithful dataset. We use a gmm with two components to cluster the input data, and color the latent representation based on the resulting labels. The point highlighted on the left panel of the plot by a cross is mapped into the green points on the right.
We now extend our experiment to labelled datasets of higher dimensionality, using the given labels for the sole purpose of assigning a color to the points in the latent space. Figure 5 shows the two-dimensional representation of four datasets, breast cancer (569 samples, 30 features), iris (150 $$\times$$ 4), wine (178 $$\times$$ 13) and digits (1797 $$\times$$ 64). For comparison, we also report the results of two manifold learning algorithms, namely t-sne (Maaten and Hinton 2008) and Probabilistic pca (Tipping and Bishop 1999). The plot shows that our algorithm yields meaningful low-dimensional representations, comparable with state-of-the-art dimensionality reduction methods.
## Conclusions
In this paper, we introduced a novel deep probabilistic model for novelty detection. The proposed dgp-ae model is an autoencoder where the encoding and the decoding mappings are governed by dgps. We make the inference of the model tractable and scalable by approximating the dgps using random feature expansions and by inferring the resulting model through stochastic variational inference that could exploit distributed and GPU computing. The proposed dgp-ae is able to flexibly model data with mixed-types feature, which is actively investigated in the recent literature (Vergari et al. 2018). Furthermore, the model is easy to implement using automatic differentiation tools, and is characterized by robust training given that, unlike most gp-based models (Dai et al. 2016), it only involves tensor products and no matrix factorizations.
Through a series of experiments, we demonstrated that dgp-ae s achieve competitive results against state-of-the-art novelty detection methods and dnn-based novelty detection methods. Crucially, dgp-ae s achieve these performance with a practical learning method, making deep probabilistic modeling as an attractive model for general novelty detection tasks. The encoded latent representation is probabilistic and it yields uncertainty that can be used to turn the proposed autoencoder into a generative model; we leave this investigation for future work, as well as the possibility to make use of dgps to model the mappings in variational autoencoders.
## References
1. Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G. S., et al. (2015). TensorFlow: Large-scale machine learning on heterogeneous systems. Software available from tensorflow.org.
2. Arjovsky, M., Chintala, S., & Bottou, L. (2017). Wasserstein gan. arXiv:1701.07875v2
3. Bache, K., & Lichman, M. (2013). UCI machine learning repository. Irvine: University of California, School of Information and Computer Sciences. http://archive.ics.uci.edu/ml.
4. Bradshaw, J., Alexander, & Ghahramani, Z. (2017). Adversarial examples, uncertainty, and transfer testing robustness in Gaussian process hybrid deep networks. arXiv:1707.02476
5. Bui, T. D., Hernández-Lobato, D., Hernández-Lobato, J. M., Li, Y., & Turner, R. E. (2016). Deep Gaussian Processes for regression using approximate expectation propagation. In M. Balcan & K. Q. Weinberger (Eds.), Proceedings of the 33nd international conference on machine learning, ICML 2016, New York City, NY, USA, June 19–24, 2016, volume 48 of JMLR workshop and conference proceedings (pp. 1472–1481). JMLR.org.
6. Cho, Y., & Saul, L. K. (2009). Kernel methods for deep learning. In Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams, & A. Culotta (Eds.), Advances in neural information processing systems (Vol. 22, pp. 342–350). Curran Associates, Inc.
7. Collobert, R., & Weston, J. (2008). A unified architecture for natural language processing: Deep neural networks with multitask learning. In Proceedings of the 25th international conference on machine learning, ICML ’08 (pp. 160–167). New York, NY: ACM.
8. Cutajar, K., Bonilla, E. V., Michiardi, P., & Filippone, M. (2017). Random feature expansions for Deep Gaussian Processes. In D. Precup & Y. W. Teh (Eds.), Proceedings of the 34th international conference on machine learning, volume 70 of proceedings of machine learning research (pp. 884–893). Sydney: International Convention Centre, PMLR.
9. Dai, Z., Damianou, A., González, J., & Lawrence, N. (2016). Variationally auto-encoded Deep Gaussian Processes. In Proceedings of the fourth international conference on learning representations (ICLR 2016).
10. Damianou, A. C., & Lawrence, N. D. (2013). Deep Gaussian Processes. In Proceedings of the sixteenth international conference on artificial intelligence and statistics, AISTATS 2013, Scottsdale, AZ, USA, April 29–May 1, 2013, volume 31 of JMLR proceedings (pp. 207–215). JMLR.org.
11. Davis, J., & Goadrich, M. (2006). The relationship between precision–recall and ROC curves. In Proceedings of the 23rd international conference on Machine learning, ICML ’06 (pp. 233–240). New York, NY: ACM.
12. Dereszynski, E. W., & Dietterich, T. G. (2011). Spatiotemporal models for data-anomaly detection in dynamic environmental monitoring campaigns. ACM Transactions on Sensor Networks (TOSN), 8(1), 3.
13. Dinh, L., Sohl-Dickstein, J., & Bengio, S. (2016). Density estimation using real NVP. arXiv:1605.08803
14. Domingues, R., Filippone, M., Michiardi, P., & Zouaoui, J. (2018). A comparative evaluation of outlier detection algorithms: Experiments and analyses. Pattern Recognition, 74, 406–421.
15. Duvenaud, D. K., Rippel, O., Adams, R. P., & Ghahramani, Z. (2014). Avoiding pathologies in very deep networks. In Proceedings of the seventeenth international conference on artificial intelligence and statistics, AISTATS 2014, Reykjavik, Iceland, April 22–25, 2014, volume 33 of JMLR workshop and conference proceedings (pp. 202–210). JMLR.org.
16. Emmott, A., Das, S., Dietterich, T., Fern, A., & Wong, W.-K. (2016). A meta-analysis of the anomaly detection problem. arXiv:1503.01158v2
17. Gal, Y., & Ghahramani, Z. (2016). Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In Proceedings of the 33rd international conference on international conference on machine learning—volume 48, ICML’16 (pp. 1050–1059). JMLR.org.
18. Gal, Y., Hron, J., & Kendall, A. (2017). Concrete dropout. arXiv:1705.07832
19. Garca, S., Fernndez, A., Luengo, J., & Herrera, F. (2010). Advanced nonparametric tests for multiple comparisons in the design of experiments in computational intelligence and data mining: Experimental analysis of power. Information Sciences, 180(10), 2044–2064.
20. Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., & Bengio, Y. (2014) Generative adversarial nets. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, & K. Q. Weinberger (Eds.), Advances in neural information processing systems (Vol. 27, pp. 2672–2680). Curran Associates, Inc.
21. Graves, A. (2011). Practical variational inference for neural networks. In J. Shawe-Taylor, R. S. Zemel, P. L. Bartlett, F. Pereira, & K. Q. Weinberger (Eds.), Advances in neural information processing systems (Vol. 24, pp. 2348–2356). Curran Associates, Inc.
22. Greensmith, J., Twycross, J., & Aickelin, U. (2006). Dendritic cells for anomaly detection. In IEEE international conference on evolutionary computation, 2006 (pp. 664–671). https://doi.org/10.1109/CEC.2006.1688374.
23. Hinton, G., Deng, L., Yu, D., Dahl, G. E., Mohamed, A.-R., Jaitly, N., et al. (2012). Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups. IEEE Signal Processing Magazine, 29(6), 82–97.
24. Hodge, V., & Austin, J. (2004). A survey of outlier detection methodologies. Artificial Intelligence Review, 22(2), 85–126.
25. Japkowicz, N., & Stephen, S. (2002). The class imbalance problem: A systematic study. Intelligent Data Analysis, 6(5), 429–449.
26. Kemmler, M., Rodner, E., Wacker, E.-S., & Denzler, J. (2013). One-class classification with Gaussian processes. Pattern Recognition, 46(12), 3507–3518.
27. Kim, J., & Scott, C. D. (2012). Robust kernel density estimation. Journal of Machine Learning Research, 13, 2529–2565.
28. Kingma, D. P., & Welling, M. (2014). Auto-encoding variational Bayes. In Proceedings of the second international conference on learning representations (ICLR 2014).
29. Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2012). ImageNet classification with deep convolutional neural networks. In Proceedings of the 25th International conference on neural information processing systems, NIPS’12 (pp. 1097–1105). Curran Associates Inc.
30. LeCun, Y., Bengio, Y., & Hinton, G. (2015). Deep learning. Nature, 521(7553), 436–444.
31. Liu, F. T., Ting, K. M., & Zhou, Z.-H. (2008). Isolation forest. In Proceedings of the 2008 eighth IEEE international conference on data mining, ICDM ’08 (pp. 413–422). IEEE Computer Society.
32. Liu, H., Shah, S., & Jiang, W. (2004). On-line outlier detection and data cleaning. Computers & Chemical Engineering, 28(9), 1635–1647.
33. Maaten, L., & Hinton, G. (2008). Visualizing data using t-SNE. Journal of Machine Learning Research, 9(Nov), 2579–2605.
34. Markou, M., & Singh, S. (2003). Novelty detection: A review-part 2: Neural network based approaches. Signal Processing, 83(12), 2499–2521.
35. Neal, R. M. (1996). Bayesian learning for neural networks (lecture notes in statistics) (1st ed.). Berlin: Springer.
36. Pimentel, M. A., Clifton, D. A., Clifton, L., & Tarassenko, L. (2014). A review of novelty detection. Signal Processing, 99, 215–249.
37. Pokrajac, D., Lazarevic, A., & Latecki, L. J. (2007). Incremental local outlier detection for data streams. In Computational intelligence and data mining, 2007. CIDM 2007. IEEE symposium on (pp. 504–515). IEEE.
38. Prastawa, M., Bullitt, E., Ho, S., & Gerig, G. (2004). A brain tumor segmentation framework based on outlier detection. Medical Image Analysis, 8(3), 275–283.
39. Rasmussen, C. E., & Williams, C. (2006). Gaussian processes for machine learning. Cambridge, MA: MIT Press.
40. Robbins, H., & Monro, S. (1951). A stochastic approximation method. The Annals of Mathematical Statistics, 22, 400–407.
41. Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R. (2014). Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(1), 1929–1958.
42. Tipping, M. E., & Bishop, C. M. (1999). Probabilistic principal component analysis. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(3), 611–622.
43. Uria, B., Côté, M.-A., Gregor, K., Murray, I., & Larochelle, H. (2016). Neural autoregressive distribution estimation. Journal of Machine Learning Research, 17(205), 1–37.
44. Vergari, A., Peharz, R., Di Mauro, N., Molina, A., Kersting, K., & Esposito, F. (2018). Sum-product autoencoding: Encoding and decoding representations using sum-product networks. In Proceedings of the AAAI conference on artificial intelligence (AAAI).
45. Worden, K., Manson, G., & Fieller, N. R. (2000). Damage detection using outlier analysis. Journal of Sound and Vibration, 229(3), 647–667.
46. Zimek, A., Schubert, E., & Kriegel, H.-P. (2012). A survey on unsupervised outlier detection in high-dimensional numerical data. Statistical Analysis and Data Mining, 5(5), 363–387.
## Acknowledgements
The authors wish to thank the Amadeus Middleware Fraud Detection team directed by Virginie Amar and Jeremie Barlet, led by the product owner Christophe Allexandre and composed of Jean-Blas Imbert, Jiang Wu, Damien Fontanes and Yang Pu for building and labeling the transactions, shared-access and payment-sub datasets. MF gratefully acknowledges support from the AXA Research Fund.
## Author information
Authors
### Corresponding author
Correspondence to Rémi Domingues.
Editors: Jesse Davis, Elisa Fromont, Derek Greene, and Bjorn Bringmaan.
## Rights and permissions
Reprints and Permissions
Domingues, R., Michiardi, P., Zouaoui, J. et al. Deep Gaussian Process autoencoders for novelty detection. Mach Learn 107, 1363–1383 (2018). https://doi.org/10.1007/s10994-018-5723-3
• Accepted:
• Published:
• Issue Date:
### Keywords
• Novelty detection
• Deep Gaussian Processes
• Autoencoder
• Unsupervised learning
• Stochastic variational inference | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.815765380859375, "perplexity": 1479.044719495571}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737319.74/warc/CC-MAIN-20200808080642-20200808110642-00538.warc.gz"} |
https://www.class-central.com/course/coursera-introduction-into-general-theory-of-relativity-6543 | Top 50 MOOCs
To support our site, Class Central may be compensated by some course providers.
Introduction into General Theory of Relativity
135
students interested
• Provider Coursera
• Subject Physics
• \$ Cost Free Online Course (Audit)
• Session Upcoming
• Language English
• Certificate Paid Certificate Available
• Start Date
• Duration 12 weeks long
Taken this course? Share your experience with other students. Write review
Overview
General Theory of Relativity or the theory of relativistic gravitation is the one which describes black holes, gravitational waves and expanding Universe. The goal of the course is to introduce you into this theory. The introduction is based on the consideration of many practical generic examples in various scopes of the General Relativity. After the completion of the course you will be able to solve basic standard problems of this theory. We assume that you are familiar with the Special Theory of Relativity and Classical Electrodynamics. However, as an aid we have recorded several complementary materials which are supposed to help you understand some of the aspects of the Special Theory of Relativity and Classical Electrodynamics and some of the calculational tools that are used in our course. Also as a complementary material we provide the written form of the lectures at the website: https://math.hse.ru/generalrelativity2015
Syllabus
General Covariance
To start with, we recall the basic notions of the Special Theory of Relativity. We explain that Minkwoskian coordinates in flat space-time correspond to inertial observers. Then we continue with transformations to non-inertial reference systems in flat space-time. We show that non-inertial observers correspond to curved coordinate systems in flat space-time. In particular, we describe in grate details Rindler coordinates that correspond to eternally homogeneously accelerating observers. This shows that our Nature allows many different types of metrics, not necessarily coincident with the Euclidian or Minkwoskain ones. We explain what means general covariance. We end up this module with the derivation of the geodesic equation for a general metric from the least action principle. In this equation we define the Christoffel symbols.
Covariant differential and Riemann tensor
We start with the definition of what is tensor in a general curved space-time. Then we define what is connection, parallel transport and covariant differential. We show that for Riemannian manifolds connection coincides with the Christoffel symbols and geodesic equations acquire a clear geometric meaning. We end up with the definition of the Riemann tensor and the description of its properties. We explain how Riemann tensor allows to distinguish flat space-time in curved coordinates from curved space-times. For this module we provide complementary video to help students to recall properties of tensors in flat space-time.
Einstein-Hilbert action and Einstein equations
We start with the explanation of how one can define Einstein equations from fundamental principles. Such as general covariance, least action principle and the proper choice of dynamical variables. Namely, the role of the latter in the General Theory of Relativity is played by the metric tensor of space-time. Then we derive the Einstein equations from the least action principle applied to the Einstein-Hilbert action. Also we define the energy-momentum tensor for matter and show that it obeys a conservation law. We describe the basic generic properties of the Einstein equations. We end up this module with some examples of energy-momentum tensors for different sorts of matter fields or bodies and particles.To help understanding this module we provide complementary video with the explanation of the least action principle in the simplest case of the scalar field in flat two-dimensional space-time.
Schwarzschild solution
With this module we start our study of the black hole type solutions. We explain how to solve the Einstein equations in the simplest settings. We find perhaps the most famous solution of these equations, which is referred to as the Schwarzschild black hole. We formulate the Birkhoff theorem. We end this module with the description of some properties of this Schwarzschild solution. We provide different types of coordinate systems for such a curved space-time.
Penrose-Carter diagrams
We start with the definition of the Penrose-Carter diagram for flat space-time. On this example we explain the uses of such diagrams. Then we continue with the definition of the Kruskal-Szekeres coordinates which cover the entire black hole space-time. With the use of these coordinates we define Penrose-Carter diagram for the Schwarzschild black hole. This diagram allows us to qualitatively understand the fundamental properties of the black hole.
Classical tests of General Theory of Relativity
We start with the definition of Killing vectors and integrals of motion, which allow one to provide conserving quantities for a particle motion in Schwarzschild space-time. We derive the explicit geodesic equation for this space-time. This equation provides a quantitative explanation of some basic properties of black holes. We use the geodesic equation to explain the precession of the Mercury perihelion and of the light deviation in curved space-time.
Interior solution and Kerr's solution
We start with the definition of the so called perfect fluid energy-momentum tensor and with the description of its properties. We use this tensor to derive the so called interior solution of the Einstein equations, which provides a simple model of a star in the General Theory of Relativity. Then we continue with a brief description of the Kerr solution, which corresponds to the rotating black hole. We end up this module with a brief description of the Cosmic Censorship hypothesis and of the black hole No Hair Theorem.
Collapse into black hole
We start with the derivation of the Oppenheimer-Snyder solution of the Einstein equations, which describes the collapse of a star into black hole. We derive the Penrose-Carter diagram for this solution. We end up this module with a brief description of the origin of the Hawking radiation and of the basic properties of the black hole formation.
Gravitational waves
With this module we start our study of gravitational waves. We explain the important difference between energy-momentum conservation laws in the absence and in the presence of the dynamical gravity. We define the gravitational energy-momentum pseudo-tensor. Then we continue with the linearized approximation to the Einstein equations which allows us to clarify the meaning of the pseudo-tensor. We end up this module with the derivation of the free monochromatic gravitational waves and of their energy-momentum pseudo-tensor. These waves are solutions of the Einstein equations in the linearized approximation.
In this module we show how moving massive bodies create gravitational waves in the linearized approximation. Then we continue with the derivation of the exact shock gravitational wave solutions of the Einstein equations. We describe their properties. To help to understand this module we provide two complementary videos. One with the explanations how to perform the averaging over directions in space. And the other video is with the derivation of the retarded Green function.
Friedman-Robertson-Walker cosmology
With this module we start our discussion of the cosmological solutions. We define constant curvature three-dimensional homogeneous spaces. Then we derive Friedman-Robertson-Walker cosmological solutions of the Einstein equations. We describe their properties. We end up this module with the derivation of the vacuum homogeneous but anisotropic cosmological Kasner solution.
Cosmological solutions with non-zero cosmological constant
In this module we derive constant curvature de Sitter and anti de Sitter solutions of the Einstein equations with non-zero cosmological constant. We describe the geometric and causal properties of such space-times and provide their Penrose-Carter diagrams. We provide coordinate systems which cover various patches of these space-times.
Emil Akhmedov
Reviews for Coursera's Introduction into General Theory of Relativity Based on 0 reviews
• 5 star 0%
• 4 star 0%
• 3 star 0%
• 2 star 0%
• 1 star 0%
Did you take this course? Share your experience with other students.
Class Central
Get personalized course recommendations, track subjects and courses with reminders, and more. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8430140018463135, "perplexity": 375.6223766099906}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039741491.47/warc/CC-MAIN-20181113194622-20181113220622-00247.warc.gz"} |
https://alpinebearing.com/bearing-categories/machine-tool-spindle/cylindrical-roller/single-row-n10/ | ## Single Row N10 Ball Bearings
#### Ball Bearing Features
• K = tapered bore (taper ratio is 1/12)
Part # Bore OD Width Width 2 Width Bi Width BO Closure Raceway Contact Angle Ball Type Lube Flange Diameter Flange Width Dynamic Kaydon Radial (lbs) Dynamic ISO Radial (lbs) Dynamic Radial (lbs) Dynamic Thrust (lbs) Static Radial (lbs) Static Thrust (lbs) Add To Quote
N1011K 55mm 90mm 18mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1012 60mm 95mm 18mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1012K 60mm 95mm 18mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1013 65mm 100mm 18mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1013K 65mm 100mm 18mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1014 70mm 110mm 20mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1014K 70mm 110mm 20mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1015 75mm 115mm 20mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1015K 75mm 115mm 20mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1016 80mm 125mm 22mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1016K 80mm 125mm 22mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1017 85mm 130mm 22mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1017K 85mm 130mm 22mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1018 90mm 140mm 24mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1018K 90mm 140mm 24mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1019 95mm 145mm 24mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1019K 95mm 145mm 24mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1020 100mm 150mm 24mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1020K 100mm 150mm 24mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1021 105mm 160mm 26mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1021K 105mm 160mm 26mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1022 110mm 170mm 28mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1022K 110mm 170mm 28mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1024 120mm 180mm 28mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1024K 120mm 180mm 28mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1026 130mm 200mm 33mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1026K 130mm 200mm 33mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1028 140mm 210mm 33mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1028K 140mm 210mm 33mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1006 30mm 55mm 13mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1006K 30mm 55mm 13mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1007 35mm 62mm 14mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1007K 35mm 62mm 14mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1008 40mm 68mm 15mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1008K 40mm 68mm 15mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1009 45mm 75mm 16mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1009K 45mm 75mm 16mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1010 50mm 80mm 16mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1010K 50mm 80mm 16mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
N1011 55mm 90mm 18mm N/A N/A N/A N/A N/A N/A Steel N/A N/A N/A N/A N/A N/A N/A N/A N/A
## Can't find the size or bearing you're looking for?
We would love to hear from you. Our knowledgeable sales staff will help guide you to the right bearing solution. Let our experts help! List your specifications and we will do our best to get you the product you need! | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9284188151359558, "perplexity": 2340.8224332252453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662619221.81/warc/CC-MAIN-20220526162749-20220526192749-00348.warc.gz"} |
http://www.gradesaver.com/textbooks/math/precalculus/precalculus-mathematics-for-calculus-7th-edition/chapter-2-section-2-1-functions-exercises-page-156/25 | ## Precalculus: Mathematics for Calculus, 7th Edition
Evaluate $g(x) = \frac{1-x}{1+x}$: $g(2) = \frac{-1}{3}$ $g(-1) =\frac{2}{0} = undefined$ $g(\frac{1}{2}) =\frac{1}{3}$ $g(a) = \frac{1-a}{1+a}$ $g(a-1) = \frac{-a+2}{a}$ $g(x^2-1) = \frac{-x^2+2}{x^2}$
For x = 2 $g(2) = \frac{1-2}{1+2}$ ... subtract numerator and add denominator $= \frac{-1}{3}$ ____________________ For x = -1 $g(-1) = \frac{1-(-1)}{1+(-1)}$ ... subtract numerator and add denominator $= \frac{2}{0} = undefined$ ____________________ For x = $\frac{1}{2}$ $g(\frac{1}{2}) = \frac{1-\frac{1}{2}}{1+\frac{1}{2}}$ ... convert whole numbers to fractions with common denominator $= \frac{\frac{2}{2}-\frac{1}{2}}{\frac{2}{2}+\frac{1}{2}}$...subtract numerator and add denominator $= \frac{\frac{1}{2}}{\frac{3}{2}}$...multiply top and bottom by 2 $=\frac{1}{3}$ ____________________ For x = a $g(a) = \frac{1-a}{1+a}$ ____________________ For x = a - 1 $g(a-1) = \frac{1-(a-1)}{1+(a-1)}$... simplify top and bottom $=\frac{1-a+1}{1+a-1}=\frac{-a+2}{a}$ ____________________ For x = $x^2 - 1$ $g( x^2 - 1) = \frac{1-( x^2 - 1)}{1+( x^2 - 1)}$... simplify top and bottom $=\frac{1- x^2 + 1}{1+ x^2 - 1}=\frac{-x^2+2}{x^2}$ ____________________ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9851173758506775, "perplexity": 1942.0175414621451}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945111.79/warc/CC-MAIN-20180421090739-20180421110739-00230.warc.gz"} |
https://byjus.com/rd-sharma-solutions/class-8-maths-chapter-7-factorization-exercise-7-9/ | # RD Sharma Solutions for Class 8 Maths Chapter - 7 Factorization Exercise 7.9
To make learning easy for students RD Sharma Solutions for Class 8 Maths Chapter 7 Factorization Exercise 7.9 are provided here. In this Exercise 7.9 of RD Sharma, we shall discuss the problems based on the factorization of quadratic polynomials by using the method of completing the perfect square. Our experts have formulated textbook solutions in a clear manner to help students prepare for their board exams and come out with flying colours in their examinations. Students can download free pdf of RD Sharma Solutions from the links provided below.
## Download the pdf of RD Sharma Solutions For Class 8 Maths Exercise 7.9 Chapter 7 Factorization
### Access Answers to RD Sharma Solutions for Class 8 Maths Exercise 7.9 Chapter 7 Factorization
#### EXERCISE 7.9 PAGE NO: 7.32
Factorize each of the following quadratic polynomials by using the method of completing the square:
1. p2 + 6p + 8
Solution:
We have,
p2 + 6p + 8
Coefficient of p2 is unity. So, we add and subtract square of half of coefficient of p.
p2 + 6p + 8 = p2 + 6p + 32 – 32 + 8 (Adding and subtracting 32)
= (p + 3)2 – 12 (By completing the square)
By using the formula (a2 – b2) = (a+b) (a-b)
= (p + 3 – 1) (p + 3 + 1)
= (p + 2) (p + 4)
2. q2 – 10q + 21
Solution:
We have,
q2 – 10q + 21
Coefficient of q2 is unity. So, we add and subtract square of half of coefficient of q.
q2 – 10q + 21 = q2 – 10q+ 52 – 52 + 21 (Adding and subtracting 52)
= (q – 5)2 – 22 (By completing the square)
By using the formula (a2 – b2) = (a+b) (a-b)
= (q – 5 – 2) (q – 5 + 2)
= (q – 3) (q – 7)
3. 4y2 + 12y + 5
Solution:
We have,
4y2 + 12y + 5
4(y2 + 3y + 5/4)
Coefficient of y2 is unity. So, we add and subtract square of half of coefficient of y.
4(y2 + 3y + 5/4) = 4 [y2 + 3y + (3/2)2 – (3/2)2 + 5/4] (Adding and subtracting (3/2)2)
= 4 [(y + 3/2)2 – 12] (Completing the square)
By using the formula (a2 – b2) = (a+b) (a-b)
= 4 (y + 3/2 + 1) (y + 3/2 – 1)
= 4 (y + 1/2) (y + 5/2) (by taking LCM)
= 4 [(2y + 1)/2] [(2y + 5)/2]
= (2y + 1) (2y + 5)
4. p2 + 6p – 16
Solution:
We have,
p2 + 6p – 16
Coefficient of p2 is unity. So, we add and subtract square of half of coefficient of p.
p2 + 6p – 16 = p2 + 6p + 32 – 32 – 16 (Adding and subtracting 32)
= (p + 3)2 – 52 (Completing the square)
By using the formula (a2 – b2) = (a+b) (a-b)
= (p + 3 + 5) (p + 3 – 5)
= (p + 8) (p – 2)
5. x2 + 12x + 20
Solution:
We have,
x2 + 12x + 20
Coefficient of x2 is unity. So, we add and subtract square of half of coefficient of x.
x2 + 12x + 20 = x2 + 12x + 62 – 62 + 20 (Adding and subtracting 62)
= (x + 6)2 – 42 (Completing the square)
By using the formula (a2 – b2) = (a+b) (a-b)
= (x + 6 + 4) (x + 6 – 4)
= (x + 2) (x + 10)
6. a2 – 14a – 51
Solution:
We have,
a2 – 14a – 51
Coefficient of a2 is unity. So, we add and subtract square of half of coefficient of a.
a2 – 14a – 51 = a2 – 14a + 72 – 72 – 51 (Adding and subtracting 72)
= (a – 7)2 – 102 (Completing the square)
By using the formula (a2 – b2) = (a+b) (a-b)
= (a – 7 + 10) (9 – 7 – 10)
= (a – 17) (a + 3)
7. a2 + 2a – 3
Solution:
We have,
a2 + 2a – 3
Coefficient of a2 is unity. So, we add and subtract square of half of coefficient of a.
a2 + 2a – 3 = a2 + 2a + 12 – 12 – 3 (Adding and subtracting 12)
= (a + 1)2 – 22 (Completing the square)
By using the formula (a2 – b2) = (a+b) (a-b)
= (a + 1 + 2) (a + 1 – 2)
= (a + 3) (a – 1)
8. 4x2 – 12x + 5
Solution:
We have,
4x2 – 12x + 5
4(x2 – 3x + 5/4)
Coefficient of x2 is unity. So, we add and subtract square of half of coefficient of x.
4(x2 – 3x + 5/4) = 4 [x2 – 3x + (3/2)2 – (3/2)2 + 5/4] (Adding and subtracting (3/2)2)
= 4 [(x – 3/2)2 – 12] (Completing the square)
By using the formula (a2 – b2) = (a+b) (a-b)
= 4 (x – 3/2 + 1) (x – 3/2 – 1)
= 4 (x – 1/2) (x – 5/2) (by taking LCM)
= 4 [(2x-1)/2] [(2x – 5)/2]
= (2x – 5) (2x – 1)
9. y2 – 7y + 12
Solution:
We have,
y2 – 7y + 12
Coefficient of y2 is unity. So, we add and subtract square of half of coefficient of y.
y2 – 7y + 12 = y2 – 7y + (7/2)2 – (7/2)2 + 12 [Adding and subtracting (7/2)2]
= (y – 7/2)2 – (7/2)2 (Completing the square)
By using the formula (a2 – b2) = (a+b) (a-b)
= (y – (7/2- 1/2)) (y – (7/2 + 1/2))
= (y – 3) (y – 4)
10. z2 – 4z – 12
Solution:
We have,
z2 – 4z – 12
Coefficient of z2 is unity. So, we add and subtract square of half of coefficient of z.
z2 – 4z – 12 = z2 – 4z + 22 – 22 – 12 [Adding and subtracting 22]
= (z – 2)2 – 42 (Completing the square)
By using the formula (a2 – b2) = (a+b) (a-b)
= (z – 2 + 4) (z – 2 – 4)
= (z – 6) (z + 2) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8452777862548828, "perplexity": 2348.324727171672}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107904039.84/warc/CC-MAIN-20201029095029-20201029125029-00495.warc.gz"} |
http://mathhelpforum.com/calculus/85756-confussing-infinite-series-problem.html | 1. confussing infinite series problem
Hi.
The problem is giving me a hard time basically because i just can't seem to figure out how to solve for it.
The problem is $\sum(-1)^n\int1/2^x$
the limits on sigma are n=1 to infinity and for the integral its n and n+1
2. Originally Posted by [s]arah
Hi.
The problem is giving me a hard time basically because i just can't seem to figure out how to solve for it.
The problem is $\sum(-1)^n\int1/2^x$
the limits on sigma are n=1 to infinity and for the integral its n and n+1
As in $\sum_{n = 0}^\infty (-1)^n \int_n^{n + 1}\frac 1{2^x}$??
Where are you getting stuck? just do the integral, you get
$\sum_{n = 0}^\infty (-1)^{n+1} \frac 1{2^x \ln 2} \bigg|_n^{n + 1}$
now, plug in the limits of integration, factor out the ln(2), and you can split the sum into two geometric series | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9868676662445068, "perplexity": 282.9244069774718}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104565.76/warc/CC-MAIN-20170818043915-20170818063915-00625.warc.gz"} |
https://boolesrings.org/davesixsmith/2016/06/07/on-permutable-meromorphic-functions/ | # On permutable meromorphic functions
Aequationes Math. 90 (2016), no. 5, 1025–1034. Also available on the arXiv. This is a joint work with John Osborne.
We study the class $\mathcal{M}$ of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in $\mathcal{M}$, with at least one essential singularity, permutes with a non-constant rational map $g$, then $g$ is a Möbius map that is not conjugate to an irrational rotation. For a given function $f \in\mathcal{M}$ which is not a Möbius map, we show that the set of functions in $\mathcal{M}$ that permute with $f$ is countably infinite. Finally, we show that there exist transcendental meromorphic functions $f: \mathbb{C} \to \mathbb{C}$ such that, among functions meromorphic in the plane, $f$ permutes only with itself and with the identity map. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9526398777961731, "perplexity": 130.48400960912906}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812758.43/warc/CC-MAIN-20180219171550-20180219191550-00753.warc.gz"} |
http://mathhelpforum.com/statistics/151261-probability-blues.html | Math Help - Probability Blues
1. Probability Blues
A sales contract between a manufacturer and a buyer requires 20 towels to be subjected to a water absorption test. If no more than 1 towel fails the test, the batch is accepted. If 2 or 3 towels fail the test, an additional 20 towels are tested. The batch is then accepted if 3 or less (out of 40) fail the test. Otherwise, the batch is rejected.
(a) If a batch of towels contains 5 % which would be rejected by the test,what is the probability that the batch is accepted.
(b) The lengths of 20 towels are measured and if the mean length is less than a value a specified in the contract, the batch is rejected. What should the value of a be to give a probability of 0.99 of accepting a batch with mean length of 106 mm and standard deviation of 6 mm.
I solved (a). The probability that the batch is accepted is 0.8961.
But I can't solve (b).
2. Originally Posted by cyt91
[snip]
(b) The lengths of 20 towels are measured and if the mean length is less than a value a specified in the contract, the batch is rejected. What should the value of a be to give a probability of 0.99 of accepting a batch with mean length of 106 mm and standard deviation of 6 mm.
[snip]
You need to find the value of a such that $\Pr(X \geq a) = p$ where X ~ Normal $(\mu = 106, \, \sigma = 6)$ and $p^{20} = 0.99 \Rightarrow p = ....$
So it's more or less a routine inverse normal problem and your class notes and textbook will have examples to review.
If you need more help, please show what you've tried and say where you get stuck. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9399057626724243, "perplexity": 546.6398717767955}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1430451423003.11/warc/CC-MAIN-20150501033703-00050-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/manifold-question.65604/ | # Manifold question?
1. Mar 2, 2005
### waht
Is the manifold a space defined by the metric tensor or is it a completetly different thing. I'm new to tensor analysis though.
Thanks.
2. Mar 2, 2005
3. Mar 2, 2005
### jcsd
No, infact a manifold doesn't necessarily need to have a (global) metric dfeined in order to be a manifold! A manifold is basically anything that can be continously parameterized or more formally, is a set of points where every point has an open neighbourhood that is homeomorphic to Rn (where n is the dimension of the manifold). The metric tensor field defines the scalar product at each point on a manifold, it doesn't define the manifold as a manifold is still a manifold whether the scalar product is defined or not.
4. Mar 2, 2005
### gvk
For comfort, think about manifold as a curve or surface embedded in 3D space. And work hard to know everything about those stuff including n-D hypersurfaces, Gauss curvature, Riemannian curvature, and parallel transport, you be ready to step up to the manifold's world without metric.
5. Mar 2, 2005
### jcsd
Actually gvk things like curvature and parallel transport dpend on the properties of the metric tensor, a mainofld at it's most basic is really just a set o fpoitns where for each point we have set of other points which are 'near to it'.
6. Mar 2, 2005
### waht
re
that makes more sense now, so basically the parametric equations define a manifold.
"Actually gvk things like curvature and parallel transport dpend on the properties of the metric tensor, a mainofld at it's most basic is really just a set o fpoitns where for each point we have set of other points which are 'near to it "
and the wolfram describes the manifold as a topological space that is locally Euclidean.
I don't get this part.
7. Mar 2, 2005
### jcsd
Okay by locally Eudlidean what is meant that for every point on the mainfold there is also a neighbourhood (if you like all the points that are less than x distance away) which is Euclidean, i.e. we can if we like treat this neighbourhood just like normal Euclidean space. So for example in general relativty the laws of special relativity are not true in a general sense but thanks to the fact that spavcetime is represnted by a manifold they are always true locally (i.e. they are true as long as we only talk about a small region of spacetime).
8. Mar 3, 2005
### Peterdevis
I agree that a manifold is a topological space (by introducing open sets) that locally lokes like the euclidian space.
In my opinion you introduce (if you want it ar not) the standard metric with this last restriction(lokes like the euclidian space) in the topological space. I agree that at this level of the manifold you don' t use it. But when you define a calculus on that manifold you use the fact that the euclidian space is equiped with the standard metric.
9. Mar 3, 2005
### mathwonk
an example of a manifold is a sphere.
If you include also the family of tangent planes to the sphere and a smoothly varying dot product on all these planes, you have a (Riemannian) metric.
a family of velocity vectors, v(p), one at each point p of the sphere, is an example of a "vector field".
The family of linear functionals, <v(p), > on tangent vectors defined by a vector field and a dot product, is an example of a "covector" field.
The family of dot products itself < , >(p), is an example of a "tensor field".
so naturally if you view your original sphere as embedded in three space, then the planes and dot product come along for free, and you do not notice they are extra structure.
And by the way, you do not need a metric to do calculus on a manifold, at least not to define derivatives, velocity vectors, and integrate differential forms. Only to measure arc lengths and curvature, volume, etc...
Last edited: Mar 3, 2005
Similar Discussions: Manifold question? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9222829341888428, "perplexity": 713.0510358701052}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948515309.5/warc/CC-MAIN-20171212060515-20171212080515-00463.warc.gz"} |
http://mymathforum.com/calculus/343046-question-binomial-theorem.html | My Math Forum Question binomial theorem
User Name Remember Me? Password
Calculus Calculus Math Forum
December 10th, 2017, 05:19 AM #1 Newbie Joined: Dec 2017 From: australia Posts: 1 Thanks: 0 Question binomial theorem Given 1/(1+cube root (x)) find its series using the binomial theorem. Use this series to approximate y at x = 1/2 and find the error between the real value and the approximated value for y at x = 1/2 (assuming n = 6). I was given a question similar to above which I have changed a bit as I still want to do the original myself. I haven't been able to work out how to do it. the best I could get have I changed the cube root into a 1/3 power and moved it to the top of the fraction making it -1/3. assuming N equals 6 I put the power together giving -2. i then plugged it into the equation (1+x)^n=1+1x+n(n-1)/n! . . . . I then used attempted to use the error equation |m/(n+1) (x-c)^n+1|. This error equation I thought was right but I don't get an answer that appears correct. can anyone explain where I am going wrong with this method? I have tried a few different ways but this seems to get me the closest. thanks in advance.
December 10th, 2017, 07:58 AM #2 Global Moderator Joined: Dec 2006 Posts: 18,416 Thanks: 1462 $\dfrac{1}{1 + y} = (1 + y)^{-1} = 1 - y + y^2 - y^3 \,+ \,.\,.\,.$ (for |$y$| < 1). Now substitute $x^{1/3}$ for $y$, etc. That was easy. Maybe you changed the original problem too much.
December 10th, 2017, 05:38 PM #3
Global Moderator
Joined: May 2007
Posts: 6,416
Thanks: 557
Quote:
Originally Posted by skipjack $\dfrac{1}{1 + y} = (1 + y)^{-1} = 1 - y + y^2 - y^3 \,+ \,.\,.\,.$ (for |$y$| < 1). Now substitute $x^{1/3}$ for $y$, etc. That was easy. Maybe you changed the original problem too much.
latex problem - it is not displaying
December 10th, 2017, 10:18 PM #4 Global Moderator Joined: Dec 2006 Posts: 18,416 Thanks: 1462 Without latex: 1/(1 + y) = (1 + y)^(-1) = 1 - y + y² - y³ + . . . (for |y| < 1). Now substitute x^(1/3) for y, etc.
December 12th, 2017, 01:57 PM #5
Global Moderator
Joined: May 2007
Posts: 6,416
Thanks: 557
Quote:
Originally Posted by skipjack Without latex: 1/(1 + y) = (1 + y)^(-1) = 1 - y + y² - y³ + . . . (for |y| < 1). Now substitute x^(1/3) for y, etc.
You now have a series in $\displaystyle x^{\frac{1}{3}}$. I wonder if the original intent was to get a series in x?
Tags binomial, question, theorem
Thread Tools Display Modes Linear Mode
Similar Threads Thread Thread Starter Forum Replies Last Post beesee Probability and Statistics 5 September 18th, 2015 02:38 PM Richard Fenton Number Theory 3 January 13th, 2015 09:29 AM SH-Rock Probability and Statistics 1 January 6th, 2011 04:08 PM rooster Applied Math 1 October 27th, 2009 09:30 AM SH-Rock Calculus 1 December 31st, 1969 04:00 PM
Contact - Home - Forums - Cryptocurrency Forum - Top | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8395742774009705, "perplexity": 1599.075876999117}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084887621.26/warc/CC-MAIN-20180118210638-20180118230638-00124.warc.gz"} |
https://infoscience.epfl.ch/record/55405 | ## Forensic automatic speaker recognition using Bayesian interpretation and statistical compensation for mismatched conditions
Nowadays, state-of-the-art automatic speaker recognition systems show very good performance in discriminating between voices of speakers under controlled recording conditions. However, the conditions in which recordings are made in investigative activities (e.g., anonymous calls and wire-tapping) cannot be controlled and pose a challenge to automatic speaker recognition. Differences in the phone handset, in the transmission channel and in the recording devices can introduce variability over and above that of the voices in the recordings. The strength of evidence, estimated using statistical models of within-source variability and between-sources variability, is expressed as a likelihood ratio, i.e., the probability of observing the features of the questioned recording in the statistical model of the suspected speaker's voice, given the two competing hypotheses: the suspected speaker is the source of the questioned recording and the speaker at the origin of the questioned recording is not the suspected speaker. The main unresolved problem in forensic automatic speaker recognition today is that of handling mismatch in recording conditions. Mismatch in recording conditions has to be considered in the estimation of the likelihood ratio. The research in this thesis mainly addresses the problem of the erroneous estimation of the strength of evidence due to the mismatch in technical conditions of encoding, transmission and recording of the databases used in a Bayesian interpretation framework. We investigate three main directions in applying the Bayesian interpretation framework to forensic automatic speaker recognition casework. The first addresses the problem of mismatched recording conditions of the databases used in the analysis. The second concerns introducing the Bayesian interpretation methodology to aural-perceptual speaker recognition as well as comparing aural-perceptual tests performed by laypersons with an automatic speaker recognition system, in matched and mismatched recording conditions. The third addresses the problem of variability in estimating the likelihood ratio, and several new solutions to cope with this variability are proposed. Firstly, we propose a new approach to estimate and statistically compensate for the effects of mismatched recording conditions using databases, in order to estimate parameters for scaling distributions to compensate for mismatch, called "scaling databases". These scaling databases reduce the need for recording large databases for potential populations in each recording condition, which is both expensive and time consuming. The compensation method is based on the principal Gaussian component in the distributions. The error in the likelihood ratios obtained after compensation increases with the deviation of the score distributions from the Gaussian distribution. We propose guidelines for the creation of a database that can be used in order to estimate and compensate for mismatch, and create a prototype of this database to validate the methodology for compensation. Secondly, we analyze the effect of mismatched recording conditions on the strength of evidence, using both aural-perceptual and automatic speaker recognition methods. We have introduced the Bayesian interpretation methodology to aural-perceptual speaker recognition from which likelihood ratios can be estimated. It was experimentally observed that in matched recording conditions of suspect and questioned recordings, the automatic systems showed better performance than the aural recognition systems. In mismatched conditions, however, the baseline automatic systems showed a comparable or slightly degraded performance as compared to the aural recognition systems. Adapting the baseline automatic system to mismatch showed comparable or better performance than aural recognition in the same conditions. Thirdly, in the application of Bayesian interpretation to real forensic case analysis, we propose several new solutions for the analysis of the variability of the strength of evidence using bootstrapping techniques, statistical significance testing and confidence intervals, and multivariate extensions of the likelihood ratio for handling cases where the suspect data is limited. In order for forensic automatic speaker recognition to be acceptable for presentation in the courts, the methodologies and techniques have to be researched, tested and evaluated for error, as well as be generally accepted in the scientific community. The methodology presented in this thesis is viewed in the light of the Daubert (USA, 1993) ruling for the admissibility of scientific evidence.
Drygajlo, Andrzej
Year:
2005
Publisher:
Lausanne, EPFL
Other identifiers:
urn: urn:nbn:ch:bel-epfl-thesis3367-7
Note: The status of this file is: EPFL only | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.803537130355835, "perplexity": 1355.957145147675}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891817908.64/warc/CC-MAIN-20180226005603-20180226025603-00720.warc.gz"} |
https://astronomy.stackexchange.com/questions/29908/what-was-the-first-astronomical-measurement-which-demonstrated-that-the-earth-i/29985 | # What was the first astronomical measurement which demonstrated that “the Earth is surrounded by vacuum”?
The question Who was the first to realize that the Earth is surrounded by vacuum? was closed because some users felt it was answered by answers to a different question in an different SE site: Who was the first to postulate that space was a vacuum?
To me the distance between realize and postulate is astronomical!
That aside, I've asked a new question: What was the first astronomical measurement which demonstrated that "the Earth is surrounded by vacuum"?
By demonstrated I'm talking about a measurement that shows it is, or at least is likely to be the case that Earth is surrounded by vacuum. It should be something such that it could be presented as scientifically convincing evidence to other scientists of the time, unencumbered by non-scientific predispositions (e.g. religion, rivalry, philosophy...)
• I'm voting to close this question as off-topic because this is a question more suited to the History of Science and Mathematics SE. – StephenG Mar 7 '19 at 6:47
• @StephenG more suited is not a recognized reason to close a question. That's why it's not presented to you as an option. Unless you can show this is clearly off-topic, you should withdraw your close vote. OP decides where to ask, you don't vote to ask somewhere else. We have a history tag with 93 questions. – uhoh Mar 7 '19 at 6:49
• @StephenG I think the last paragraph makes a clear argument why this is a different question, give it another read-through. In the mean time your close reason is not valid, the site has a long series of well-received questions about the history of astronomy. – uhoh Mar 7 '19 at 7:01
• @StephenG answers to Are Questions related to Cosmology and ancient history of astronomy allowed in Astronomy SE? and Do questions about the history and discoveries of astronomy belong here? and 93 questions tagged with history seem to argue in my favor here. – uhoh Mar 7 '19 at 7:08
• Is anyway surprising that a more or less clear answer cannot be found quickly. – Alchimista Mar 7 '19 at 9:45
Torricelli, the inventor of the Mercury Barometer (~1644) argued that the height of the column of mercury was governed by atmospheric pressure (the "weight of the atmosphere" as he would have put it). He asserted that the space above the mercury in his tube was a vacuum, a totally anti-Aristotlean concept at that time.
To test this he enlisted the help of Blaise Pascal and Florin Perier in carrying one of two barometers up a mountain, the Puy de Dome in central France. His prediction was that the pressure at the top of the mountain would be less than at the base because of a lesser weight of air.
The experiment showed a drop in height (ie of pressure) and Torriceli himself concluded that the height of the atmosphere would be about 20km, above which there would be vacuum - the same vacuum as in the top of the barometer tube. The experiment was IMHO one of the greatest experiments in the history of physics. See for example https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3768090/
I would argue the famous Michelson-Morley experiment.
## Luminiferous Aether
A bit of background first before getting into the experiment itself. It is quite easy to surmise, via Newton's laws of motion, that Earth should be in a vacuum, otherwise the constant drag of traveling through some medium must eventually cause us to crash into the Sun. Despite this Isaac Newton himself proposed the concept of a Luminiferous Aether which pervaded all of space and was the medium through which light propagated. Before this point there were various concepts of an "Aether" pervading space, but this, I believe, was the first truly scientific approach to the concept as a way of explaining physical phenomenon rather than a simple supposition of existence (such as the Greeks had done). The Luminiferous Aether was proposed as an almost magical concept to avoid numerous physical issues. I think Wikipedia describes it best.
The mechanical qualities of the aether had become more and more magical: it had to be a fluid in order to fill space, but one that was millions of times more rigid than steel in order to support the high frequencies of light waves. It also had to be massless and without viscosity, otherwise it would visibly affect the orbits of planets. Additionally it appeared it had to be completely transparent, non-dispersive, incompressible, and continuous at a very small scale.
The concept of the Luminiferous Aether was accepted on the authority of Newton and inability to explain the propogation of light otherwise.
## Michelson-Morley Experiment (1887)
It wasn't until the Michelson-Morley Experiment that the Luminiferous Aether was seriously shot down. The goal of the Michelson-Morley Experiment was to find evidence of this Luminiferous Aether and the null result was very strong evidence that this Aether did not exist.
The experiment itself was set up to measure the speed of light through this Aether. The idea was that as the Earth moved through the Aether, it would cause a sort of "Aether wind" that would slow down the speed of light. Thus if one measured the speed of light in the direction of the wind and perpendicular to it, one should get different speeds. The Michelson-Morley experiment set up exactly this scenario and used increasingly accurate measurements to try and find this difference in speed. Ultimate no difference was found and the Aether was ruled out as a material which existed.
From that point onwards it was assumed that space was a vacuum, completely devoid of anything. Oddly enough, this assumption was so strong, that people didn't even really believe in the concept of the Solar Wind at first and were quite perplexed initially when they sent the first rockets into space with particle detectors that ended up detecting all sorts of charged particles in space. Regardless, I would say the Michelson-Morley experiment is the first time scientists had scientific evidence that space was a vacuum.
• oh this is a really interesting angle, thanks! – uhoh Mar 8 '19 at 16:35
• but the M-M experiment was before Einstein, and the Lorentz contraction explains the null result -- so the Aether is no longer ruled out by M-M. – amI Jun 16 '19 at 3:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8462316989898682, "perplexity": 656.3227102791838}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989693.19/warc/CC-MAIN-20210512100748-20210512130748-00585.warc.gz"} |
https://infoscience.epfl.ch/record/159850 | ## A Simple Alternative Derivation of the Expectation Correction Algorithm
The switching linear dynamical system (SLDS) is a popular model in time-series analysis. However, the complexity of inferring the state of the latent variables scales exponentially with the length of the time-series, resulting in many approximation strategies in the literature. We focus on the recently devised expectation correction (EC) approximation which can be considered a form of Gaussian sum smoother. The algorithm has excellent numerical performance compared to a wide range of competing techniques, exploiting more fully the available information than, for example, generalised pseudo Bayes. We show that EC can be seen as an extension to the SLDS of the Rauch, Tung, Striebel inference algorithm for the linear dynamical system. This yields a simpler derivation of the EC algorithm and facilitates comparison with existing, similar approaches.
Published in:
Ieee Signal Processing Letters, 16, 121-124
Year:
2009
Keywords:
Laboratories: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9345625638961792, "perplexity": 861.2671769131099}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267161350.69/warc/CC-MAIN-20180925083639-20180925104039-00503.warc.gz"} |
http://crypto.stackexchange.com/questions/11776/can-we-build-authenticated-encryption-using-feistel-networks | Can we build authenticated encryption using Feistel Networks?
Most of the encryption modes of Feistel Networks especially the ones used to build fixed length block ciphers just provide confidentiality .
Can we build authenticated encryption using Feistel Networks ?
-
One useful rule of thumb in cryptography is that even the meanest primitive can be used to build virtually everything. For example, you can take a PRG and build a PRF and PRP from that, and so on: pretty much, once you have a one-way function of some kind, you can build almost everything in symmetric cryptography. This is a pretty neat result, actually, since it boils down the whole of secure symmetric cryptography to the existence of one-way functions. – Reid Nov 18 '13 at 17:30
Yes, you certainly can. If you want a variable-length authenticated encryption mode, then simply take any Feistel cipher in the OCB mode. If fixed-length is fine, then the following idea should work. Build a wide Feistel-based permutation (fixed-key blockcipher) $G$ and encrypt $$C = G(P||N||K)\oplus K,$$ where $N$ is nonce, $P$ is plaintext, $C$ is ciphertext, $K$ is key. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8907279968261719, "perplexity": 1670.7047402975784}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394678690318/warc/CC-MAIN-20140313024450-00072-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://quant.stackexchange.com/users/3554/user3554?tab=reputation&sort=post | # user3554
Unregistered less info
reputation
2
bio website location age member for 1 year, 6 months seen Jan 6 '13 at 13:08 profile views 0
# 36 Reputation
5 May 24
+5 14:25 upvote What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?
5 Feb 25
+5 21:56 upvote What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?
5 Mar 29 '13
+5 13:47 upvote What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?
2 Jan 24 '13
5 Jan 9 '13
5 Jan 7 '13
8 Jan 6 '13 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.863717257976532, "perplexity": 4738.911518709967}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997877881.80/warc/CC-MAIN-20140722025757-00231-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://annals.math.princeton.edu/2009/169-2/p04 | # Curvature of vector bundles associated to holomorphic fibrations
### Abstract
Let $L$ be a (semi)-positive line bundle over a Kähler manifold, $X$, fibered over a complex manifold $Y$. Assuming the fibers are compact and nonsingular we prove that the hermitian vector bundle $E$ over $Y$ whose fibers over points $y$ are the spaces of global sections over $X_y$ to $L\otimes K_{X/Y}$, endowed with the $L^2$-metric, is (semi)-positive in the sense of Nakano. We also discuss various applications, among them a partial result on a conjecture of Griffiths on the positivity of ample bundles.
## Authors
Bo Berndtsson
Department of Mathematical Sciences
Chalmers University of Technology and the University of Göteborg
412 96 Göteborg
Sweden | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9633464217185974, "perplexity": 432.9468568342493}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218193288.61/warc/CC-MAIN-20170322212953-00036-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://perimeterinstitute.ca/seminar/polynomial-time-classical-simulation-quantum-ferromagnets | # Polynomial-time classical simulation of quantum ferromagnets
We consider a broad family of quantum spin systems which includes as special cases the ferromagnetic XY model and ferromagnetic Ising model on any graph, with or without a transverse magnetic field. We prove that the partition function (and consequently the free energy or ground energy) of any model in this family can be efficiently approximated using a classical randomized algorithm. We first show how to approximate the partition function by the perfect matching sum of a finite graph with positive edge weights. Although the perfect matching sum is not known to be efficiently approximable in general, the graphs obtained by our method have a special structure which facilitates efficient approximation via a randomized algorithm due to Jerrum and Sinclair. This is joint work with Sergey Bravyi (arXiv:1612.05602).
Event Type:
Seminar
Scientific Area(s):
Speaker(s):
Event Date:
Wednesday, April 19, 2017 - 16:00 to 17:30
Location:
Time Room | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8653405904769897, "perplexity": 356.41974414604385}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218191396.90/warc/CC-MAIN-20170322212951-00185-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://nrich.maths.org/4932/solution | ### Pattern Recognition
When does a pattern start to exhibit structure? Can you crack the code used by the computer?
### Statistics - Maths of Real Life
This pilot collection of resources is designed to introduce key statistical ideas and help students to deepen their understanding.
### Binomial Conditions
When is an experiment described by the binomial distribution? Why do we need both the condition about independence and the one about constant probability?
# Over-booking
##### Age 16 to 18 Challenge Level:
Andrei fromTudor Vianu National College, Romania, gives a very clear account of the use of the binomial and normal distributions to solve this problem.
The passengers who have bought tickets either turn up for the flight or do not turn up. Taking $X$ as the random variable for the number of passengers who turn up for the flight, then $X$ is binomially distributed with parameters $p$, the probability of arriving for the flight, and $n$ as the number of tickets sold. The probability distribution is: $$P(x;n,p)= {n \choose x}p^x(1-p)^{n-x},\ x=0,1,...,n.$$ The mean of the distribution is $E(x)=np$ and the variance $\sigma ^2 ={np(1-p)}.$
In this problem $n=400$ and $p=0.95$.
So, $E(X)=380$ and $\sigma = 4.36$ and the expected number of empty seats is $20$.
It is known that, if the value of $n$ is large, the variable $X$ could be considered to have a probability distribution that approximates to the standard normal distribution, with the same mean and variance. \par To verify that the normal distribution could be, in the conditions of the problem, a good approximation for the binomial distribution, I have to verify that both the mean $\mu =np$ and the variance $\sigma^2 = n(1-p)$ are greater than 5. Here $np=380$ and $n(1- p)=20$. So, the use of the normal distribution is acceptable.
Using the applet at http://davidmlane.com/hyperstat/z_table.html , I tried to find the number of tickets, $x$, that the airline should sell to satisfy the conditions of the problem.
Let $x$ be the number of tickets sold, which, as explained before, could be considered to have a normal distribution $N(\mu,\sigma^2)$. The mean of the distribution is $x\times 0.95$, and the standard deviation is $\sqrt{x\times 0.95\times 0.05}$. The area under the curve and below $400$ is $98$ per cent or $0.98$ and the area above $400$ is $2$ per cent or $0.02$ (the probability that too many passengers will turn up for the flight).
Trying for some values of $x$ I obtained the number of tickets that the airline must sell. Put $$y = {x-np\over \sqrt{np(1-p)}};$$ then $y$ has distribution $N(0,1)$. The probability that all passengers who arrive for the flight can actually get a seat is ${\rm Prob}\{x \leq 400.5\}$ (because $x=400$ is fine, but $x=401$ is not). Thus $${\rm Prob}\{x \leq 400.5\} = {\rm Prob}\left\{y\leq {400.5-np\over \sqrt{np(1-p)}}\right\}$$ and this can now be found from tables of the normal distribution.
We find that if $411$ tickets are sold then the probability of too many passengers arriving is less than $2$ percent but for $412$ it is more than $2$ percent so the ideal number of tickets to be sold is $411$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9348940849304199, "perplexity": 163.18741243710167}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578534596.13/warc/CC-MAIN-20190422035654-20190422061654-00178.warc.gz"} |
https://converths.com/how-many-feet-is-6-meters-calculator-for-units/ | Is this the same way as how you convert between units – 6 meters to feet? How many feet is 6 m?
Online Numerator
## How much is 6 meters in feet?
We usually use different units for length in different countries. There are several internationally agreed systems of measurements. For example, the metric system, Imperial units (also known as British Imperial), and the Chinese system of weights and measures. Each and every system of unit and conversion is common in various countries and regions.
## 6 meters equal how many feet?
But, how much is 6 m in feet? As we know, based on the basic formula that there are 3.28084 feet in 1 meter. We can multiply them when we want to convert meters to feet, that is 1 meter multiply by 3.28084 feet.
Also check out the video below for details about the conversion, and you can always round up decimals to their nearest whole number, for instance, 3.28084 feet can be rounded up to 3.29 or 3 feet. Anyway, 6 ft how many meters?
So,
. 1 meter = 3.28084 feet
. Or 1 m = 3.28084 ft
6 meters = 6 x 1 m = 6 x 3.28084 ft = 19.68504 feet
6 meters = 19.68504 feet
(PS: m = meter (plural: meters), ft = foot (plural: feet))
More example:
. 1 meter = 3.28084 feet
. Or 1 m = 3.28084 ft
20 meters = 20 x 1 m = 20 x 3.28084 ft = 65.6168 feet
20 meters = 65.6168 feet
## How Many Feet is 6 Meters – Video
Got a different answer? Which unit system do you use or prefer?
Leave your comment below, share with a friend and never stop wondering.❤️ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8553348183631897, "perplexity": 1513.6280045165943}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337680.35/warc/CC-MAIN-20221005234659-20221006024659-00682.warc.gz"} |
http://math.stackexchange.com/questions/209673/if-a-poissona-then-what-is-joint-probability-mass-function-of-male-and-fema?answertab=oldest | If A ~ Poisson(a), then what is joint probability mass function of male and female fish?
Suppose that the number of fish in a big sea $A$ adheres to a Poisson distribution. What is joint probability mass function of having $x$ male and $y$ female fish?
I thought that I could just divide A by $\frac{1}{2}$ and add the two together... but that didn't make very much sense on second thought...:
$$M \rightarrow \text{ the number of male fish.} \\ F \rightarrow \text{ the number of female fish.} \\ P(M = x, Y = y) = 2\frac{e^{-a}a^{x+y}}{2(x+y!)}$$
-
Note that the probability of having $x+y$ fish (in presumably a small defined area) is $$e^{-\lambda}\frac{\lambda^{x+y}}{(x+y)!},\tag{1}$$ where $\lambda$ is the parameter you called $a$.
Given that there are $x+y$ fish in the area, the probability that $x$ are male and $y$ are female is equal to $$\binom{x+y}{x}\left(\frac{1}{2}\right)^{x+y}.\tag{2}$$ (The "fact" that the conditional distribution is binomial is based on a not necessarily reasonable independence assumption.)
Multiply $(1)$ and $(2)$. The expression can be simplified somewhat.
It is easy to modify the second expression if instead of equidistribution we assume that a fish is male with probability $p$, and female with probability $q=1-p$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8274757862091064, "perplexity": 138.22844545207212}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802778013.38/warc/CC-MAIN-20141217075258-00121-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/what-does-it-mean-to-not-have-a-pivot-in-every-row.578691/ | # What does it mean to not have a pivot in every row
1. Feb 17, 2012
### Instinctlol
I am trying to fully understand this theorem
Theorem: Let A be an m x n matrix. The following are all true or all false.
1. For each b in Rm, the equation Ax has a solution
2. Each b in Rm is a linear combination of the columns of A.
3. The columns of A span Rm
4. A has a pivot position in every row.
So when A does not have a pivot in every row, it disproves (1) because each b will not have a solution.
How would you disprove (2) with (4)?
2. Feb 17, 2012
### HallsofIvy
Staff Emeritus
If A does not have a pivot in every row, then its determinant is 0 and A is not invertible.
When you say "disprove (2) with (4)" do you mean disprove (2) assuming (4) is NOT true?
If A does not have a pivot in every row, then A maps Rn[/itex] into a propersubspace of Rm so there exist b in Rm not in that subspace. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8623058795928955, "perplexity": 530.3800114166014}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461862047707.47/warc/CC-MAIN-20160428164727-00102-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://brilliant.org/problems/integral-limit-2/ | # Integral limit
Calculus Level 3
$\large \displaystyle \lim_{x \rightarrow 0} \dfrac{ ( 1 - \cos x)(\cos x - e^x)}{x^{\color{blue}n}}$
Find the integer $$\color{blue}n$$ for which above limit is a finite non-zero number.
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9976937770843506, "perplexity": 3599.8899470186357}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948551162.54/warc/CC-MAIN-20171214222204-20171215002204-00775.warc.gz"} |
http://math.stackexchange.com/questions/27363/find-the-matrix-of-a-linear-transformation-relative-to-a-basis | # Find the Matrix of a Linear Transformation Relative to a Basis
Our book gives this problem:
Find the $\mathcal{B}$-matrix for the transformation $\vec{x} \rightarrow A\vec{x}$ when the basis $\mathcal{B} = \{ \vec{b}_1, \vec{b}_2 \}$, where $A = \left[\begin{array}{cc} 3 & 4 \\ -1 & -1 \\ \end{array} \right]$, $\vec{b}_1 = \left[\begin{array}{cc} 2 \\ -1 \\ \end{array} \right]$, and $\vec{b}_2 = \left[\begin{array}{cc} 1 \\ 2 \\ \end{array} \right]$.
From what I understand, it's asking us to find the matrix for the same exact transformation as $A$, except relative to to the given bases. I can't figure out where to go from here, though... any thoughts?
-
What you need to do is form the matrix $B = (\vec{b}_1|\vec{b}_2)$, where $\vec{b}_i$ is the $i$th column of B, and note that this matrix converts vectors from the standard basis into the basis $\mathcal{B}$, while the inverse $B^{-1}$ will convert vectors in the basis $\mathcal{B}$ into the standard basis. Thus if you have a vector already in the basis $\mathcal{B}$, you can convert it to standard basis by multiplying by $B^{-1}$, multiply it by $A$, and finally convert back to $\mathcal{B}$ by multiplying by $B$, so your overall matrix is $BAB^{-1}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9896405339241028, "perplexity": 83.1076787588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928754.15/warc/CC-MAIN-20150521113208-00201-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://baylor-ir.tdl.org/baylor-ir/handle/2104/7979?show=full | dc.contributor.author Robinson, Matthew dc.contributor.author Ali, Tibra dc.contributor.author Cleaver, Gerald dc.date.accessioned 2010-07-09T15:12:17Z dc.date.available 2010-07-09T15:12:17Z dc.date.issued 2010-07-09T15:12:17Z dc.identifier.uri http://hdl.handle.net/2104/7979 dc.description.abstract This is the second in a series of papers intended to provide a basic overview of some of the major ideas in particle physics. Part I [40] was primarily an algebraic exposition of gauge theories. We developed the group theoretic tools needed to understand the basic construction of gauge theory, as well as the physical concepts and tools to understand the structure of the Standard Model of Particle Physics as a gauge theory. en In this paper (and the paper to follow), we continue our emphasis on gauge theories, but we do so with a more geometrical approach. We will conclude this paper with a brief discussion of general relativity, and save more advanced topics (including fibre bundles, characteristic classes, etc.) for the next paper in the series. We wish to reiterate that these notes are not intended to be a comprehensive introduc- tion to any of the ideas contained in them. Their purpose is to introduce the “forest" rather than the “trees". The primary emphasis is on the algebraic/geometric/mathematical un- derpinnings rather than the calculational/phenomenological details. The topics were chosen according to the authors’ preferences and agenda. These notes are intended for a student who has completed the standard undergradu- ate physics and mathematics courses, as well as the material contained in the first paper in this series. Having studied the material in the “Further Reading" sections of [40] would be ideal, but the material in this series of papers is intended to be self-contained, and familiarity with the first paper will suffice. dc.format.extent 4127862 bytes dc.format.mimetype application/pdf dc.language.iso en en dc.title A Simple Introduction to Particle Physics: Part II Geometric Foundations and Relativity en dc.type Article en dc.description.keywords Elementary Particle Physics en dc.description.keywords Geometric Foundatioms en dc.description.keywords Relativity en
| {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8506687879562378, "perplexity": 881.4970200950811}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541361.65/warc/CC-MAIN-20161202170901-00097-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/1197056/evaluate-the-following-limit-without-lhopital | # Evaluate the following limit without L'Hopital
I tried to evaluate the following limits but I just couldn't succeed, basically I can't use L'Hopital to solve this...
for the second limit I tried to transform it into $e^{\frac{2n\sqrt{n+3}ln(\frac{3n-1}{2n+3})}{(n+4)\sqrt{n+1}}}$ but still with no success...
$$\lim_{n \to \infty } \frac{2n^2-3}{-n^2+7}\frac{3^n-2^{n-1}}{3^{n+2}+2^n}$$
$$\lim_{n \to \infty } \frac{3n-1}{2n+3}^{\frac{2n\sqrt{n+3}}{(n+4)\sqrt{n+1}}}$$
Any suggestions/help? :)
Thanks
For the first limit, it breaks into 2 factors with finite limits.
$$\lim{n \to \infty} \frac{2n^2-3}{7-n^2} = \frac{2n^2}{-n^2} =-2\\ \lim{n \to \infty} \frac{3^n-2^{n-1}}{3^{n+2}+2^n2} = \frac{3^n}{3^{n+2}} = \frac{1}{9}$$ so the answer is $-\frac{2}{9}$.
For the second, rewrite it as $$\left(\frac{(3n-1)(2n-3)}{4n^2-9} \right) ^{\frac{\sqrt{n}2n(1+\frac{3}{2n}+\ldots)}{\sqrt{n}(n+4)(1+\frac{1}{2n}+\ldots)}}$$ and expand to next-lowest order in $1/n$ to get $$\left( \frac{3}{2} \left[ 1-\frac{11}{6n}+\ldots\right] \right)^{2(1+\frac{3}{2n}+\ldots-\frac{9}{2n}+\ldots)}$$ Since the exponent does not go to infinity we can in fact just use the lowest order terms, getting $$\left( \frac{3}{2} \right)^2 = \frac{9}{4}$$
• Really bad form to use $=$ that way. In particular, there is no $n$ variable in the left hand side. – Thomas Andrews Mar 19 '15 at 15:26
Hints : $$\frac{2n^2-3}{-n^2+7} = \frac{2 - \frac{3}{n^2}}{-1+\frac{7}{n^2}},$$ and $$\frac{3^n-2^{n-1}}{3^{n+2}+2^n} = \frac{1-\frac{1}{2}\left( \frac{2}{3} \right)^n}{3^2+\left( \frac{2}{3} \right)^n}.$$
For the second one pay attention to the order of the numerator and denominator: the largest terms converge to some constant, the rest to 0, so you should get $(\frac{3}{2})^2$.
• I think that it could be $(\frac{3}{2})^2$. If I am right, you have a typo. What do you think ? – Claude Leibovici Mar 19 '15 at 15:05
• OK fixed, thanks. – Alex Mar 19 '15 at 15:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9702168107032776, "perplexity": 282.07120548510943}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370519111.47/warc/CC-MAIN-20200404011558-20200404041558-00401.warc.gz"} |
http://www.newton.ac.uk/programmes/RMA/Abstract4/reznikov.html | # Subconvexity of L-functions and the uniqueness principle.
Author: Andre Reznikov (Bar-IIan University)
### Abstract
We consider the triple L-function $L(1/2,f\times g\times \phi_i)$ for fixed Maass forms f and g as the eigenvalue of $\phi_i$ goes to infinity.
We deduce a subconvexity bound for this L-function from the uniqueness principle in representation theory and from simple geometric properties of the corresponding invariant functional. Joint with J. Bernstein. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9426423907279968, "perplexity": 1467.254354611432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368702454815/warc/CC-MAIN-20130516110734-00059-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://math.stackexchange.com/help/badges/31?page=66 | # Help Center > Badges > Commentator
Leave 10 comments.
Awarded 19637 times
Awarded jan 23 at 20:07 to
Awarded jan 23 at 19:47 to
Awarded jan 23 at 19:47 to
Awarded jan 23 at 19:17 to
Awarded jan 23 at 18:33 to
Awarded jan 23 at 16:51 to
Awarded jan 23 at 16:23 to
Awarded jan 23 at 16:18 to
Awarded jan 23 at 15:48 to
Awarded jan 23 at 15:16 to
Awarded jan 23 at 15:06 to
Awarded jan 23 at 14:38 to
Awarded jan 23 at 13:12 to
Awarded jan 23 at 12:47 to
Awarded jan 23 at 11:33 to
Awarded jan 23 at 10:19 to
Awarded jan 23 at 9:05 to
Awarded jan 23 at 8:13 to
Awarded jan 23 at 7:17 to
Awarded jan 23 at 6:00 to
Awarded jan 23 at 5:55 to
Awarded jan 23 at 4:37 to
Awarded jan 23 at 4:28 to
Awarded jan 23 at 4:12 to
Awarded jan 22 at 23:11 to
Awarded jan 22 at 22:51 to
Awarded jan 22 at 22:46 to
Awarded jan 22 at 22:17 to
Awarded jan 22 at 21:56 to
Awarded jan 22 at 20:23 to
Awarded jan 22 at 19:10 to
Awarded jan 22 at 18:53 to
Awarded jan 22 at 18:48 to
Awarded jan 22 at 16:51 to
Awarded jan 22 at 16:41 to
Awarded jan 22 at 16:41 to
Awarded jan 22 at 16:36 to
Awarded jan 22 at 15:33 to
Awarded jan 22 at 13:25 to
Awarded jan 22 at 11:27 to
Awarded jan 22 at 10:25 to
Awarded jan 22 at 9:25 to
Awarded jan 22 at 9:25 to
Awarded jan 22 at 9:18 to
Awarded jan 22 at 8:41 to
Awarded jan 22 at 7:44 to
Awarded jan 22 at 5:50 to
Awarded jan 22 at 4:21 to
Awarded jan 22 at 4:21 to
Awarded jan 22 at 3:36 to
Awarded jan 21 at 23:01 to
Awarded jan 21 at 20:04 to
Awarded jan 21 at 19:44 to
Awarded jan 21 at 19:44 to
Awarded jan 21 at 18:57 to
Awarded jan 21 at 18:42 to
Awarded jan 21 at 17:37 to
Awarded jan 21 at 15:12 to
Awarded jan 21 at 15:02 to
Awarded jan 21 at 14:36 to | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8894700407981873, "perplexity": 1822.6790341263657}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644068749.35/warc/CC-MAIN-20150827025428-00302-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/101008/calculating-a-distributional-derivative | # Calculating a distributional derivative
Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to distributions and therefore has a distributional derivative. What is the explicit formula for $DN_j$? Is it related to the classical formula $2\langle u_j , Du_j\rangle$?
-
What is the norm $\|\cdot\|$ here? – Andrew Jun 30 '12 at 17:12
Euclidean norm. – dcs24 Jul 1 '12 at 9:16
First, I do not understand why do you need a sequence of functions when the question involves an individual function. Suppose that $u$ is real valued. Then the product of the distributions $u$ and $u'$ may not even be defined. (This is the case when $u$ is the Heaviside function.) However, if the distributional derivative of $u$ is Lebesgue integrable, then
$$\frac{d}{dt}(\; u^2\;) = 2u u'.$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9694554805755615, "perplexity": 114.50818320238247}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049276131.97/warc/CC-MAIN-20160524002116-00057-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/179762/examples-of-continuous-differential-equations-with-no-solution | # Examples of continuous differential equations with no solution
Several theorems regarding differential equations are true not only in finite dimension but also in infinite dimension Banach spaces. This is in particular the case for the Cauchy–Lipschitz theorem.
On the other end the Peano existence theorem is false for Banach space with infinite dimensions. See here for a counterexample.
Do you know other counterexamples in "classical" Banach spaces that are different from $c_0$ (the space of sequences of reals converging to $0$)? In particular, is there "an easy example" in the space $C([0,1],\mathbb{R})$ with $\sup$ norm?
-
Here is an example in $C([-1,1],R)$, which is a continuous analogue to the discrete example you pointed to: $${du(t,x)\over dt} = \operatorname{sign}(u(t,x))\sqrt{|u(t,x)|} + x\;,\qquad u(0,x) = 0\;.$$ For any $t > 0$, the solution (in $L^\infty$) develops a discontinuity at the origin, so that it doesn't belong to $C([-1,1],R)$.
Hello Martin, are you sure of your statement? Because for $x=0$ the equation has 2 solutions. The first one is the constant function equal to $0$ and the second one, the function equal to $0$ for $t < 0$ and to $\frac{t^2}{4}$ for $t \ge 0$. The solution of the differential equation seems continuous if you pick-up the second solution for $x=0$. – mathcounterexamples.net Aug 31 '14 at 19:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9233753681182861, "perplexity": 159.56633443762888}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398454553.89/warc/CC-MAIN-20151124205414-00007-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.meritnation.com/cbse-class-10/math/rd-sharma-2019/co-ordinate-geometry/textbook-solutions/12_1_3513_24214_6.4_27037 | Rd Sharma 2019 Solutions for Class 10 Math Chapter 1 Real Numbers are provided here with simple step-by-step explanations. These solutions for Real Numbers are extremely popular among Class 10 students for Math Real Numbers Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma 2019 Book of Class 10 Math Chapter 1 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma 2019 Solutions. All Rd Sharma 2019 Solutions for class Class 10 Math are prepared by experts and are 100% accurate.
#### Question 1:
Find the distance between the following pair of points:
(a) (−6, 7) and (−1, −5)
(b) (a+b, b+c) and (ab, cb)
(c) (asinα, −bcosα) and (−acos α, bsin α)
(d) (a, 0) and (0, b)
The distance d between two points and is given by the formula
(i) The two given points are (−6, 7) and (−1, −5)
The distance between these two points is
Hence the distance is.
(ii) The two given points are
The distance between these two points is
Hence the distance is.
(iii) The two given points are and
The distance between these two points is
Hence the distance is.
(iv) The two given points are (a, 0) and (0, b)
The distance between these two points is
Hence the distance is.
#### Question 2:
Find the value of a when the distance between the points (3, a) and (4, 1) is $\sqrt{10}$.
The distance d between two points and is given by the formula
The distance between two points (3, a) and (4, 1) is given as. Substituting these values in the formula for distance between two points we have,
Now, squaring the above equation on both sides of the equals sign
Thus we arrive at a quadratic equation. Let us solve this now,
The roots of the above quadratic equation are thus 4 and −2.
Thus the value of ‘a’ could either be.
#### Question 3:
If the points (2, 1) and (1, −2) are equidistant from the point (x, y), show that x + 3y = 0.
The distance d between two points and is given by the formula
Here it is said that the points (2, 1) and (1, 2) are equidistant from (x, y).
Let be the distance between (2, 1) and (x, y).
Let be the distance between (1, −2) and (x, y).
So, using the distance formula for both these pairs of points we have
Now since both these distances are given to be the same, let us equate both and
Squaring on both sides we have,
Hence we have proved that when the points (2, 1) and (1,−2) are equidistant from (x, y) we have .
#### Question 4:
Find the values of x, y if the distances of the point (x, y) from (−3, 0) as well as from (3, 0) are 4.
The distance d between two points and is given by the formula
It is said that (x, y) is equidistant from both (−3,0) and (3,0).
Let be the distance between (x, y) and (−3,0).
Let be the distance between (x, y) and (3,0).
So, using the distance formula for both these pairs of points we have
Now since both these distances are given to be the same, let us equate both and .
Squaring on both sides we have,
It is also said that the value of both and is 4 units.
Substituting the value of ‘x’ in the equation for either or we can get the value of ‘y’.
Squaring on both sides of the equation we have,
Hence the values of ‘x’ and ‘y’ are.
#### Question 5:
The length of a line segment is of 10 units and the coordinates of one end-point are (2, −3). If the abscissa of the other end is 10, find the ordinate of the other end.
The distance d between two points and is given by the formula
Here it is given that one end of a line segment has co−ordinates (2,−3). The abscissa of the other end of the line segment is given to be 10. Let the ordinate of this point be ‘y’.
So, the co−ordinates of the other end of the line segment is (10, y).
The distance between these two points is given to be 10 units.
Substituting these values in the formula for distance between two points we have,
Squaring on both sides of the equation we have,
We have a quadratic equation for ‘y’. Solving for the roots of this equation we have,
The roots of the above equation are ‘9’ and ‘3’
Thus the ordinates of the other end of the line segment could be.
#### Question 6:
Show that the points (−4, −1), (−2, −4) (4, 0) and (2, 3) are the vertices points of a rectangle.
The distance d between two points and is given by the formula
In a rectangle, the opposite sides are equal in length. The diagonals of a rectangle are also equal in length.
Here the four points are A(4,1), B(2,4), C(4,0) and D(2,3).
First let us check the length of the opposite sides of the quadrilateral that is formed by these points.
We have one pair of opposite sides equal.
Now, let us check the other pair of opposite sides.
The other pair of opposite sides are also equal. So, the quadrilateral formed by these four points is definitely a parallelogram.
For a parallelogram to be a rectangle we need to check if the diagonals are also equal in length.
Now since the diagonals are also equal we can say that the parallelogram is definitely a rectangle.
Hence we have proved that the quadrilateral formed by the four given points is a.
#### Question 7:
Show that the points A (1, −2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.
The distance d between two points and is given by the formula
In a parallelogram the opposite sides are equal in length.
Here the four points are A(1,2), B(3, 6), C(5, 10) and D(3, 2).
Let us check the length of the opposite sides of the quadrilateral that is formed by these points.
We have one pair of opposite sides equal.
Now, let us check the other pair of opposite sides.
The other pair of opposite sides is also equal. So, the quadrilateral formed by these four points is definitely a parallelogram.
Hence we have proved that the quadrilateral formed by the given four points is a .
#### Question 8:
Prove that the points A(1, 7), B (4, 2), C(−1, −1) D (−4, 4) are the vertices of a square.
The distance d between two points and is given by the formula
In a square all the sides are equal in length. Also, the diagonals are equal in length in a square.
Here the four points are A(1, 7), B(4, 2), C(1,1) and D(4, 4).
First let us check if all the four sides are equal.
Since all the sides of the quadrilateral are the same it is a rhombus.
For the rhombus to be a square the diagonals also have to be equal to each other.
Since the diagonals of the rhombus are also equal to each other the rhombus is a square.
Hence the quadrilateral formed by the given points is a.
#### Question 9:
Prove that the points (3, 0), (6, 4) and (−1, 3) are vertices of a right-angled isosceles triangle.
The distance d between two points and is given by the formula
In an isosceles triangle there are two sides which are equal in length.
Here the three points are A(3, 0), B(6, 4) and C(1, 3).
Let us check the length of the three sides of the triangle.
Here, we see that two sides of the triangle are equal. So the triangle formed should be an isosceles triangle.
We can also observe that
Hence proved that the triangle formed by the three given points is an.
#### Question 10:
Prove that (2, −2) (−2, 1) and (5, 2) are the vertices of a right angled triangle. Find the area of the triangle and the length of the hypotenuse.
The distance d between two points and is given by the formula
In a right-angled triangle, by Pythagoras theorem, the square of the longest side is equal to the sum of squares of the other two sides in the triangle.
Here the three points are A(2,2), B(2,1) and C(5,2).
Let us find out the lengths of all the sides of the triangle.
Here we have,
Since the square of the longest side is equal to the sum of squares of the other two sides the given triangle is a .
In a right angled triangle the area of the triangle ‘A’ is given by,
In a right angled triangle the sides containing the right angle will not be the longest side.
Hence the area of the given right angled triangle is,
Hence the area of the given right-angled triangle is.
In a right-angled triangle the hypotenuse will be the longest side. Here the longest side is ‘BC’.
Hence the hypotenuse of the given right-angled triangle is
#### Question 11:
Prove that the points (2a, 4a), (2a, 6a) and (2a+$\sqrt{3}a$, 5a) are the vertices of an equilateral triangle.
The distance d between two points and is given by the formula
In an equilateral triangle all the sides have equal length.
Here the three points are, and.
Let us now find out the lengths of all the three sides of the given triangle.
Since all the three sides have equal lengths the triangle has to be an equilateral triangle.
#### Question 12:
Prove that the points (2,3), (−4, −6) and (1, 3/2) do not form a triangle.
The distance d between two points and is given by the formula
In any triangle the sum of lengths of any two sides need to be greater than the third side.
Here the three points are, and
Let us now find out the lengths of all the three sides of the given triangle.
Here we see that,
This is in violation of the basic property of any triangle to exist. Therefore these points cannot give rise to a triangle.
Hence we have proved that the given three points do not form a triangle.
#### Question 13:
The points A(2, 9), B(a, 5) and C(5, 5) are the vertices of a triangle ABC right angled at B. Find the values of a and hence the area of ΔABC.
Consider the figure.
Using distance formula,
We are given that ABC is a right triangle right angled at B.
By Pythagoras theorem, we have;
We cannot put a = 5 as it will make BC = 0. So, we ignore a = 5 and accept a = 2.
#### Question 14:
Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.
The distance d between two points and is given by the formula
In a rhombus all the sides are equal in length.
Here the four points are A (2, 1), B (34), C (2, 3) and D (3, 2).
First let us check if all the four sides are equal.
Here, we see that all the sides are equal, so it has to be a rhombus.
#### Question 15:
Two vertices of an isosceles triangle are (2, 0) and (2, 5). Find the third vertex if the length of the equal sides is 3.
The distance d between two points and is given by the formula
In an isosceles triangle two sides will be of equal length.
Here two vertices of the triangle is given as A (2, 0) and B (2, 5). Let the third side of the triangle be C(x, y)
It is given that the length of the equal sides is 3 units.
Let us now find the length of the side in which both the vertices are known.
So, now we know that the side ‘AB’ is not one of the equal sides of the isosceles triangle.
So, we have
Equating these two equations we have,
Squaring on both sides of the equation we have,
We know that the length of the equal sides is 3 units. So substituting the value of ‘y’ in equation for either ‘AC’ or ‘BC’ we can get the value of ‘x’.
Squaring on both sides,
We have a quadratic equation for ‘x’. Solving for roots of the above equation we have,
Hence the possible co−ordinates of the third vertex of the isosceles triangle are.
#### Question 16:
Which point on x-axis is equidistant from (5, 9) and (−4, 6) ?
The distance d between two points and is given by the formula
Here we are to find out a point on the x−axis which is equidistant from both the points A (5, 9) and B (4, 6)
Let this point be denoted as C(x, y)
Since the point lies on the x-axis the value of its ordinate will be 0. Or in other words we have.
Now let us find out the distances from ‘A’ and ‘B’ to ‘C
We know that both these distances are the same. So equating both these we get,
Squaring on both sides we have,
Hence the point on the x-axis which lies at equal distances from the mentioned points is.
#### Question 17:
Prove that the points (−2, 5), (0, 1) and (2, −3) are collinear.
The distance d between two points and is given by the formula
For three points to be collinear the sum of distances between two pairs of points should be equal to the third pair of points.
The given points are A (2, 5), B (0, 1) and C (2, 3)
Let us find the distances between the possible pairs of points.
We see that
Since sum of distances between two pairs of points equals the distance between the third pair of points the three points must be collinear.
Hence we have proved that the three given points are.
#### Question 18:
The coordinates of the point P are (−3, 2). Find the coordinates of the point Q which lies on the line joining P and origin such that OP = OQ.
If and are given as two points, then the co-ordinates of the midpoint of the line joining these two points is given as
It is given that the point ‘P’ has co-ordinates (3, 2)
Here we are asked to find out the co-ordinates of point ‘Q’ which lies along the line joining the origin and point ‘P’. Thus we can see that the points ‘P’, ‘Q’ and the origin are collinear.
Let the point ‘Q’ be represented by the point (x, y)
Further it is given that the
This implies that the origin is the midpoint of the line joining the points ‘P’ and ‘Q’.
So we have that
Substituting the values in the earlier mentioned formula we get,
Equating individually we have, and.
Thus the co−ordinates of the point ‘Q’ is
#### Question 19:
Which point on y-axis is equidistant from (2, 3) and (−4, 1)?
The distance d between two points and is given by the formula
Here we are to find out a point on the y-axis which is equidistant from both the points A (2, 3) and B (4, 1).
Let this point be denoted as C(x, y).
Since the point lies on the y-axis the value of its ordinate will be 0. Or in other words we have.
Now let us find out the distances from ‘A’ and ‘B’ to ‘C
We know that both these distances are the same. So equating both these we get,
Squaring on both sides we have,
Hence the point on the y-axis which lies at equal distances from the mentioned points is.
#### Question 20:
The three vertices of a parallelogram are (3, 4) (3, 8) and (9, 8). Find the fourth vertex.
We are given three vertices of a parallelogram A(3, 4), B(3, 8) and C(9, 8).
We know that diagonals of a parallelogram bisect each other. Let the fourth vertex be D(x, y).
Mid point of BD =
Mid point of AC = =
Since the mid point of BD = mid point of AC
So, = (6, 6)
Thus, x = 9, y = 4.
So, the fourth vertex is (9, 4).
#### Question 21:
Find a point which is equidistant from the points How many such points are there ?
Let A(x, y) be the point which is equidistant from the points P(−5, 4) and Q(−1, 6).
Then AP = AQ.
Hence all the points satisfying the equation (i) are equidistant from the points P and Q.
There are infinite such points.
#### Question 22:
The centre of a circle is (. Find the values of a if the circle passes through the point (11,$-$9) and has diameter $10\sqrt{2}$ units.
The length of the diameter is .
The centre of the circle be C(2a, a−7).
Suppose it passes through the point P(11, −9).
Therefore, PC = r
Hence the values of a are 3 or 5.
#### Question 23:
Ayush starts walking from his house to office . Instead of going to the office directly , he goes to a bank first , from there to his daughter 's school and then reaches the office. what is the extra distance travelled by Ayush in reaching the office ? ( Assume that all distances covered are in straight lines) . If the house is situated at (2,4) , bank at (5,8) , school at (13,14) and office at ( 13,26) and coordinates are in kilometers .
The position of the ayush's house is (2, 4) and that of the bank is (5, 8).
The distance between the house and the bank is
The position of the the bank is (5, 8) and that of the school is (13, 14).
The distance between the bank and the school is
The position of the school is (13, 14) and that of the office is (13, 6)
The distance between the bank and the school is
Suppose d be the total distance travelled by ayush
Now, let D be the shortest distance between ayush house and the office,
Thus the extra distance covered by ayush is d − D = 27 − 24.6 = 2.4 km.
#### Question 24:
Find the value of k, if the point P (0, 2) is equidistant from (3, k) and (k, 5).
The distance d between two points and is given by the formula
It is said that P(0,2) is equidistant from both A(3,k) and B(k,5).
So, using the distance formula for both these pairs of points we have
Now since both these distances are given to be the same, let us equate both.
Squaring on both sides we have,
Hence the value of ‘k’ for which the point ‘P’ is equidistant from the other two given points is.
#### Question 25:
If are two vertices of an equilateral triangle , find the coordinates of the third vertex , given that the origin lies in the
(i) interior (ii) exterior of the triangle .
Suppose A(−4, 3) and B(4, 3) be the vertices of an equilateral triangle.
The distance AB is
Consider the figure below:
Here, D is point such that CD is perpendicular to AB.
By pythagoras theorem, we have
$A{D}^{2}+C{D}^{2}=A{C}^{2}\phantom{\rule{0ex}{0ex}}⇒{4}^{2}+C{D}^{2}={8}^{2}\phantom{\rule{0ex}{0ex}}⇒C{D}^{2}=64-16\phantom{\rule{0ex}{0ex}}⇒C{D}^{2}=48\phantom{\rule{0ex}{0ex}}⇒CD=4\sqrt{3}$
Since C is on the negative y axis, so coordinate of C are $3-4\sqrt{3}$ units.
If C is on the positive y axis, so coordinate of C are $3+4\sqrt{3}$ units.
#### Question 26:
Show that the points (−3, 2), (−5,−5), (2, −3) and (4, 4) are the vertices of a rhombus. Find the area of this rhombus.
The distance d between two points and is given by the formula
In a rhombus all the sides are equal in length. And the area ‘A’ of a rhombus is given as
Here the four points are A(3,2), B(5,5), C(2,3) and D(4,4)
First let us check if all the four sides are equal.
Here, we see that all the sides are equal, so it has to be a rhombus.
Hence we have proved that the quadrilateral formed by the given four vertices is a.
Now let us find out the lengths of the diagonals of the rhombus.
Now using these values in the formula for the area of a rhombus we have,
Thus the area of the given rhombus is.
#### Question 27:
Find the coordinates of the circumcentre of the triangle whose vertices are (3, 0), (−1, −6) and (4, −1). Also, find its circumradius.
The distance d between two points and is given by the formula
The circumcentre of a triangle is the point which is equidistant from each of the three vertices of the triangle.
Here the three vertices of the triangle are given to be A(3,0), B(1,6) and C(4,1)
Let the circumcentre of the triangle be represented by the point R(x, y).
So we have
Equating the first pair of these equations we have,
Squaring on both sides of the equation we have,
Equating another pair of the equations we have,
Squaring on both sides of the equation we have,
Now we have two equations for ‘x’ and ‘y’, which are
From the second equation we have. Substituting this value of ‘y’ in the first equation we have,
Therefore the value of ‘y’ is,
Hence the co-ordinates of the circumcentre of the triangle with the given vertices are.
The length of the circumradius can be found out substituting the values of ‘x’ and ‘y’ in ‘AR
Thus the circumradius of the given triangle is.
#### Question 28:
Find a point on the x-axis which is equidistant from the points (7, 6) and (−3, 4).
The distance d between two points and is given by the formula
Here we are to find out a point on the x−axis which is equidistant from both the points A(7,6) and B(3,4).
Let this point be denoted as C(x, y).
Since the point lies on the x-axis the value of its ordinate will be 0. Or in other words we have.
Now let us find out the distances from ‘A’ and ‘B’ to ‘C
We know that both these distances are the same. So equating both these we get,
Squaring on both sides we have,
Hence the point on the x-axis which lies at equal distances from the mentioned points is.
#### Question 29:
(i) Show that the points A(5, 6), B(1, 5), C(2, 1) and D(6,2) are the vertices of a square.
(ii) Prove hat the points A (2, 3) B(−2,2) C(−1,−2), and D(3, −1) are the vertices of a square ABCD.
The distance d between two points and is given by the formula
In a square all the sides are equal to each other. And also the diagonals are also equal to each other.
Here the four points are A(5,6), B(1,5), C(2,1) and D(6,2).
First let us check if all the four sides are equal.
Here, we see that all the sides are equal, so it has to be a rhombus.
Now let us find out the lengths of the diagonals of this rhombus.
Now since the diagonals of the rhombus are also equal to each other this rhombus has to be a square.
Hence we have proved that the quadrilateral formed by the given four points is a.
#### Question 30:
Find the point on x-axis which is equidistant from the points (−2, 5) and (2,−3).
The distance d between two points and is given by the formula
Here we are to find out a point on the x-axis which is equidistant from both the points A(2,5) and B(2,3)
Let this point be denoted as C(x, y).
Since the point lies on the x-axis the value of its ordinate will be 0. Or in other words we have.
Now let us find out the distances from ‘A’ and ‘B’ to ‘C
We know that both these distances are the same. So equating both these we get,
Squaring on both sides we have,
Hence the point on the x-axis which lies at equal distances from the mentioned points is.
#### Question 31:
Find the value of x such that PQ = QR where the coordinates of P, Q and R are (6, −1) , (1, 3) and (x, 8) respectively.
The distance d between two points and is given by the formula
The three given points are P(6,1), Q(1,3) and R(x,8).
Now let us find the distance between ‘P’ and ‘Q’.
Now, let us find the distance between ‘Q’ and ‘R’.
It is given that both these distances are equal. So, let us equate both the above equations,
Squaring on both sides of the equation we get,
Now we have a quadratic equation. Solving for the roots of the equation we have,
Thus the roots of the above equation are 5 and −3.
Hence the values of ‘x’ are.
#### Question 32:
Prove that the points (0, 0), (5, 5) and (−5, 5) are the vertices of a right isosceles triangle.
The distance d between two points and is given by the formula
In an isosceles triangle there are two sides which are equal in length.
By Pythagoras Theorem in a right-angled triangle the square of the longest side will be equal to the sum of squares of the other two sides.
Here the three points are A(0,0), B(5,5) and C(5,5).
Let us check the length of the three sides of the triangle.
Here, we see that two sides of the triangle are equal. So the triangle formed should be an isosceles triangle.
Further it is seen that
If in a triangle the square of the longest side is equal to the sum of squares of the other two sides then the triangle has to be a right-angled triangle.
Hence proved that the triangle formed by the three given points is an.
#### Question 33:
If the point P(x, y) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.
The distance d between two points and is given by the formula
The three given points are P(x, y), A(5,1) and B(1,5).
Now let us find the distance between ‘P’ and ‘A’.
Now, let us find the distance between ‘P’ and ‘B’.
$PB=\sqrt{{\left(x-1\right)}^{2}+{\left(y-5\right)}^{2}}\phantom{\rule{0ex}{0ex}}$
It is given that both these distances are equal. So, let us equate both the above equations,
PA = PB
$\sqrt{{\left(x-5\right)}^{2}+{\left(y-1\right)}^{2}}=\sqrt{{\left(x-1\right)}^{2}+{\left(y-5\right)}^{2}}\phantom{\rule{0ex}{0ex}}$
Squaring on both sides of the equation we get,
${\left(x-5\right)}^{2}+{\left(y-1\right)}^{2}={\left(x-1\right)}^{2}+{\left(y-5\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒{x}^{2}+25-10x+{y}^{2}+1-2y={x}^{2}+1-2x+{y}^{2}+25-10y\phantom{\rule{0ex}{0ex}}⇒26-10x-2y=26-10y-2x\phantom{\rule{0ex}{0ex}}⇒10y-2y=10x-2x\phantom{\rule{0ex}{0ex}}⇒8y=8x\phantom{\rule{0ex}{0ex}}⇒y=x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$
Hence we have proved that x y.
#### Question 34:
If Q (0,1) is equidistant from P (5, −3) and R (x, 6), find the values of x. Also, find the distances QR and PR.
The distance d between two points and is given by the formula
The three given points are Q(0,1), P(5,3) and R(x,6).
Now let us find the distance between ‘P’ and ‘Q’.
Now, let us find the distance between ‘Q’ and ‘R’.
It is given that both these distances are equal. So, let us equate both the above equations,
Squaring on both sides of the equation we get,
Hence the values of ‘x’ are.
Now, the required individual distances,
Hence the length of ‘QR’ is.
For ‘PR’ there are two cases. First when the value of ‘x’ is 4,
Then when the value of ‘x’ is −4,
Hence the length of ‘PR’ can beunits
#### Question 35:
Find the values of y for which the distance between the points P (2, −3) and Q(10,y) is 10 units.
The distance d between two points and is given by the formula
The distance between two points P(2,3) and Q(10,y) is given as 10 units. Substituting these values in the formula for distance between two points we have,
Now, squaring the above equation on both sides of the equals sign
Thus we arrive at a quadratic equation. Let us solve this now,
The roots of the above quadratic equation are thus 3 and −9.
Thus the value of ‘y’ could either be.
#### Question 36:
If the point P(k − 1, 2) is equidistant from the points A(3, k) and B(k, 5), find the values of k. [CBSE 2014]
It is given that P(k − 1, 2) is equidistant from the points A(3, k) and B(k, 5).
∴ AP = BP
Squaring on both sides, we get
${k}^{2}-8k+16+4-4k+{k}^{2}=10\phantom{\rule{0ex}{0ex}}⇒2{k}^{2}-12k+10=0\phantom{\rule{0ex}{0ex}}⇒{k}^{2}-6k+5=0\phantom{\rule{0ex}{0ex}}⇒\left(k-1\right)\left(k-5\right)=0$
Thus, the value of k is 1 or 5.
#### Question 37:
If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also, find the length of AB. [CBSE 2014]
It is given that A(0, 2) is equidistant from the points B(3, p) and C(p, 5).
∴ AB = AC
$⇒\sqrt{{\left(3-0\right)}^{2}+{\left(p-2\right)}^{2}}=\sqrt{{\left(p-0\right)}^{2}+{\left(5-2\right)}^{2}}$ (Distance formula)
Squaring on both sides, we get
$9+{p}^{2}-4p+4={p}^{2}+9\phantom{\rule{0ex}{0ex}}⇒-4p+4=0\phantom{\rule{0ex}{0ex}}⇒p=1$
Thus, the value of p is 1.
∴ AB = $\sqrt{{\left(3-0\right)}^{2}+{\left(1-2\right)}^{2}}=\sqrt{{3}^{2}+{\left(-1\right)}^{2}}=\sqrt{9+1}=\sqrt{10}$ units
#### Question 38:
(i) A(−1,−2) B(1, 0), C (−1, 2), D(−3, 0)
(ii) A(−3, 5) B(3, 1), C (0, 3), D(−1, −4)
(iii) A(4, 5) B(7, 6), C (4, 3), D(1, 2)
(i) A (−1,−2) , B(1,0), C(−1,2), D(−3,0)
Let A, B, C and D be the four vertices of the quadrilateral ABCD.
We know the distance between two points Pand Qis given by distance formula:
Hence
Similarly,
Similarly,
Also,
Hence from above we see that all the sides of the quadrilateral are equal. Hence it is a square.
(ii) A (−3,5) , B(3,1), C(0,3), D(−1,−4)
Let A, B, C and D be the four vertices of the quadrilateral ABCD.
We know the distance between two points Pand Qis given by distance formula:
Hence
Similarly,
Similarly,
Also,
Hence from the above we see that it is not a quadrilateral
(iii) A (4, 5), B (7,6), C(4,3), D(1,2)
Let A, B, C and D be the four vertices of the quadrilateral ABCD.
We know the distance between two points Pand Qis given by distance formula:
Hence
Similarly,
Similarly,
Also,
Hence from above we see that
AB = CD and BC = DA
Here opposite sides of the quadrilateral is equal. Hence it is a parallelogram.
#### Question 39:
Find the equation of the perpendicular bisector of the line segment joining points (7, 1) and (3,5).
TO FIND: The equation of perpendicular bisector of line segment joining points (7, 1) and (3, 5)
Let P(x, y) be any point on the perpendicular bisector of AB. Then,
PA=PB
Hence the equation of perpendicular bisector of line segment joining points (7, 1) and (3, 5) is
#### Question 40:
Prove that the points (3, 0), (4, 5), (−1, 4) and (−2 −1), taken in order, form a rhombus. Also, find its area.
The distance d between two points and is given by the formula
In a rhombus all the sides are equal in length. And the area ‘A’ of a rhombus is given as
Here the four points are A(3,0), B(4,5), C(1,4) and D(2,1).
First let us check if all the four sides are equal.
Here, we see that all the sides are equal, so it has to be a rhombus.
Hence we have proved that the quadrilateral formed by the given four vertices is a.
Now let us find out the lengths of the diagonals of the rhombus.
Now using these values in the formula for the area of a rhombus we have,
Thus the area of the given rhombus is.
#### Question 41:
In the seating arrangement of desks in a classroom three students Rohini, Sandhya and Bina are seated at A(3, 1), B(6, 4), and C(8, 6). Do you think they are seated in a line?
The distance d between two points and is given by the formula
For three points to be collinear the sum of distances between any two pairs of points should be equal to the third pair of points.
The given points are A(3,1), B(6,4) and C(8,6).
Let us find the distances between the possible pairs of points.
We see that.
Since sum of distances between two pairs of points equals the distance between the third pair of points the three points must be collinear.
Hence, the three given points are.
#### Question 42:
Find a point on y-axis which is equidistant form the points (5, −2) and (−3, 2).
The distance d between two points and is given by the formula
Here we are to find out a point on the y−axis which is equidistant from both the points A(5,2) and B(3,2).
Let this point be denoted as C(x, y).
Since the point lies on the y-axis the value of its ordinate will be 0. Or in other words we have.
Now let us find out the distances from ‘A’ and ‘B’ to ‘C
We know that both these distances are the same. So equating both these we get,
Squaring on both sides we have,
Hence the point on the y-axis which lies at equal distances from the mentioned points is.
#### Question 43:
Find a relation between x and y such that the point (x, y) is equidistant from the points (3, 6) and (−3, 4).
The distance d between two points and is given by the formula
Let the three given points be P(x, y), A(3,6) and B(3,4).
Now let us find the distance between ‘P’ and ‘A’.
Now, let us find the distance between ‘P’ and ‘B’.
It is given that both these distances are equal. So, let us equate both the above equations,
Squaring on both sides of the equation we get,
Hence the relationship between ‘x’ and ‘y’ based on the given condition is.
#### Question 44:
If a point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), then find the value of p. [CBSE 2012, 2013]
It is given that A(0, 2) is equidistant from the points B(3, p) and C(p, 5).
∴ AB = AC
$⇒\sqrt{{\left(3-0\right)}^{2}+{\left(p-2\right)}^{2}}=\sqrt{{\left(p-0\right)}^{2}+{\left(5-2\right)}^{2}}$ (Distance formula)
Squaring on both sides, we get
$9+{p}^{2}-4p+4={p}^{2}+9\phantom{\rule{0ex}{0ex}}⇒-4p+4=0\phantom{\rule{0ex}{0ex}}⇒p=1$
Thus, the value of p is 1.
#### Question 45:
Prove that the points (7, 10), (−2, 5) and (3, −4) are the vertices of an isosceles right triangle. [CBSE 2013]
Let the given points be A(7, 10), B(−2, 5) and C(3, −4).
Using distance formula, we have
$\mathrm{AB}=\sqrt{{\left(-2-7\right)}^{2}+{\left(5-10\right)}^{2}}=\sqrt{{\left(-9\right)}^{2}+{\left(-5\right)}^{2}}=\sqrt{81+25}=\sqrt{106}$ units
$\mathrm{BC}=\sqrt{{\left[3-\left(-2\right)\right]}^{2}+{\left(-4-5\right)}^{2}}=\sqrt{{5}^{2}+{\left(-9\right)}^{2}}=\sqrt{25+81}=\sqrt{106}$ units
$\mathrm{CA}=\sqrt{{\left(3-7\right)}^{2}+{\left(-4-10\right)}^{2}}=\sqrt{{\left(-4\right)}^{2}+{\left(-14\right)}^{2}}=\sqrt{16+196}=\sqrt{212}$ units
Thus, AB = BC = $\sqrt{106}$ units
∴ ∆ABC is an isosceles triangle.
Also,
AB2 + BC2 = 106 + 106 = 212
and CA2 = 212
∴ AB2 + BC2 = CA2
So, ∆ABC is right angled at B. (Converse of Pythagoras theorem)
Hence, the given points are the vertices of an isosceles right triangle.
#### Question 46:
If the point P(x, 3) is equidistant from the point A(7, −1) and B(6, 8), then find the value of x and find the distance AP. [CBSE 2014]
It is given that P(x, 3) is equidistant from the point A(7, −1) and B(6, 8).
∴ AP = BP
$⇒\sqrt{{\left(x-7\right)}^{2}+{\left[3-\left(-1\right)\right]}^{2}}=\sqrt{{\left(x-6\right)}^{2}+{\left(8-3\right)}^{2}}$ (Distance formula)
Squaring on both sides, we get
${\left(x-7\right)}^{2}+16={\left(x-6\right)}^{2}+25\phantom{\rule{0ex}{0ex}}⇒{x}^{2}-14x+49+16={x}^{2}-12x+36+25\phantom{\rule{0ex}{0ex}}⇒-14x+12x=61-65\phantom{\rule{0ex}{0ex}}⇒-2x=-4\phantom{\rule{0ex}{0ex}}⇒x=2$
Thus, the value of x is 2.
$\therefore \mathrm{AP}=\sqrt{{\left(2-7\right)}^{2}+{\left[3-\left(-1\right)\right]}^{2}}=\sqrt{{\left(-5\right)}^{2}+{4}^{2}}=\sqrt{25+16}=\sqrt{41}$ units
#### Question 47:
If A(3, y) is equidistant from points P(8, −3) and Q(7, 6), find the value of y and find the distance AQ. [CBSE 2014]
It is given that A(3, y) is equidistant from points P(8, −3) and Q(7, 6).
∴ AP = AQ
$⇒\sqrt{{\left(3-8\right)}^{2}+{\left[y-\left(-3\right)\right]}^{2}}=\sqrt{{\left(3-7\right)}^{2}+{\left(y-6\right)}^{2}}$ (Distance formula)
Squaring on both sides, we get
$25+{\left(y+3\right)}^{2}=16+{\left(y-6\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒25+{y}^{2}+6y+9=16+{y}^{2}-12y+36\phantom{\rule{0ex}{0ex}}⇒12y+6y=52-34\phantom{\rule{0ex}{0ex}}⇒18y=18\phantom{\rule{0ex}{0ex}}⇒y=1$
Thus, the value of y is 1.
$\therefore \mathrm{AQ}=\sqrt{{\left(3-7\right)}^{2}+{\left(1-6\right)}^{2}}=\sqrt{{\left(-4\right)}^{2}+{\left(-5\right)}^{2}}=\sqrt{16+25}=\sqrt{41}$ units
#### Question 48:
If (0, −3) and (0, 3) are the two vertices of an equilateral triangle, find the coordinates of its third vertex. [CBSE 2014]
Let the given points be A(0, −3) and B(0, 3). Suppose the coordinates of the third vertex be C(x, y).
Now, ∆ABC is an equilateral triangle.
∴ AB = BC = CA
$\sqrt{{\left(0-0\right)}^{2}+{\left(-3-3\right)}^{2}}=\sqrt{{\left(x-0\right)}^{2}+{\left(y-3\right)}^{2}}=\sqrt{{\left(x-0\right)}^{2}+{\left[y-\left(-3\right)\right]}^{2}}$ (Distance formula)
Squaring on both sides, we get
$36={x}^{2}+{\left(y-3\right)}^{2}={x}^{2}+{\left(y+3\right)}^{2}$
${x}^{2}+{\left(y-3\right)}^{2}={x}^{2}+{\left(y+3\right)}^{2}$ and ${x}^{2}+{\left(y-3\right)}^{2}=36$
Now,
${x}^{2}+{\left(y-3\right)}^{2}={x}^{2}+{\left(y+3\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒{y}^{2}-6y+9={y}^{2}+6y+9\phantom{\rule{0ex}{0ex}}⇒-12y=0\phantom{\rule{0ex}{0ex}}⇒y=0$
Putting y = 0 in ${x}^{2}+{\left(y-3\right)}^{2}=36$, we get
${x}^{2}+{\left(0-3\right)}^{2}=36\phantom{\rule{0ex}{0ex}}⇒{x}^{2}=36-9=27\phantom{\rule{0ex}{0ex}}⇒x=±\sqrt{27}=±3\sqrt{3}$
Thus, the coordinates of the third vertex are or .
#### Question 49:
If the point P(2, 2) is equidistant from the points A(−2, k) and B(−2k, −3), find k. Also, find the length of AP. [CBSE 2014]
It is given that P(2, 2) is equidistant from the points A(−2, k) and B(−2k, −3).
∴ AP = BP
$⇒\sqrt{{\left(-2-2\right)}^{2}+{\left(k-2\right)}^{2}}=\sqrt{{\left(-2k-2\right)}^{2}+{\left(-3-2\right)}^{2}}$ (Distance formula)
Squaring on both sides, we get
$16+{k}^{2}-4k+4=4{k}^{2}+8k+4+25\phantom{\rule{0ex}{0ex}}⇒3{k}^{2}+12k+9=0\phantom{\rule{0ex}{0ex}}⇒{k}^{2}+4k+3=0\phantom{\rule{0ex}{0ex}}⇒\left(k+3\right)\left(k+1\right)=0$
Thus, the value of k is −1 or −3.
When k = −1,
$\mathrm{AP}=\sqrt{{\left(-2-2\right)}^{2}+{\left(-1-2\right)}^{2}}=\sqrt{{\left(-4\right)}^{2}+{\left(-3\right)}^{2}}=\sqrt{16+9}=\sqrt{25}=5$ units
When k = −3,
$\mathrm{AP}=\sqrt{{\left(-2-2\right)}^{2}+{\left(-3-2\right)}^{2}}=\sqrt{{\left(-4\right)}^{2}+{\left(-5\right)}^{2}}=\sqrt{16+25}=\sqrt{41}$ units
#### Question 50:
Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles.
In ΔABC, the coordinates of the vertices are A(–2, 0), B(2, 0), C(0, 2).
$AB=\sqrt{{\left(2+2\right)}^{2}+{\left(0-0\right)}^{2}}=4\phantom{\rule{0ex}{0ex}}BC=\sqrt{{\left(0-2\right)}^{2}+{\left(2-0\right)}^{2}}=\sqrt{8}=2\sqrt{2}\phantom{\rule{0ex}{0ex}}CA=\sqrt{{\left(0+2\right)}^{2}+{\left(2-0\right)}^{2}}=\sqrt{8}=2\sqrt{2}$
In ΔPQR, the coordinates of the vertices are P(–4, 0), Q(4, 0), R(0, 4).
$PQ=\sqrt{{\left(4+4\right)}^{2}+{\left(0-0\right)}^{2}}=8\phantom{\rule{0ex}{0ex}}QR=\sqrt{{\left(0-4\right)}^{2}+{\left(4-0\right)}^{2}}=4\sqrt{2}\phantom{\rule{0ex}{0ex}}PR=\sqrt{{\left(0+4\right)}^{2}+{\left(4-0\right)}^{2}}=4\sqrt{2}$
Now, for ΔABC and ΔPQR to be similar, the corresponding sides should be proportional.
Thus, ΔABC is similar to ΔPQR.
#### Question 51:
An equilateral triangle has two vertices at the points (3, 4) and (−2, 3), find the coordinates of the third vertex.
The distance d between two points and is given by the formula
In an equilateral triangle all the sides are of equal length.
Here we are given that A (3, 4) and B (2, 3) are two vertices of an equilateral triangle. Let C(x, y) be the third vertex of the equilateral triangle.
First let us find out the length of the side of the equilateral triangle.
Hence the side of the equilateral triangle measures units.
Now, since it is an equilateral triangle, all the sides need to measure the same length.
Hence we have
Equating both these equations we have,
Squaring on both sides we have,
From the above equation we have,
Substituting this and the value of the side of the triangle in the equation for one of the sides we have,
Squaring on both sides,
Now we have a quadratic equation for ‘x’. Solving for the roots of this equation,
We know that. Substituting the value of ‘x’ we have,
Hence the two possible values of the third vertex are.
#### Question 52:
Find the circumcentre of the triangle whose vertices are (−2, −3), (−1, 0), (7, −6).
The distance d between two points and is given by the formula
The circumcentre of a triangle is the point which is equidistant from each of the three vertices of the triangle.
Here the three vertices of the triangle are given to be A(2.−3), B(1,0) and C(7,6).
Let the circumcentre of the triangle be represented by the point R(x, y).
So we have
Equating the first pair of these equations we have,
Squaring on both sides of the equation we have,
Equating another pair of the equations we have,
Squaring on both sides of the equation we have,
Now we have two equations for ‘x’ and ‘y’, which are
From the second equation we have. Substituting this value of ‘y’ in the first equation we have,
Therefore the value of ‘y’ is,
Hence the co−ordinates of the circumcentre of the triangle with the given vertices are.
#### Question 53:
Find the angle subtended at the origin by the line segment whose end points are (0, 100) and (10, 0).
The distance d between two points and is given by the formula
In a right angled triangle the angle opposite the hypotenuse subtends an angle of.
Here let the given points be A(0,100), B(10,0). Let the origin be denoted by O(0,0).
Let us find the distance between all the pairs of points
Here we can see that.
So, is a right angled triangle with ‘AB’ being the hypotenuse. So the angle opposite it has to be. This angle is nothing but the angle subtended by the line segment ‘AB’ at the origin.
Hence the angle subtended at the origin by the given line segment is.
#### Question 54:
Find the centre of the circle passing through (5, −8), (2, −9) and (2, 1).
The distance d between two points and is given by the formula
The centre of a circle is at equal distance from all the points on its circumference.
Here it is given that the circle passes through the points A(5,8), B(2,9) and C(2,1).
Let the centre of the circle be represented by the point O(x, y).
So we have
Equating the first pair of these equations we have,
Squaring on both sides of the equation we have,
Equating another pair of the equations we have,
Squaring on both sides of the equation we have,
Now we have two equations for ‘x’ and ‘y’, which are
From the second equation we have. Substituting this value of ‘y’ in the first equation we have,
Therefore the value of ‘y’ is,
Hence the co-ordinates of the centre of the circle are.
#### Question 55:
If two opposite vertices of a square are (5, 4) and (1, −6), find the coordinates of its remaining two vertices.
The distance d between two points and is given by the formula
In a square all the sides are of equal length. The diagonals are also equal to each other. Also in a square the diagonal is equal to times the side of the square.
Here let the two points which are said to be the opposite vertices of a diagonal of a square be A(5,4) and C(1,6).
Let us find the distance between them which is the length of the diagonal of the square.
Now we know that in a square,
Substituting the value of the diagonal we found out earlier in this equation we have,
Now, a vertex of a square has to be at equal distances from each of its adjacent vertices.
Let P(x, y) represent another vertex of the same square adjacent to both ‘A’ and ‘C’.
But these two are nothing but the sides of the square and need to be equal to each other.
Squaring on both sides we have,
From this we have,
Substituting this value of ‘x’ and the length of the side in the equation for ‘AP’ we have,
Squaring on both sides,
We have a quadratic equation. Solving for the roots of the equation we have,
The roots of this equation are −3 and 1.
Now we can find the respective values of ‘x’ by substituting the two values of ‘y
When
When
Therefore the other two vertices of the square are.
#### Question 56:
Find the centre of the circle passing through (6, −6), (3, −7) and (3, 3).
The distance d between two points and is given by the formula
The centre of a circle is at equal distance from all the points on its circumference.
Here it is given that the circle passes through the points A(6,6), B(3,7) and C(3,3).
Let the centre of the circle be represented by the point O(x, y).
So we have
Equating the first pair of these equations we have,
Squaring on both sides of the equation we have,
Equating another pair of the equations we have,
Squaring on both sides of the equation we have,
Now we have two equations for ‘x’ and ‘y’, which are
From the second equation we have. Substituting this value of ‘y’ in the first equation we have,
Therefore the value of ‘y’ is,
Hence the co-ordinates of the centre of the circle are.
#### Question 57:
Two opposite vertices of a square are (−1, 2) and (3, 2). Find the coordinates of other two vertices.
The distance d between two points and is given by the formula
In a square all the sides are of equal length. The diagonals are also equal to each other. Also in a square the diagonal is equal to times the side of the square.
Here let the two points which are said to be the opposite vertices of a diagonal of a square be A(1,2) and C(3,2).
Let us find the distance between them which is the length of the diagonal of the square.
Now we know that in a square,
Substituting the value of the diagonal we found out earlier in this equation we have,
Now, a vertex of a square has to be at equal distances from each of its adjacent vertices.
Let P(x, y) represent another vertex of the same square adjacent to both ‘A’ and ‘C’.
But these two are nothing but the sides of the square and need to be equal to each other.
Squaring on both sides we have,
Substituting this value of ‘x’ and the length of the side in the equation for ‘AP’ we have,
Squaring on both sides,
We have a quadratic equation. Solving for the roots of the equation we have,
The roots of this equation are 0 and 4.
Therefore the other two vertices of the square are.
#### Question 1:
Find the coordinates of the point which divides the line segment joining (−1,3) and (4, −7) internally in the ratio 3 : 4
We have A (−1, 3) and B (4,−7) be two points. Let a pointdivide the line segment joining the points A and B in the ratio 3:4 internally.
Now according to the section formula if point a point P divides a line segment joining andin the ratio m: n internally than,
Now we will use section formula to find the co-ordinates of unknown point P as,
Therefore, co-ordinates of point P is
#### Question 2:
Find the points of trisection of the line segment joining the points:
(a) 5, −6 and (−7, 5),
(b) (3, −2) and (−3, −4),
(c) (2, −2) and (−7, 4).
The co-ordinates of a point which divided two points and internally in the ratio is given by the formula,
The points of trisection of a line are the points which divide the line into the ratio.
(i) Here we are asked to find the points of trisection of the line segment joining the points A(5,−6) and B(−7,5).
So we need to find the points which divide the line joining these two points in the ratio and 2 : 1.
Let P(x, y) be the point which divides the line joining ‘AB’ in the ratio 1 : 2.
Let Q(e, d) be the point which divides the line joining ‘AB’ in the ratio 2 : 1.
Therefore the points of trisection of the line joining the given points are .
(ii) Here we are asked to find the points of trisection of the line segment joining the points A(3,−2) and B(−3,−4).
So we need to find the points which divide the line joining these two points in the ratio and 2 : 1.
Let P(x, y) be the point which divides the line joining ‘AB’ in the ratio 1 : 2.
Let Q(e, d) be the point which divides the line joining ‘AB’ in the ratio 2 : 1.
Therefore the points of trisection of the line joining the given points are.
(iii) Here we are asked to find the points of trisection of the line segment joining the points A(2,−2) and B(−7,4).
So we need to find the points which divide the line joining these two points in the ratio and 2 : 1.
Let P(x, y) be the point which divides the line joining ‘AB’ in the ratio 1 : 2.
Let Q(e, d) be the point which divides the line joining ‘AB’ in the ratio 2 : 1.
Therefore the points of trisection of the line joining the given points are .
#### Question 3:
Find the coordinates of the point where the diagonals of the parallelogram formed by joining the points (−2, −1), (1, 0), (4, 3) and(1, 2) meet.
The co-ordinates of the midpoint between two points and is given by,
In a parallelogram the diagonals bisect each other. That is the point of intersection of the diagonals is the midpoint of either of the diagonals.
Here, it is given that the vertices of a parallelogram are A(−2,−1), B(1,0) and C(4,3) and D(1,2).
We see that ‘AC’ and ‘BD’ are the diagonals of the parallelogram.
The midpoint of either one of these diagonals will give us the point of intersection of the diagonals.
Let this point be M(x, y).
Let us find the midpoint of the diagonal ‘AC’.
Hence the co-ordinates of the point of intersection of the diagonals of the given parallelogram are.
#### Question 4:
Prove that the points (3, −2), (4, 0), (6, −3) and (5, −5) are the vertices of a parallelogram.
Let A (3,−2); B (4, 0); C (6,−3) and D (5,−5) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a parallelogram.
We should proceed with the fact that if the diagonals of a quadrilateral bisect each other than the quadrilateral is a parallelogram.
Now to find the mid-point of two pointsand we use section formula as,
So the mid-point of the diagonal AC is,
Similarly mid-point of diagonal BD is,
Therefore the mid-points of the diagonals are coinciding and thus diagonal bisects each other.
Hence ABCD is a parallelogram.
#### Question 5:
If P ( 9a $-$2 ,$-$b) divides the line segment joining A (3a + 1 , $-$3 ) and B (8a, 5) in the ratio 3 : 1 , find the values of a and b .
It is given that P divides AB in the ratio 3 : 1.
Therefore, by section formula we have
$⇒9a-2=\frac{3\left(8a\right)+1\left(3a+1\right)}{3+1}\phantom{\rule{0ex}{0ex}}⇒4\left(9a-2\right)=24a+3a+1\phantom{\rule{0ex}{0ex}}⇒36a-8=27a+1\phantom{\rule{0ex}{0ex}}⇒9a=9\phantom{\rule{0ex}{0ex}}⇒a=1$
And,
$⇒-b=\frac{3\left(5\right)+1\left(-3\right)}{3+1}\phantom{\rule{0ex}{0ex}}⇒-4b=15-3\phantom{\rule{0ex}{0ex}}⇒b=-3$
#### Question 6:
If (a,b) is the mid-point of the line segment joining the points A (10, $-$6) , B (k,4) and a$-$2b = 18 , find the value of k and the distance AB.
It is given that A(10, −6) and B(k, 4).
Suppose (a, b) be midpoint of AB. Then,
#### Question 7:
Find the ratio in which the point (2, y) divides the line segment joining the points A (−2, 2) and B (3, 7). Also, find the value of y.
The co-ordinates of a point which divided two points and internally in the ratio is given by the formula,
Here we are given that the point P(2,y) divides the line joining the points A(−2,2) and B(3,7) in some ratio.
Let us substitute these values in the earlier mentioned formula.
Equating the individual components we have
We see that the ratio in which the given point divides the line segment is.
Let us now use this ratio to find out the value of ‘y’.
Equating the individual components we have
Thus the value of ‘y’ is.
#### Question 8:
If A (−1, 3), B (1, −1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.
The distance d between two points and is given by the formula
The co-ordinates of the midpoint between two points and is given by,
Here, it is given that the three vertices of a triangle are A(−1,3), B(1,−1) and C(5,1).
The median of a triangle is the line joining a vertex of a triangle to the mid-point of the side opposite this vertex.
Let ‘D’ be the mid-point of the side ‘BC’.
Let us now find its co-ordinates.
Thus we have the co-ordinates of the point as D(3,0).
Now, let us find the length of the median ‘AD’.
Thus the length of the median through the vertex ‘A’ of the given triangle is.
#### Question 9:
If the points P, Q(x, 7), R, S(6, y) in this order divide the line segment joining A(2, p) and B(7, 10) in 5 equal parts, find x, y and p. [CBSE 2015]
It is given that P, Q(x, 7), R, S(6, y) divides the line segment joining A(2, p) and B(7, 10) in 5 equal parts.
∴ AP = PQ = QR = RS = SB .....(1)
Now,
AP + PQ + QR + RS + SB = AB
⇒ SB + SB + SB + SB + SB = AB [From (1)]
⇒ 5SB = AB
⇒ SB = $\frac{1}{5}$AB .....(2)
Now,
AS = AP + PQ + QR + RS = $\frac{1}{5}$AB +$\frac{1}{5}$AB + $\frac{1}{5}$AB + $\frac{1}{5}$AB = $\frac{4}{5}$AB .....(3)
From (2) and (3), we get
AS : SB = $\frac{4}{5}$AB : $\frac{1}{5}$AB = 4 : 1
Similarly,
AQ : QB = 2 : 3
Using section formula, we get
Coordinates of Q = $\left(\frac{2×7+3×2}{2+3},\frac{2×10+3×p}{2+3}\right)=\left(\frac{20}{5},\frac{20+3p}{5}\right)=\left(4,\frac{20+3p}{5}\right)$
$\therefore \left(x,7\right)=\left(4,\frac{20+3p}{5}\right)$
$⇒x=4$ and $7=\frac{20+3p}{5}$
Now,
$7=\frac{20+3p}{5}\phantom{\rule{0ex}{0ex}}⇒20+3p=35\phantom{\rule{0ex}{0ex}}⇒3p=15\phantom{\rule{0ex}{0ex}}⇒p=5$
Coordinates of S = $\left(\frac{4×7+1×2}{4+1},\frac{4×10+1×p}{4+1}\right)=\left(\frac{30}{5},\frac{40+5}{5}\right)=\left(6,9\right)$
$\therefore \left(6,y\right)=\left(6,9\right)\phantom{\rule{0ex}{0ex}}⇒y=9$
Thus, the values of x, y and p are 4, 9 and 5, respectively.
#### Question 10:
If a vertex of a triangle be (1, 1) and the middle points of the sides through it be (−2,−3) and (5 2) find the other vertices.
Let a in which P and Q are the mid-points of sides AB and AC respectively. The coordinates are: A (1, 1); P (−2, 3) and Q (5, 2).
We have to find the co-ordinates of and.
In general to find the mid-point of two pointsand we use section formula as,
Therefore mid-point P of side AB can be written as,
Now equate the individual terms to get,
So, co-ordinates of B is (−5, 5)
Similarly, mid-point Q of side AC can be written as,
Now equate the individual terms to get,
So, co-ordinates of C is (9, 3)
#### Question 11:
(i) In what ratio is the line segment joining the points (−2,−3) and (3, 7) divided by the y-axis? Also, find the coordinates of the point of division.
(ii) In what ratio is the line segment joining (−3, −1) and (−8, −9) divided at the point (−5, −21/5)?
(i) The ratio in which the y-axis divides two points and is $\lambda :1$
The co-ordinates of the point dividing two points and in the ratio is given as,
; where
Here the two given points are A(−2,−3) and B(3,7).
Since, the point is on the y-axis so, x coordinate is 0.
$\frac{3\lambda -2}{1}=0\phantom{\rule{0ex}{0ex}}⇒\lambda =\frac{2}{3}$
Thus the given points are divided by the y-axis in the ratio.
The co-ordinates of this point (x, y) can be found by using the earlier mentioned formula.
Thus the co-ordinates of the point which divides the given points in the required ratio are.
(ii) The co-ordinates of a point which divided two points and internally in the ratio is given by the formula,
Here it is said that the point divides the points (−3,−1) and (−8,−9). Substituting these values in the above formula we have,
Equating the individual components we have,
Therefore the ratio in which the line is divided is.
#### Question 12:
If the mid-point of the line joining (3, 4) and (k, 7) is (x, y) and 2x + 2y + 1 = 0 find the value of k.
We have two points A (3, 4) and B (k, 7) such that its mid-point is.
It is also given that point P lies on a line whose equation is
In general to find the mid-point of two pointsand we use section formula as,
Therefore mid-point P of side AB can be written as,
Now equate the individual terms to get,
Since, P lies on the given line. So,
Put the values of co-ordinates of point P in the equation of line to get,
On further simplification we get,
So,
#### Question 13:
Find the ratio in which the point P$\left(\frac{3}{4},\frac{5}{12}\right)$ divides the line segment joining the points A$\left(\frac{1}{2},\frac{3}{2}\right)$ and B$\left(2,-5\right)$. [CBSE 2015]
Suppose P$\left(\frac{3}{4},\frac{5}{12}\right)$ divides the line segment joining the points A$\left(\frac{1}{2},\frac{3}{2}\right)$ and B$\left(2,-5\right)$ in the ratio k : 1.
Using section formula, we get
Coordinates of P = $\left(\frac{2k+\frac{1}{2}}{k+1},\frac{-5k+\frac{3}{2}}{k+1}\right)$
$\therefore \left(\frac{2k+\frac{1}{2}}{k+1},\frac{-5k+\frac{3}{2}}{k+1}\right)=\left(\frac{3}{4},\frac{5}{12}\right)$
$⇒\frac{2k+\frac{1}{2}}{k+1}=\frac{3}{4}$ and $\frac{-5k+\frac{3}{2}}{k+1}=\frac{5}{12}$
Now,
$\frac{2k+\frac{1}{2}}{k+1}=\frac{3}{4}\phantom{\rule{0ex}{0ex}}⇒8k+2=3k+3\phantom{\rule{0ex}{0ex}}⇒5k=1\phantom{\rule{0ex}{0ex}}⇒k=\frac{1}{5}$
Putting k = $\frac{1}{5}$ in $\frac{-5k+\frac{3}{2}}{k+1}=\frac{5}{12}$, we get
LHS = $\frac{-5×\frac{1}{5}+\frac{3}{2}}{\frac{1}{5}+1}=\frac{-1+\frac{3}{2}}{\frac{1}{5}+1}=\frac{\frac{1}{2}}{\frac{6}{5}}=\frac{5}{12}$ = RHS
Thus, the required ratio is $\frac{1}{5}$ : 1 or 1 : 5.
#### Question 14:
Find the ratio in which the line segment joining (−2, −3) and (5, 6) is divided by (i) x-axis (ii) y-axis. Also, find the coordinates of the point of division in each case.
The ratio in which the x−axis divides two points and is $\lambda :1$
The ratio in which the y-axis divides two points and is $\mu :1$
The co-ordinates of the point dividing two points and in the ratio is given as,
Where
Here the two given points are A(−2,−3) and B(5,6).
1. The ratio in which the x-axis divides these points is
$\frac{6\lambda -3}{3}=0\phantom{\rule{0ex}{0ex}}\lambda =\frac{1}{2}$
Let point P(x, y) divide the line joining ‘AB’ in the ratio
Substituting these values in the earlier mentioned formula we have,
Thus the ratio in which the x−axis divides the two given points and the co-ordinates of the point is.
1. The ratio in which the y-axis divides these points is
$\frac{5\mu -2}{3}=0\phantom{\rule{0ex}{0ex}}⇒\mu =\frac{2}{5}$
Let point P(x, y) divide the line joining ‘AB’ in the ratio
Substituting these values in the earlier mentioned formula we have,
Thus the ratio in which the x-axis divides the two given points and the co-ordinates of the point is.
#### Question 15:
Prove that the points (4, 5) (7, 6), (6, 3) (3, 2) are the vertices of a parallelogram. Is it a rectangle.
Let A (4, 5); B (7, 6); C (6, 3) and D (3, 2) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a parallelogram.
We should proceed with the fact that if the diagonals of a quadrilateral bisect each other than the quadrilateral is a parallelogram.
Now to find the mid-point of two pointsand we use section formula as,
So the mid-point of the diagonal AC is,
Similarly mid-point of diagonal BD is,
Therefore the mid-points of the diagonals are coinciding and thus diagonal bisects each other.
Hence ABCD is a parallelogram.
Now to check if ABCD is a rectangle, we should check the diagonal length.
Similarly,
Diagonals are of different lengths.
Hence ABCD is not a rectangle.
#### Question 16:
Prove that (4, 3), (6, 4) (5, 6) and (3, 5) are the angular points of a square.
Let A (4, 3); B (6, 4); C (5, 6) and D (3, 5) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a square.
So we should find the lengths of sides of quadrilateral ABCD.
All the sides of quadrilateral are equal.
So now we will check the lengths of the diagonals.
All the sides as well as the diagonals are equal. Hence ABCD is a square.
#### Question 17:
Prove that the points (−4,−1), (−2, 4), (4, 0) and (2, 3) are the vertices of a rectangle.
Let A (−4,−1); B (−2,−4); C (4, 0) and D (2, 3) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a rectangle.
So we should find the lengths of opposite sides of quadrilateral ABCD.
Opposite sides are equal. So now we will check the lengths of the diagonals.
Opposite sides are equal as well as the diagonals are equal. Hence ABCD is a rectangle.
#### Question 18:
Find the lengths of the medians of a triangle whose vertices are A (−1,3), B(1,−1) and C(5, 1).
We have to find the lengths of the medians of a triangle whose co-ordinates of the vertices are A (−1, 3); B (1,−1) and C (5, 1).
So we should find the mid-points of the sides of the triangle.
In general to find the mid-point of two pointsand we use section formula as,
Therefore mid-point P of side AB can be written as,
Now equate the individual terms to get,
So co-ordinates of P is (0, 1)
Similarly mid-point Q of side BC can be written as,
Now equate the individual terms to get,
So co-ordinates of Q is (3, 0)
Similarly mid-point R of side AC can be written as,
Now equate the individual terms to get,
So co-ordinates of Q is (2, 2)
Therefore length of median from A to the side BC is,
Similarly length of median from B to the side AC is,
Similarly length of median from C to the side AB is
#### Question 19:
Find the ratio in which the line segment joining the points A(3, −3) and B(−2, 7) is divided by the x-axis. Also, find the coordinates of the point of division. [CBSE 2014]
Suppose the x-axis divides the line segment joining the points A(3, −3) and B(−2, 7) in the ratio k : 1.
Using section formula, we get
Coordinates of the point of division = $\left(\frac{-2k+3}{k+1},\frac{7k-3}{k+1}\right)$
Since the point of division lies on the x-axis, so its y-coordinate is 0.
$\therefore \frac{7k-3}{k+1}=0\phantom{\rule{0ex}{0ex}}⇒7k-3=0\phantom{\rule{0ex}{0ex}}⇒k=\frac{3}{7}$
So, the required ratio is $\frac{3}{7}$ : 1 or 3 : 7.
Putting k = $\frac{3}{7}$, we get
Coordinates of the point of division = $\left(\frac{-2×\frac{3}{7}+3}{\frac{3}{7}+1},0\right)=\left(\frac{-6+21}{3+7},0\right)=\left(\frac{15}{10},0\right)=\left(\frac{3}{2},0\right)$
Thus, the coordinates of the point of division are $\left(\frac{3}{2},0\right)$.
#### Question 20:
Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, −3). Also, find the value of x. [CBSE 2014]
Suppose P(x, 2) divides the line segment joining the points A(12, 5) and B(4, −3) in the ratio k : 1.
Using section formula, we get
Coordinates of P = $\left(\frac{4k+12}{k+1},\frac{-3k+5}{k+1}\right)$
$\therefore \left(\frac{4k+12}{k+1},\frac{-3k+5}{k+1}\right)=\left(x,2\right)$
$⇒x=\frac{4k+12}{k+1}$ and $\frac{-3k+5}{k+1}=2$
Now,
$\frac{-3k+5}{k+1}=2\phantom{\rule{0ex}{0ex}}⇒-3k+5=2k+2\phantom{\rule{0ex}{0ex}}⇒5k=3\phantom{\rule{0ex}{0ex}}⇒k=\frac{3}{5}$
So, P divides the line segment AB in the ratio 3 : 5.
Putting k = $\frac{3}{5}$ in $x=\frac{4k+12}{k+1}$, we get
$x=\frac{4×\frac{3}{5}+12}{\frac{3}{5}+1}=\frac{12+60}{3+5}=\frac{72}{8}=9$
Thus, the value of x is 9.
#### Question 21:
Find the ratio in which the point P(−1, y) lying on the line segment joining A(−3, 10) and B(6 −8) divides it. Also find the value of y. [CBSE 2013]
Suppose P(−1, y) divides the line segment joining A(−3, 10) and B(6 −8) in the ratio k : 1.
Using section formula, we get
Coordinates of P = $\left(\frac{6k-3}{k+1},\frac{-8k+10}{k+1}\right)$
$\therefore \left(\frac{6k-3}{k+1},\frac{-8k+10}{k+1}\right)=\left(-1,y\right)$
$⇒\frac{6k-3}{k+1}=-1$ and $y=\frac{-8k+10}{k+1}$
Now,
$\frac{6k-3}{k+1}=-1\phantom{\rule{0ex}{0ex}}⇒6k-3=-k-1\phantom{\rule{0ex}{0ex}}⇒7k=2\phantom{\rule{0ex}{0ex}}⇒k=\frac{2}{7}$
So, P divides the line segment AB in the ratio 2 : 7.
Putting k = $\frac{2}{7}$ in $y=\frac{-8k+10}{k+1}$, we get
$y=\frac{-8×\frac{2}{7}+10}{\frac{2}{7}+1}=\frac{-16+70}{2+7}=\frac{54}{9}=6$
Hence, the value of y is 6.
#### Question 22:
Find the coordinates of a point A, where AB is a diameter of the circle whose centre is (2, −3) and B is (1, 4).
Let the co-ordinates of point A be.
Centre lies on the mid-point of the diameter. So applying the mid-point formula we get,
Similarly,
So the co-ordinates of A are (3,−10)
#### Question 23:
If the points (−2, −1), (1, 0), (x, 3) and (1, y) form a parallelogram, find the values of x and y.
Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (−2,−1); B (1, 0); C (x, 3) and D (1, y).
Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.
In general to find the mid-point of two pointsand we use section formula as,
The mid-point of the diagonals of the parallelogram will coincide.
So,
Therefore,
Now equate the individual terms to get the unknown value. So,
Similarly,
Therefore,
#### Question 24:
The points A(2, 0), B(9, 1) C(11, 6) and D(4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.
Let A (2, 0); B (9, 1); C (11, 6) and D (4, 4) be the vertices of a quadrilateral. We have to check if the quadrilateral ABCD is a rhombus or not.
So we should find the lengths of sides of quadrilateral ABCD.
All the sides of quadrilateral are unequal. Hence ABCD is not a rhombus.
#### Question 25:
In what ratio does the point (−4, 6) divide the line segment joining the points A(−6, 10) and B(3,−8)?
The co-ordinates of a point which divided two points and internally in the ratio is given by the formula,
Here it is said that the point (−4,6) divides the points A(−6,10) and B(3,−8). Substituting these values in the above formula we have,
Equating the individual components we have,
Therefore the ratio in which the line is divided is
#### Question 26:
Find the ratio in which the y-axis divides the line segment joining the points (5, −6) and (−1,−4). Also, find the coordinates of the point of division.
The ratio in which the y-axis divides two points and is $\lambda :1$
The co-ordinates of the point dividing two points and in the ratio is given as,
where,
Here the two given points are A(5,−6) and B(−1,−4).
Since, the y-axis divided the given line, so the x coordinate will be 0.
$\frac{-\lambda +5}{\lambda +1}=0\phantom{\rule{0ex}{0ex}}\lambda =\frac{5}{1}$
Thus the given points are divided by the y-axis in the ratio.
The co-ordinates of this point (x, y) can be found by using the earlier mentioned formula.
Thus the co-ordinates of the point which divides the given points in the required ratio are.
#### Question 27:
Show that A (−3, 2), B (−5, −5), C (2,−3), and D (4, 4) are the vertices of a rhombus.
Let A (−3, 2); B (−5,−5); C (2,−3) and D (4, 4) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a rhombus.
So we should find the lengths of sides of quadrilateral ABCD.
All the sides of quadrilateral are equal. Hence ABCD is a rhombus.
#### Question 28:
Find the length of the medians of a ΔABC having vertices at A(0, −1), B(2, 1) and C(0, 3).
We have to find the lengths of the medians of a triangle whose co-ordinates of the vertices are A (0,−1); B (2, 1) and C (0, 3).
So we should find the mid-points of the sides of the triangle.
In general to find the mid-point of two pointsand we use section formula as,
Therefore mid-point P of side AB can be written as,
Now equate the individual terms to get,
So co-ordinates of P is (1, 0)
Similarly mid-point Q of side BC can be written as,
Now equate the individual terms to get,
So co-ordinates of Q is (1, 2)
Similarly mid-point R of side AC can be written as,
Now equate the individual terms to get,
So co-ordinates of R is (0, 1)
Therefore length of median from A to the side BC is,
Similarly length of median from B to the side AC is,
Similarly length of median from C to the side AB is
#### Question 29:
Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, –3). Hence, find m.
Let P divides AB in a ratio of λ : 1
Therefore, coordinates of the point P are
Given that coordinates of the point P are (4, m).
$⇒\frac{6\lambda +2}{\lambda +1}=4\phantom{\rule{0ex}{0ex}}⇒6\lambda +2=4\lambda +4\phantom{\rule{0ex}{0ex}}⇒\lambda =1$
Hence, the point P divides AB in a ratio of 1 : 1.
Replacing the value of λ = 1 in y-coordinate of P, we get
$\frac{-3\left(1\right)+3}{1+1}=m\phantom{\rule{0ex}{0ex}}⇒m=0$
Thus, y-coordinate of P is equal to 0.
#### Question 30:
Find the coordinates of the points which divide the line segment joining the points (−4, 0) and (0, 6) in four equal parts.
The co-ordinates of the midpoint between two points and is given by,
Here we are supposed to find the points which divide the line joining A(−4,0) and B(0,6) into 4 equal parts.
We shall first find the midpoint M(x, y) of these two points since this point will divide the line into two equal parts
So the point M(−2,3) splits this line into two equal parts.
Now, we need to find the midpoint of A(−4,0) and M(−2,3) separately and the midpoint of B(0,6) and M(−2,3). These two points along with M(−2,3) split the line joining the original two points into four equal parts.
Let be the midpoint of A(−4,0) and M(−2,3).
Now let bet the midpoint of B(0,6) and M(−2,3).
Hence the co-ordinates of the points which divide the line joining the two given points are.
#### Question 31:
Show that the mid-point of the line segment joining the points (5, 7) and (3, 9) is also the mid-point of the line segment joining the points (8, 6) and (0, 10).
We have two points A (5, 7) and B (3, 9) which form a line segment and similarly
C (8, 6) and D (0, 10) form another line segment.
We have to prove that mid-point of AB is also the mid-point of CD.
In general to find the mid-point of two pointsand we use section formula as,
Therefore mid-point P of line segment AB can be written as,
Now equate the individual terms to get,
So co-ordinates of P is (4, 8)
Similarly mid-point Q of side CD can be written as,
Now equate the individual terms to get,
So co-ordinates of Q is (4, 8)
Hence the point P and Q coincides.
Thus mid-point of AB is also the mid-point of CD.
#### Question 32:
Find the distance of the point (1, 2) from the mid-point of the line segment joining the points (6, 8) and (2, 4).
We have to find the distance of a point A (1, 2) from the mid-point of the line segment joining P (6, 8) and Q (2, 4).
In general to find the mid-point of any two pointsand we use section formula as,
Therefore mid-point B of line segment PQ can be written as,
Now equate the individual terms to get,
So co-ordinates of B is (4, 6)
Therefore distance between A and B,
#### Question 33:
If A and B are (1, 4) and (5, 2) respectively, find the coordinates of P when AP/BP = 3/4.
The co-ordinates of the point dividing two points and in the ratio is given as,
where,
Here the two given points are A(1,4) and B(5,2). Let point P(x, y) divide the line joining ‘AB’ in the ratio
Substituting these values in the earlier mentioned formula we have,
Thus the co-ordinates of the point which divides the given points in the required ratio are.
#### Question 34:
Show that the points A (1, 0), B (5, 3), C (2, 7) and D (−2, 4) are the vertices of a parallelogram.
Let A (1, 0); B (5, 3); C (2, 7) and D (−2, 4) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a parallelogram.
We should proceed with the fact that if the diagonals of a quadrilateral bisect each other than the quadrilateral is a parallelogram.
Now to find the mid-point of two pointsand we use section formula as,
So the mid-point of the diagonal AC is,
Similarly mid-point of diagonal BD is,
Therefore the mid-points of the diagonals are coinciding and thus diagonal bisects each other.
Hence ABCD is a parallelogram.
#### Question 35:
Determine the ratio in which the point P (m, 6) divides the join of A(−4, 3) and B(2, 8). Also, find the value of m.
The co-ordinates of a point which divided two points and internally in the ratio is given by the formula,
Here we are given that the point P(m,6) divides the line joining the points A(−4,3) and B(2,8) in some ratio.
Let us substitute these values in the earlier mentioned formula.
Equating the individual components we have
We see that the ratio in which the given point divides the line segment is.
Let us now use this ratio to find out the value of ‘m’.
Equating the individual components we have
Thus the value of ‘m’ is.
#### Question 36:
Determine the ratio in which the point (−6, a) divides the join of A (−3, 1) and B (−8, 9). Also find the value of a.
The co-ordinates of a point which divided two points and internally in the ratio is given by the formula,
Here we are given that the point P(−6,a) divides the line joining the points A(−3,1) and B(−8,9) in some ratio.
Let us substitute these values in the earlier mentioned formula.
Equating the individual components we have
We see that the ratio in which the given point divides the line segment is.
Let us now use this ratio to find out the value of ‘a’.
Equating the individual components we have
Thus the value of ‘a’ is.
#### Question 37:
ABCD is a rectangle formed by joining the points A (−1, −1), B(−1 4) C (5 4) and D (5, −1). P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.
We have a rectangle ABCD formed by joining the points A (−1,−1); B (−1, 4); C (5, 4) and D (5,−1). The mid-points of the sides AB, BC, CD and DA are P, Q, R, S respectively.
We have to find that whether PQRS is a square, rectangle or rhombus.
In general to find the mid-point of two pointsand we use section formula as,
Therefore mid-point P of side AB can be written as,
Now equate the individual terms to get,
So co-ordinates of P is
Similarly mid-point Q of side BC can be written as,
Now equate the individual terms to get,
So co-ordinates of Q is (2, 4)
Similarly mid-point R of side CD can be written as,
Now equate the individual terms to get,
So co-ordinates of R is
Similarly mid-point S of side DA can be written as,
Now equate the individual terms to get,
So co-ordinates of S is (2,−1)
So we should find the lengths of sides of quadrilateral PQRS.
All the sides of quadrilateral are equal.
So now we will check the lengths of the diagonals.
All the sides are equal but the diagonals are unequal. Hence ABCD is a rhombus.
#### Question 38:
Points P, Q, R and S divides the line segment joining A(1, 2) and B(6, 7) in 5 equal parts. Find the coordinates of the points P, Q and R. [CBSE 2014]
It is given that P, Q, R and S divides the line segment joining A(1, 2) and B(6, 7) in 5 equal parts.
∴ AP = PQ = QR = RS = SB .....(1)
Now,
AP + PQ + QR + RS + SB = AB
⇒ AP + AP + AP + AP + AP = AB [From (1)]
⇒ 5AP = AB
⇒ AP = $\frac{1}{5}$AB .....(2)
Now,
PB = PQ + QR + RS + SB = $\frac{1}{5}$AB +$\frac{1}{5}$AB + $\frac{1}{5}$AB + $\frac{1}{5}$AB = $\frac{4}{5}$AB .....(3)
From (2) and (3), we get
AP : PB = $\frac{1}{5}$AB : $\frac{4}{5}$AB = 1 : 4
Similarly,
AQ : QB = 2 : 3 and AR : RB = 3 : 2
Using section formula, we get
Coordinates of P = $\left(\frac{1×6+4×1}{1+4},\frac{1×7+4×2}{1+4}\right)=\left(\frac{10}{5},\frac{15}{5}\right)=\left(2,3\right)$
Coordinates of Q = $\left(\frac{2×6+3×1}{2+3},\frac{2×7+3×2}{2+3}\right)=\left(\frac{15}{5},\frac{20}{5}\right)=\left(3,4\right)$
Coordinates of R = $\left(\frac{3×6+2×1}{3+2},\frac{3×7+2×2}{3+2}\right)=\left(\frac{20}{5},\frac{25}{5}\right)=\left(4,5\right)$
#### Question 39:
If A and B are two points having coordinates (−2, −2) and (2, −4) respectively, find the coordinates of P such that AP = $\frac{3}{7}$AB.
We have two points A (−2,−2) and B (2,−4). Let P be any point which divide AB as,
Since,
So,
Now according to the section formula if any point P divides a line segment joining andin the ratio m: n internally than,
Therefore P divides AB in the ratio 3: 4. So,
#### Question 40:
Find the coordinates of the points which divide the line segment joining A(−2, 2) and B (2, 8) into four equal parts.
The co-ordinates of the midpoint between two points and is given by,
Here we are supposed to find the points which divide the line joining A(−2,2) and B(2,8) into 4 equal parts.
We shall first find the midpoint M(x, y) of these two points since this point will divide the line into two equal parts.
So the point M(0,5) splits this line into two equal parts.
Now, we need to find the midpoint of A(−2,2) and M(0,5) separately and the midpoint of B(2,8) and M(0,5). These two points along with M(0,5) split the line joining the original two points into four equal parts.
Let be the midpoint of A(−2,2) and M(0,5).
Now let bet the midpoint of B(2,8) and M(0,5).
Hence the co-ordinates of the points which divide the line joining the two given points are.
#### Question 41:
Three consecutive vertices of a parallelogram are (−2,−1), (1, 0) and (4, 3). Find the fourth vertex.
Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (−2,−1); B (1, 0) and C (4, 3). We have to find the co-ordinates of the forth vertex.
Let the forth vertex be
Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.
Now to find the mid-point of two pointsand we use section formula as,
The mid-point of the diagonals of the parallelogram will coincide.
So,
Therefore,
Now equate the individual terms to get the unknown value. So,
So the forth vertex is
#### Question 42:
The points (3, −4) and (−6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (−1,−3). Find the coordinates of the fourth vertex.
Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (3,−4); B (−1,−3) and C (−6, 2). We have to find the co-ordinates of the forth vertex.
Let the forth vertex be
Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.
Now to find the mid-point of two pointsand we use section formula as,
The mid-point of the diagonals of the parallelogram will coincide.
So,
Therefore,
Now equate the individual terms to get the unknown value. So,
So the forth vertex is
#### Question 43:
If the coordinates of the mid-points of the sides of a triangle are (1, 1) (2, −3) and (3, 4), find the vertices of the triangle.
The co-ordinates of the midpoint between two points and is given by,
Let the three vertices of the triangle be, and.
The three midpoints are given. Let these points be, and.
Let us now equate these points using the earlier mentioned formula,
Equating the individual components we get,
Using the midpoint of another side we have,
Equating the individual components we get,
Using the midpoint of the last side we have,
Equating the individual components we get,
Adding up all the three equations which have variable ‘x’ alone we have,
Substituting in the above equation we have,
Therefore,
And
Adding up all the three equations which have variable ‘y’ alone we have,
Substituting in the above equation we have,
Therefore,
And
Therefore the co-ordinates of the three vertices of the triangle are.
#### Question 44:
Determine the ratio in which the straight line x − y − 2 = 0 divides the line segment joining (3, −1) and (8, 9).
Let the line divide the line segment joining the points A (3,−1) and B (8, 9) in the ratio at any point
Now according to the section formula if point a point P divides a line segment joining andin the ratio m: n internally than,
So,
Since, P lies on the given line. So,
Put the values of co-ordinates of point P in the equation of line to get,
On further simplification we get,
So,
So the line divides the line segment joining A and B in the ratio 2: 3 internally.
#### Question 45:
Three vertices of a parallelogram are (a+b, a−b), (2a+b, 2a−b), (a−b, a+b). Find the fourth vertex.
Let ABCD be a parallelogram in which the co-ordinates of the vertices are;and. We have to find the co-ordinates of the forth vertex.
Let the forth vertex be
Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.
In general to find the mid-point of two pointsand we use section formula as,
The mid-point of the diagonals of the parallelogram will coincide.
So,
Therefore,
Now equate the individual terms to get the unknown value. So,
So the forth vertex is
#### Question 46:
If two vertices of a parallelogram are (3, 2) (−1, 0) and the diagonals cut at (2, −5), find the other vertices of the parallelogram.
We have a parallelogram ABCD in which A (3, 2) and B (−1, 0) and the co-ordinate of the intersection of diagonals is M (2,−5).
We have to find the co-ordinates of vertices C and D.
So let the co-ordinates be and
In general to find the mid-point of two pointsand we use section formula as,
The mid-point of the diagonals of the parallelogram will coincide.
So,
Therefore,
Now equate the individual terms to get the unknown value. So,
So the co-ordinate of vertex C is (1,−12)
Similarly,
Therefore,
Now equate the individual terms to get the unknown value. So,
So the co-ordinate of vertex C is (5,−10)
#### Question 47:
If the coordinates of the mid-points of the sides of a triangle are (3, 4) (4, 6) and (5, 7), find its vertices.
The co-ordinates of the midpoint between two points and is given by,
Let the three vertices of the triangle be, and.
The three midpoints are given. Let these points be, and.
Let us now equate these points using the earlier mentioned formula,
Equating the individual components we get,
Using the midpoint of another side we have,
Equating the individual components we get,
Using the midpoint of the last side we have,
Equating the individual components we get,
Adding up all the three equations which have variable ‘x’ alone we have,
Substituting in the above equation we have,
Therefore,
And
Adding up all the three equations which have variable ‘y’ alone we have,
Substituting in the above equation we have,
Therefore,
And
Therefore the co-ordinates of the three vertices of the triangle are.
#### Question 48:
The line segment joining the points P(3, 3) and Q(6, −6) is trisected at the points A and B such that A is nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.
We have two points P (3, 3) and Q (6,−6). There are two points A and B which trisect the line segment joining P and Q.
Let the co-ordinate of A be
Now according to the section formula if any point P divides a line segment joining andin the ratio m: n internally than,
The point A is the point of trisection of the line segment PQ. So, A divides PQ in the ratio 1: 2
Now we will use section formula to find the co-ordinates of unknown point A as,
Therefore, co-ordinates of point A is(4, 0)
It is given that point A lies on the line whose equation is
So point A will satisfy this equation.
So,
#### Question 49:
If three consecutive vertices of a parallelogram are (1, −2), (3, 6) and (5, 10), find its fourth vertex.
Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (1,−2);
B (3, 6) and C(5, 10). We have to find the co-ordinates of the forth vertex.
Let the forth vertex be
Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.
Now to find the mid-point of two pointsand we use section formula as,
The mid-point of the diagonals of the parallelogram will coincide.
So,
Therefore,
Now equate the individual terms to get the unknown value. So,
Similarly,
So the forth vertex is
#### Question 50:
If the points A (a, −11), B (5, b), C(2, 15) and D (1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.
Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (a,−11); B (5, b); C (2, 15) and D (1, 1).
Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.
In general to find the mid-point of two pointsand we use section formula as,
The mid-point of the diagonals of the parallelogram will coincide.
So,
Therefore,
Now equate the individual terms to get the unknown value. So,
Similarly,
Therefore,
#### Question 51:
If the coordinates of the mid-points of the sides of a triangle be (3, −2), (−3, 1) and (4, −3), then find the coordinates of its vertices.
The co-ordinates of the midpoint between two points and is given by,
Let the three vertices of the triangle be, and.
The three midpoints are given. Let these points be, and.
Let us now equate these points using the earlier mentioned formula,
Equating the individual components we get,
Using the midpoint of another side we have,
Equating the individual components we get,
Using the midpoint of the last side we have,
Equating the individual components we get,
Adding up all the three equations which have variable ‘x’ alone we have,
Substituting in the above equation we have,
Therefore,
And
Adding up all the three equations which have variable ‘y’ alone we have,
Substituting in the above equation we have,
Therefore,
And
Therefore the co-ordinates of the three vertices of the triangle are.
#### Question 52:
The line segment joining the points (3, −4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, −2) and (5/3, q) respectively. Find the values of p and q.
We have two points A (3,−4) and B (1, 2). There are two points P (p,−2) and Q which trisect the line segment joining A and B.
Now according to the section formula if any point P divides a line segment joining andin the ratio m: n internally than,
The point P is the point of trisection of the line segment AB. So, P divides AB in the ratio 1: 2
Now we will use section formula to find the co-ordinates of unknown point A as,
Equate the individual terms on both the sides. We get,
Similarly, the point Q is the point of trisection of the line segment AB. So, Q divides AB in the ratio 2: 1
Now we will use section formula to find the co-ordinates of unknown point A as,
Equate the individual terms on both the sides. We get,
#### Question 53:
The line joining the points (2, 1) and (5, −8) is trisected at the points P and Q. If point P lies on the line 2xy + k = 0. Find the value of k.
We have two points A (2, 1) and B (5,−8). There are two points P and Q which trisect the line segment joining A and B.
Now according to the section formula if any point P divides a line segment joining andin the ratio m: n internally than,
The point P is the point of trisection of the line segment AB. So, P divides AB in the ratio 1: 2
Now we will use section formula to find the co-ordinates of unknown point A as,
Therefore, co-ordinates of point P is(3,−2)
It is given that point P lies on the line whose equation is
So point A will satisfy this equation.
So,
#### Question 54:
A (4, 2), B(6, 5) and C (1, 4) are the vertices of ΔABC.
(i) The median from A meets BC in D. Find the coordinates of the point D.
(ii) Find the coordinates of point P and AD such that AP : PD = 2 : 1.
(iii) Find the coordinates of the points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1
(iv) What do you observe?
We have triangle in which the co-ordinates of the vertices are A (4, 2); B (6, 5) and C (1, 4)
(i)It is given that median from vertex A meets BC at D. So, D is the mid-point of side BC.
In general to find the mid-point of two pointsand we use section formula as,
Therefore mid-point D of side BC can be written as,
Now equate the individual terms to get,
So co-ordinates of D is
(ii)We have to find the co-ordinates of a point P which divides AD in the ratio 2: 1 internally.
Now according to the section formula if any point P divides a line segment joining andin the ratio m: n internally than,
P divides AD in the ratio 2: 1. So,
(iii)We need to find the mid-point of sides AB and AC. Let the mid-points be F and E for the sides AB and AC respectively.
Therefore mid-point F of side AB can be written as,
So co-ordinates of F is
Similarly mid-point E of side AC can be written as,
So co-ordinates of E is
Q divides BE in the ratio 2: 1. So,
Similarly, R divides CF in the ratio 2: 1. So,
(iv)We observe that that the point P, Q and R coincides with the centroid. This also shows that centroid divides the median in the ratio 2: 1.
#### Question 55:
If the points A (6, 1), B (8, 2), C (9, 4) and D (k, p) are the vertices of a parallelogram taken in order, then find the values of k and p.
Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (6, 1); B (8, 2); C (9, 4) and D (k, p).
Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.
In general to find the mid-point of two pointsand we use section formula as,
The mid-point of the diagonals of the parallelogram will coincide.
So,
Therefore,
Now equate the individual terms to get the unknown value. So,
Similarly,
Therefore,
#### Question 56:
A point P divides the line segment joining the points A(3, −5) and B(−4, 8) such that $\frac{\mathrm{AP}}{\mathrm{PB}}=\frac{k}{1}.$ If P lies on the line x + y = 0, then find the value of k. [CBSE 2012]
It is given that $\frac{\mathrm{AP}}{\mathrm{PB}}=\frac{k}{1}.$
So, P divides the line segment joining the points A(3, −5) and B(−4, 8) in the ratio k : 1.
Using the section formula, we get
Coordinates of P = $\left(\frac{-4k+3}{k+1},\frac{8k-5}{k+1}\right)$
Since P lies on the line x + y = 0, so
$\frac{-4k+3}{k+1}+\frac{8k-5}{k+1}=0\phantom{\rule{0ex}{0ex}}⇒\frac{-4k+3+8k-5}{k+1}=0\phantom{\rule{0ex}{0ex}}⇒4k-2=0\phantom{\rule{0ex}{0ex}}⇒k=\frac{1}{2}$
Hence, the value of k is $\frac{1}{2}$.
#### Question 57:
The mid-point P of the line segment joining the points A(−10, 4) and B(−2, 0) lies on the line segment joining the points C(−9, −4) and D(−4, y). Find the ratio in which P divides CD. Also, find the value of y. [CBSE 2014]
It is given that P is the mid-point of the line segment joining the points A(−10, 4) and B(−2, 0).
∴ Coordinates of P = $\left(\frac{-10+\left(-2\right)}{2},\frac{4+0}{2}\right)=\left(\frac{-12}{2},\frac{4}{2}\right)=\left(-6,2\right)$
Suppose P divides the line segment joining the points C(−9, −4) and D(−4, y) in the ratio k : 1.
Using section formula, we get
Coordinates of P = $\left(\frac{-4k-9}{k+1},\frac{ky-4}{k+1}\right)$
Now,
$\frac{-4k-9}{k+1}=-6\phantom{\rule{0ex}{0ex}}⇒-4k-9=-6k-6\phantom{\rule{0ex}{0ex}}⇒2k=3\phantom{\rule{0ex}{0ex}}⇒k=\frac{3}{2}$
So, P divides the line segment CD in the ratio 3 : 2.
Putting k = $\frac{3}{2}$ in $\frac{ky-4}{k+1}=2$, we get
$\frac{\frac{3y}{2}-4}{\frac{3}{2}+1}=2\phantom{\rule{0ex}{0ex}}⇒\frac{3y-8}{5}=2\phantom{\rule{0ex}{0ex}}⇒3y-8=10\phantom{\rule{0ex}{0ex}}⇒3y=18\phantom{\rule{0ex}{0ex}}⇒y=6$
Hence, the value of y is 6.
#### Question 58:
If the point divides internally the line segment joining the points A (2, 5) and B( xy ) in the ratio 3 : 4 , find the value of x2 + y2 .
It is given that the point C(–1, 2) divides the line segment joining the points A(2, 5) and B(xy) in the ratio 3 : 4 internally.
Using the section formula, we get
⇒ 3x + 8 = –7 and 3y + 20 = 14
⇒ 3x = –15 and 3y = –6
x = –5 and y = –2
x2 + y2 = 25 + 4 = 29
Hence, the value of x2 + y2 is 29.
#### Question 59:
ABCD is a parallelogram with vertices . Find the coordinates of the fourth vertex D in terms of .
Suppose the coordinates of D be (x, y).
Since diagonals of a parallelogram bisect each other.
Therefore the midpoint of AC is the midpoint of BD, i.e respectively.
#### Question 60:
The points are the vertices of $∆$ ABC .
(i) The median from A meets BC at D . Find the coordinates of the point D.
(ii) Find the coordinates of the point P on AD such that AP : PD = 2 : 1.
(iii) Find the points of coordinates Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.
(iv) What are the coordinates of the centropid of the triangle ABC
(i) Median AD of the triangle will divide the side BC in two equal parts.
Therefore, D is the midpoint of side BC.
Coordinates of D are
$\left(\frac{{x}_{2}+{x}_{3}}{2},\frac{{y}_{2}+{y}_{3}}{2}\right)$
(ii)
THe point P divided the side AD in the ratio 2: 1.
Coordinates of P are
(iii)
Median BE of the triangle will divide the side AC in two equal parts.
Therefore, E is the midpoint of side AC.
Coordinates of E are
$\left(\frac{{x}_{1}+{x}_{3}}{2},\frac{{y}_{1}+{y}_{3}}{2}\right)$
The point Q divided the side BE in the ratio 2: 1.
Coordinates of Q are
Similarly, Coordinates of Q are R are $\left(\frac{{x}_{1}+{x}_{2}+{x}_{3}}{3},\frac{{y}_{1}+{y}_{2}+{y}_{3}}{3}\right)$.
(iv)
The points P, Q and R coincides and is the centroid of the triangle ABC.
So, coordinates of the centroid is $\left(\frac{{x}_{1}+{x}_{2}+{x}_{3}}{3},\frac{{y}_{1}+{y}_{2}+{y}_{3}}{3}\right)$.
#### Question 1:
Find the centroid of the triangle whose vertices are:
(i) (1, 4) (−1,−1), (3, −2)
(ii) (−2, 3) (2, −1) (4, 0)
We know that the co-ordinates of the centroid of a triangle whose vertices are is-
(i) The co-ordinates of the centroid of a triangle whose vertices are (1, 4); (−1,−1); (3,−2) are-
(ii) The co-ordinates of the centroid of a triangle whose vertices are (−2, 3); (2,−1); (4, 0) are-
#### Question 2:
Two vertices of a triangle are (1, 2), (3, 5) and its centroid is at the origin. Find the coordinates of the third vertex.
We have to find the co-ordinates of the third vertex of the given triangle. Let the co-ordinates of the third vertex be.
The co-ordinates of other two vertices are (1, 2) and (3, 5)
The co-ordinate of the centroid is (0, 0)
We know that the co-ordinates of the centroid of a triangle whose vertices are is−
So,
Compare individual terms on both the sides-
So,
Similarly,
So,
So the co-ordinate of third vertex
#### Question 3:
Find the third vertex of a triangle, if two of its vertices are at (−3, 1) and (0, −2) and the centroid is at the origin.
We have to find the co-ordinates of the third vertex of the given triangle. Let the co-ordinates of the third vertex be.
The co-ordinates of other two vertices are (−3, 1) and (0, −2)
The co-ordinate of the centroid is (0, 0)
We know that the co-ordinates of the centroid of a triangle whose vertices are is-
So,
Compare individual terms on both the sides-
So,
Similarly,
So,
So the co-ordinate of third vertex
#### Question 4:
A (3, 2) and B (−2, 1) are two vertices of a triangle ABC whose centroid G has the coordinates $\left(\frac{5}{3},-\frac{1}{3}\right)$. Find the coordinates of the third vertex C of the triangle.
We have to find the co-ordinates of the third vertex of the given triangle. Let the co-ordinates of the third vertex be.
The co-ordinates of other two vertices are A (3, 2) and C (−2, 1)
The co-ordinate of the centroid is
We know that the co-ordinates of the centroid of a triangle whose vertices are is-
So,
Compare individual terms on both the sides-
So,
Similarly,
So,
So the co-ordinate of third vertex
#### Question 5:
If (−2, 3), (4, −3) and (4, 5) are the mid-points of the sides of a triangle, find the coordinates of its centroid.
Letbe ant triangle such that P (−2, 3); Q (4,−3) and R (4, 5) are the mid-points of the sides AB, BC, CA respectively.
We have to find the co-ordinates of the centroid of the triangle.
Let the vertices of the triangle be
In general to find the mid-point of two pointsand we use section formula as,
So, co-ordinates of P,
Equate the x component on both the sides to get,
…… (1)
Similarly,
…… (2)
Similarly, co-ordinates of Q,
Equate the x component on both the sides to get,
…… (3)
Similarly,
…… (4)
Similarly, co-ordinates of R,
Equate the x component on both the sides to get,
…… (5)
Similarly,
…… (6)
Add equation (1) (3) and (5) to get,
Similarly, add equation (2) (4) and (6) to get,
We know that the co-ordinates of the centroid G of a triangle whose vertices are is-
So, centroid G of a triangle is,
#### Question 6:
Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.
Letbe any triangle such that O is the origin and the other co-ordinates are. P and R are the mid-points of the sides OA and OB respectively.
We have to prove that line joining the mid-point of any two sides of a triangle is equal to half of the third side which means,
In general to find the mid-point of two pointsand we use section formula as,
So,
Co-ordinates of P is,
Similarly, co-ordinates of R is,
In general, the distance between A and B is given by,
Similarly,
Hence,
#### Question 7:
Prove that the lines joining the middle points of the opposite sides of a quadrilateral and the join of the middle points of its diagonals meet in a point and bisect one another.
Let us consider a Cartesian plane having a parallelogram OABC in which O is the origin.
We have to prove that middle point of the opposite sides of a quadrilateral and the join of the mid-points of its diagonals meet in a point and bisect each other.
Let the co-ordinate of A be. So the coordinates of other vertices of the quadrilateral are- O (0, 0); B; C
Let P, Q, R and S be the mid-points of the sides AB, BC, CD, DA respectively.
In general to find the mid-point of two pointsand we use section formula as,
So co-ordinate of point P,
Similarly co-ordinate of point Q,
Similarly co-ordinate of point R,
Similarly co-ordinate of point S,
Let us find the co-ordinates of mid-point of PR as,
Similarly co-ordinates of mid-point of QS as,
Now the mid-point of diagonal AC,
Similarly the mid−point of diagonal OA,
Hence the mid-points of PR, QS, AC and OA coincide.
Thus, middle point of the opposite sides of a quadrilateral and the join of the mid-points of its diagonals meet in a point and bisect each other.
#### Question 8:
If G be the centroid of a triangle ABC and P be any other point in the plane, prove that PA2 + PB2 + PC2 = GA2 + GB2 + GC2 + 3GP2.
Letbe any triangle whose coordinates are. Let P be the origin and G be the centroid of the triangle.
We have to prove that,
…… (1)
We know that the co-ordinates of the centroid G of a triangle whose vertices are is−
In general, the distance between A and B is given by,
So,
Now,
So we get the value of left hand side of equation (1) as,
Similarly we get the value of right hand side of equation (1) as,
Hence,
#### Question 9:
If G be the centroid of a triangle ABC, prove that:
AB2 + BC2 + CA2 = 3 (GA2 + GB2 + GC2)
#### Question 10:
In Fig. 14.36, a right triangle BOA is given C is the mid-point of the hypotenuse AB. Show that it is equidistant from the vertices O, A and B.
We have a right angled triangle, right angled at O. Co-ordinates are B (0,2b); A (2a, 0) and C (0, 0).
We have to prove that mid-point C of hypotenuse AB is equidistant from the vertices.
In general to find the mid-point of two pointsand we use section formula as,
So co-ordinates of C is,
In general, the distance between A and B is given by,
So,
Hence, mid−point C of hypotenuse AB is equidistant from the vertices.
#### Question 1:
On which axis do the following points lie?
(a) P(5, 0)
(b) Q(0−2)
(c) R(−4,0)
(d) S(0,5)
According to the Rectangular Cartesian Co-ordinate system of representing a point (x, y),
If then the point lies in the 1st quadrant
If then the point lies in the 2nd quadrant
If then the point lies in the 3rd quadrant
If then the point lies in the 4th quadrant
But in case
If then the point lies on the y-axis
If then the point lies on the x-axis
(i) Here the point is given to be P (5, 0). Comparing this with the standard form of
(x, y) we have
Here we see that
Hence the given point lies on the
(ii) Here the point is given to be Q (0, -2). Comparing this with the standard form of (x, y) we have
Here we see that
Hence the given point lies on the
(iii) Here the point is given to be R (-4, 0). Comparing this with the standard form of (x, y) we have
Here we see that
Hence the given point lies on the
(iv) Here the point is given to be S (0, 5). Comparing this with the standard form of (x, y) we have
Here we see that
Hence the given point lies on the
#### Question 2:
Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when
(i) A coincides with the origin and AB and AD are along OX and OY respectively.
(ii) The centre of the square is at the origin and coordinate axes are parallel to the sides AB and AD respectively.
The distance between any two adjacent vertices of a square will always be equal. This distance is nothing but the side of the square.
Here, the side of the square ‘ABCD’ is given to be ‘2a’.
(i) Since it is given that the vertex ‘A’ coincides with the origin we know that the co-ordinates of this point is (0, 0).
We also understand that the side ‘AB’ is along the x-axis. So, the vertex ‘B’ has got to be at a distance of ‘2a’ from ‘A’.
Hence the vertex ‘B’ has the co-ordinates (2a, 0).
Also it is said that the side ‘AD’ is along the y-axis. So, the vertex ‘D’ it has got to be at a distance of ‘2a’ from ‘A’.
Hence the vertex ‘D’ has the co-ordinates (0, 2a)
Finally we have vertex ‘C’ at a distance of ‘2a’ both from vertex ‘B’ as well as ‘D’.
Hence the vertex of ‘C’ has the co-ordinates (2a, 2a)
So, the co-ordinates of the different vertices of the square are
(ii) Here it is said that the centre of the square is at the origin and that the sides of the square are parallel to the axes.
Moving a distance of half the side of the square in either the ‘upward’ or ‘downward’ direction and also along either the ‘right’ or ‘left’ direction will give us all the four vertices of the square.
Half the side of the given square is ‘a’.
The centre of the square is the origin and its vertices are (0, 0). Moving a distance of ‘a’ to the right as well as up will lead us to the vertex ‘A’ and it will have vertices (a, a).
Moving a distance of ‘a’ to the left as well as up will lead us to the vertex ‘B’ and it will have vertices (-(a, a).
Moving a distance of ‘a’ to the left as well as down will lead us to the vertex ‘C’ and it will have vertices (-(a, -a).
Moving a distance of ‘a’ to the right as well as down will lead us to the vertex ‘D’ and it will have vertices (a,-,−a).
So, the co-ordinates of the different vertices of the square are
#### Question 3:
The base PQ of two equilateral triangles PQR and PQR' with side 2a lies along y-axis such that the mid-point of PQ is at the origin. Find the coordinates of the vertices R and R' of the triangles.
In an equilateral triangle, the height ‘h’ is given by
Here it is given that ‘PQ’ forms the base of two equilateral triangles whose side measures ‘2a’ units.
The height of these two equilateral triangles has got to be
In an equilateral triangle the height drawn from one vertex meets the midpoint of the side opposite this vertex.
So here we have ‘PQ’ being the base lying along the y-axis with its midpoint at the origin, that is at (0, 0).
So the vertices ‘R’ and ‘R’’ will lie perpendicularly to the y-axis on either sides of the origin at a distance of ‘’ units.
Hence the co-ordinates of ‘R’ and ‘R’’ are
#### Question 1:
Find the area of a triangle whose vertices are
(i) (6, 3) (−3, 5) and (4, −2)
(ii)
(iii) (a, c + a), (a, c) and (−a, c − a)
We know area of triangle formed by three points is given by
(i) The vertices are given as (6, 3), (−3, 5), (4, −2).
(ii) The vertices are given as .
(iii)
The vertices are given as .
#### Question 2:
Find the area of the quadrilaterals, the coordinates of whose vertices are
(i) (−3, 2), (5, 4), (7, −6) and (−5, −4)
(ii) (1, 2), (6, 2), (5, 3) and (3, 4)
(iii) (−4, −2, (−3, −5), (3, −2), (2, 3)
(i)
Let the vertices of the quadrilateral be A (−3, 2), B (5, 4), C (7, −6), and D (−5, −4). Join AC to form two triangles ΔABC and ΔACD.
(ii)
Let the vertices of the quadrilateral be A (1, 2), B (6, 2), C (5, 3), and D (3, 4). Join AC to form two triangles ΔABC and ΔACD.
(iii)
Let the vertices of the quadrilateral be A (−4, −2), B (−3, −5), C (3, −2), and D (2, 3). Join AC to form two triangles ΔABC and ΔACD.
#### Question 3:
The four vertices of a quadrilateral are (1, 2), (−5, 6), (7, −4) and (k, −2) taken in order. If the area of the quadrilateral is zero, find the value of k.
GIVEN: The four vertices of quadrilateral are (1, 2), (−5, 6), (7, −4) and D (k, −2) taken in order. If the area of the quadrilateral is zero
TO FIND: value of k
PROOF: Let four vertices of quadrilateral are A (1, 2) and B (−5, 6) and C (7, −4) and D (k, −2)
We know area of triangle formed by three points is given by
Now Area of ΔABC
Taking three points when A (1, 2) and B (−5, 6) and C (7, −4)
Also,
Now Area of ΔACD
Taking three points when A (1, 2) and C (7, −4) and D (k, −2)
Hence
#### Question 4:
The vertices of ΔABC are (−2, 1), (5, 4) and (2, −3) respectively. Find the area of the triangle and the length of the altitude through A.
GIVEN: The vertices of triangle ABC are A (−2, 1) and B (5, 4) and C (2, −3)
TO FIND: The area of triangle ABC and length if the altitude through A
PROOF: We know area of triangle formed by three points is given by
Now Area of ΔABC
Taking three points A (−2, 1) and B (5, 4) and C(2, −3)
We have
Now,
#### Question 5:
Show that the following sets of points are collinear.
(a) (2, 5), (4, 6) and (8, 8)
(b) (1, −1), (2, 1) and (4, 5)
The formula for the area ‘A’ encompassed by three points, and is given by the formula,
We know area of triangle formed by three points is given by
If three points are collinear the area encompassed by them is equal to 0.
The three given points are A(2, 5), B(4, 6) and C(8, 8). Substituting these values in the earlier mentioned formula we have,
Since the area enclosed by the three points is equal to 0, the three points need to be.
The three given points are A(1, −1), B(2, 1) and C(4, 5). Substituting these values in the earlier mentioned formula we have,
Since the area enclosed by the three points is equal to 0, the three points need to be.
#### Question 6:
Q
Let the vertices of the quadrilateral be A (−3, 2), B (5, 4), C (7, −6), and D (−5, −4). Join AC to form two triangles ΔABC and ΔACD.
#### Question 7:
In $∆$ABC , the coordinates of vertex A are (0, $-$1) and D (1,0) and E(0,10) respectively the mid-points of the sides AB and AC . If F is the mid-points of the side BC , find the area of $∆$ DEF.
Let the coordinates of B and C be and , respectively.
D is the midpoint of AB.
So,
Thus, the coordinates of B are (2, 1).
Similarly, E is the midpoint of AC.
So,
Thus, the coordinates of C are (0, 3).
Also, F is the midpoint of BC. So, its coordinates are
Now,
Area of a triangle = $\frac{1}{2}\left[{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)\right]$
Thus, the area of $∆$ABC is
And the area of $∆$DEF is
#### Question 8:
Find the area of the triangle PQR with Q(3, 2) and the mid-points of the sides through Q being (2, −1) and (1, 2). [CBSE 2015]
Let P(x1, y1), Q(3, 2) and R(x2, y2) be the vertices of the ∆PQR.
Suppose S(2, −1) and T(1, 2) be the mid-points of sides QR and PQ, respectively.
Using mid-point formula, we have
∴ Coordinates of P = (−1, 2)
Also,
∴ Coordinates of R = (1, −4)
So, P(−1, 2), Q(3, 2) and R(1, −4) are the vertices of ∆PQR.
Hence, the area of the triangle is 12 square units.
#### Question 9:
If P(−5, −3), Q(−4, −6), R(2, −3) and S(1, 2) are the vertices of a quadrilateral PQRS, find its area. [CBSE 2015]
It is given that P(−5, −3), Q(−4, −6), R(2, −3) and S(1, 2) are the vertices of a quadrilateral PQRS.
Area of the quadrilateral PQRS = Area of ∆PQR + Area of ∆PRS
∴ Area of the quadrilateral PQRS = $\frac{21}{2}+\frac{35}{2}=\frac{56}{2}=28$ square units
Hence, the area of the given quarilateral is 28 square units.
#### Question 10:
If A(−3, 5), B(−2, −7), C(1, −8) and D(6, 3) are the vertices of a quadrilateral ABCD, find its area. [CBSE 2014]
It is given that A(−3, 5), B(−2, −7), C(1, −8) and D(6, 3) are the vertices of a quadrilateral ABCD.
Area of the quadrilateral ABCD = Area of ∆ABC + Area of ∆ACD
∴ Area of the quadrilateral ABCD = $\frac{35}{2}+\frac{109}{2}=\frac{144}{2}=72$ square units
Hence, the area of the given quadrilateral is 72 square units.
#### Question 11:
For what value of a the point (a, 1), (1, −1), and (11, 4) are collinear?
The formula for the area ‘A’ encompassed by three points, and is given by the formula,
$∆=\frac{1}{2}\left|\left({x}_{1}{y}_{2}+{x}_{2}{y}_{3}+{x}_{3}{y}_{1}\right)-\left({x}_{2}{y}_{1}+{x}_{3}{y}_{2}+{x}_{1}{y}_{3}\right)\right|$
If three points are collinear the area encompassed by them is equal to 0.
The three given points are A(a, 1), B(1, −1) and C(11, 4). It is also said that they are collinear and hence the area enclosed by them should be 0.
Hence the value of ‘a’ for which the given points are collinear is.
#### Question 12:
Prove that the points (a, b), (a1, b1) and (a −a1, b −b1) are collinear if ab1 = a1b.
The formula for the area ‘A’ encompassed by three points, and is given by the formula,
$∆=\frac{1}{2}\left|\left({x}_{1}{y}_{2}+{x}_{2}{y}_{3}+{x}_{3}{y}_{1}\right)-\left({x}_{2}{y}_{1}+{x}_{3}{y}_{2}+{x}_{1}{y}_{3}\right)\right|$
If three points are collinear the area encompassed by them is equal to 0.
The three given points are, and. If they are collinear then the area enclosed by them should be 0.
Hence we have proved that for the given conditions to be satisfied we need to have.
#### Question 13:
If the vertices of a triangle are (1, −3), (4, p) and (−9, 7) and its area is 15 sq. units, find the value(s) of p. [CBSE 2012]
Let A(1, −3), B(4, p) and C(−9, 7) be the vertices of the ∆ABC.
Here, x1 = 1, y1 = −3; x2 = 4, y2 = p and x3 = −9, y3 = 7
ar(∆ABC) = 15 square units
$⇒\frac{1}{2}\left|{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)\right|=15\phantom{\rule{0ex}{0ex}}⇒\frac{1}{2}\left|1\left(p-7\right)+4\left[7-\left(-3\right)\right]+\left(-9\right)\left(-3-p\right)\right|=15\phantom{\rule{0ex}{0ex}}⇒\frac{1}{2}\left|p-7+40+27+9p\right|=15\phantom{\rule{0ex}{0ex}}⇒\left|10p+60\right|=30$
$⇒10p+60=30$ or $10p+60=-30$
$⇒10p=-30$ or $10p=-90$
$⇒p=-3$ or $p=-9$
Hence, the value of p is −3 or −9.
#### Question 14:
If (x, y) be on the line joining the two points (1, −3) and (−4, 2) , prove that x + y + 2= 0.
Since the point (x, y) lie on the line joining the points (1, −3) and (−4, 2); the area of triangle formed by these points is 0.
That is,
Thus, the result is proved.
#### Question 15:
Find the value of k if points A(k, 3), B(6, −2) and C(−3, 4) are collinear.
The formula for the area ‘A’ encompassed by three points, and is given by the formula,
$∆=\frac{1}{2}\left|\left({x}_{1}{y}_{2}+{x}_{2}{y}_{3}+{x}_{3}{y}_{1}\right)-\left({x}_{2}{y}_{1}+{x}_{3}{y}_{2}+{x}_{1}{y}_{3}\right)\right|$
If three points are collinear the area encompassed by them is equal to 0.
The three given points are A(k, 3), B(6, −2) and C(3, 4). It is also said that they are collinear and hence the area enclosed by them should be 0.
Hence the value of ‘k’ for which the given points are collinear is.
#### Question 16:
Find the value of k, if the points A(7, −2), B (5, 1) and C (3, 2k) are collinear.
The formula for the area ‘A’ encompassed by three points, and is given by the formula,
$∆=\frac{1}{2}\left|\left({x}_{1}{y}_{2}+{x}_{2}{y}_{3}+{x}_{3}{y}_{1}\right)-\left({x}_{2}{y}_{1}+{x}_{3}{y}_{2}+{x}_{1}{y}_{3}\right)\right|$
If three points are collinear the area encompassed by them is equal to 0.
The three given points are A(7, −2), B(5, 1) and C(3, 2k). It is also said that they are collinear and hence the area enclosed by them should be 0.
Hence the value of ‘k’ for which the given points are collinear is.
#### Question 17:
If the point P (m, 3) lies on the line segment joining the points and B (2, 8), find the value of m.
The formula for the area ‘A’ encompassed by three points, and is given by the formula,
$∆=\frac{1}{2}\left|\left({x}_{1}{y}_{2}+{x}_{2}{y}_{3}+{x}_{3}{y}_{1}\right)-\left({x}_{2}{y}_{1}+{x}_{3}{y}_{2}+{x}_{1}{y}_{3}\right)\right|$
If three points are collinear the area encompassed by them is equal to 0.
It is said that the point P(m,3) lies on the line segment joining the points and B(2,8). Hence we understand that these three points are collinear. So the area enclosed by them should be 0.
Hence the value of ‘m’ for which the given condition is satisfied is.
#### Question 18:
If R (x, y) is a point on the line segment joining the points P (a, b) and Q (b, a), then prove that x + y = a + b.
The formula for the area ‘A’ encompassed by three points, and is given by the formula,
$∆=\frac{1}{2}\left|\left({x}_{1}{y}_{2}+{x}_{2}{y}_{3}+{x}_{3}{y}_{1}\right)-\left({x}_{2}{y}_{1}+{x}_{3}{y}_{2}+{x}_{1}{y}_{3}\right)\right|$
If three points are collinear the area encompassed by them is equal to 0.
It is said that the point R(x, y) lies on the line segment joining the points P(a, b) and Q(b, a). Hence we understand that these three points are collinear. So the area enclosed by them should be 0.
Hence under the given conditions we have proved that.
#### Question 19:
Find the value of k, if the points A (8, 1) B(3, −4) and C(2, k) are collinear.
The formula for the area ‘A’ encompassed by three points, and is given by the formula,
$∆=\frac{1}{2}\left|\left({x}_{1}{y}_{2}+{x}_{2}{y}_{3}+{x}_{3}{y}_{1}\right)-\left({x}_{2}{y}_{1}+{x}_{3}{y}_{2}+{x}_{1}{y}_{3}\right)\right|$
If three points are collinear the area encompassed by them is equal to 0.
The three given points are A(8,1), B(3,4) and C(2,k). It is also said that they are collinear and hence the area enclosed by them should be 0.
Hence the value of ‘k’ for which the given points are collinear is.
#### Question 20:
Find the value of a for which the area of the triangle formed by the points A(a, 2a), B(−2, 6) and C(3, 1) is 10 square units.
The formula for the area ‘A’ encompassed by three points, and is given by the formula,
$∆=\frac{1}{2}\left|\left({x}_{1}{y}_{2}+{x}_{2}{y}_{3}+{x}_{3}{y}_{1}\right)-\left({x}_{2}{y}_{1}+{x}_{3}{y}_{2}+{x}_{1}{y}_{3}\right)\right|$
The three given points are A(a,2a), B(2,6) and C(3,1). It is also said that the area enclosed by them is 10 square units. Substituting these values in the above mentioned formula we have,
$∆=\frac{1}{2}\left|\left(a×6+\left(-2\right)×1+3×2a\right)-\left(\left(-2\right)×2a+3×6+a×1\right)\right|\phantom{\rule{0ex}{0ex}}10=\frac{1}{2}\left|\left(6a-2+6a\right)-\left(-4a+18+a\right)\right|\phantom{\rule{0ex}{0ex}}10=\frac{1}{2}\left|15a-20\right|\phantom{\rule{0ex}{0ex}}20=\left|15a-20\right|\phantom{\rule{0ex}{0ex}}4=\left|3a-4\right|$
We have. Hence either
Or
Hence the values of ‘a’ which satisfies the given conditions are.
#### Question 21:
If $a\ne b\ne 0$, prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.
Let A(a, a2), B(b, b2) and C(0, 0) be the coordinates of the given points.
We know that the area of triangle having vertices is $\left|\frac{1}{2}\left[{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)\right]\right|$ square units.
So,
Area of ∆ABC
Since the area of the triangle formed by the points (a, a2), (b, b2) and (0, 0) is not zero, so the given points are not collinear.
#### Question 22:
The area of a triangle is 5 sq units. Two of its vertices are (2, 1) and (3, –2). If the third vertex is find the value of y.
Let
Now
Hence, y =
#### Question 23:
Prove that the points (a, 0), (0, b) and (1, 1) are collinear if $\frac{1}{a}+\frac{1}{b}=1$.
The formula for the area ‘A’ encompassed by three points, and is given by the formula,
We know area of triangle formed by three points is given by
If three points are collinear the area encompassed by them is equal to 0.
The three given points are A(a,0), B(0,b) and C(1,1).
It is given that
So we have,
Using this in the previously arrived equation for area we have,
Since the area enclosed by the three points is equal to 0, the three points need to be.
#### Question 24:
The point A divides the join of P (−5, 1) and Q(3, 5) in the ratio k:1. Find the two values of k for which the area of ΔABC where B is (1, 5) and C(7, −2) is equal to 2 units.
GIVEN: point A divides the line segment joining P (−5, 1) and Q (3, −5) in the ratio k: 1
Coordinates of point B (1, 5) and C (7, −2)
TO FIND: The value of k
PROOF: point A divides the line segment joining P (−5, 1) and Q (3, −5) in the ratio k: 1
So the coordinates of A are
We know area of triangle formed by three points is given by $∆=\frac{1}{2}\left|\left({x}_{1}{y}_{2}+{x}_{2}{y}_{3}+{x}_{3}{y}_{1}\right)-\left({x}_{2}{y}_{1}+{x}_{3}{y}_{2}+{x}_{1}{y}_{3}\right)\right|$
Now Area of ΔABC= 2 sq units.
Taking three points A , B (1, 5) and C (7, −2)
Hence
#### Question 25:
The area of a triangle is 5. Two of its vertices are (2, 1) and (3, −2). The third vertex lies on y = x + 3. Find the third vertex.
GIVEN: The area of triangle is 5.Two of its vertices are (2, 1) and (3, −2). The third vertex lies on y = x+3
TO FIND: The third vertex.
PROOF: Let the third vertex be (x, y)
We know area of triangle formed by three points is given by $∆=\frac{1}{2}\left|\left({x}_{1}{y}_{2}+{x}_{2}{y}_{3}+{x}_{3}{y}_{1}\right)-\left({x}_{2}{y}_{1}+{x}_{3}{y}_{2}+{x}_{1}{y}_{3}\right)\right|$
Now
Taking three points(x, y), (2, 1) and (3, −2)
Also it is given the third vertex lies on y = x+3
Substituting the value in equation (1) and (2) we get
Hence the coordinates of and
#### Question 26:
If , prove that the points (a, a2), (b, b2), (c, c2) can never be collinear.
GIVEN: If
TO PROVE: that the points can never be collinear.
PROOF:
We know three points are collinear when
Now taking three points
Also it is given that
Hence area of triangle made by these points is never zero. Hence given points are never collinear.
#### Question 27:
Four points A (6, 3), B (−3, 5), C(4, −2) and D (x, 3x) are given in such a way that find x.
GIVEN: four points A (6, 3), B (−3, 5) C (4, −2) and D(x, 3x) such that
TO FIND: the value of x
PROOF:
We know area of the triangles formed by three points is given by
Now
Area of triangle DBC taking D(x, 3x), B (−3, 5), C (4, −2)
Area of triangle ABC taking, A (6, 3), B (−3, 5), C (4, −2)
Also it is given that
Substituting the values from (1) and (2) we get
#### Question 28:
If three points (x1, y1) (x2, y2), (x3, y3) lie on the same line, prove that
$\frac{{y}_{2}-{y}_{3}}{{x}_{2}{x}_{3}}+\frac{{y}_{3}-{y}_{1}}{{x}_{3}{x}_{1}}+\frac{{y}_{1}-{y}_{2}}{{x}_{1}{x}_{2}}=0$
GIVEN: If three points lie on the same line
TO PROVE:
PROOF:
We know that three points are collinear if
Hence proved.
#### Question 29:
Find the area of a parallelogram ABCD if three of its vertices are A(2, 4), B(2 + $\sqrt{3}$, 5) and C(2, 6). [CBSE 2013]
It is given that A(2, 4), B(2 + $\sqrt{3}$, 5) and C(2, 6) are the vertices of the parallelogram ABCD.
We know that the diagonal of a parallelogram divides it into two triangles having equal area.
∴ Area of the parallogram ABCD = 2 × Area of the ∆ABC
Now,
∴ Area of the parallogram ABCD = 2 × Area of the ∆ABC = 2 × $\sqrt{3}$ = 2$\sqrt{3}$ square units
Hence, the area of given parallelogram is 2$\sqrt{3}$ square units.
#### Question 30:
Find the value(s) of k for which the points (3k − 1, k − 2), (k, k − 7) and (k − 1, −k − 2) are collinear. [CBSE 2014]
Let A(3k − 1, k − 2), B(k, k − 7) and C(k − 1, −k − 2) be the given points.
The given points are collinear. Then,
$\mathrm{ar}\left(∆\mathrm{ABC}\right)=0\phantom{\rule{0ex}{0ex}}⇒\frac{1}{2}\left|{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)\right|=0\phantom{\rule{0ex}{0ex}}⇒{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)=0\phantom{\rule{0ex}{0ex}}$
$⇒\left(3k-1\right)\left[\left(k-7\right)-\left(-k-2\right)\right]+k\left[\left(-k-2\right)-\left(k-2\right)\right]+\left(k-1\right)\left[\left(k-2\right)-\left(k-7\right)\right]=0\phantom{\rule{0ex}{0ex}}⇒\left(3k-1\right)\left(2k-5\right)+k\left(-2k\right)+5\left(k-1\right)=0\phantom{\rule{0ex}{0ex}}⇒6{k}^{2}-17k+5-2{k}^{2}+5k-5=0\phantom{\rule{0ex}{0ex}}⇒4{k}^{2}-12k=0$
Hence, the value of k is 0 or 3.
#### Question 31:
If the points A(−1, −4), B(b, c) and C(5, −1) are collinear and 2b + c = 4, find the values of b and c. [CBSE 2014]
The given points A(−1, −4), B(b, c) and C(5, −1) are collinear.
$\therefore \mathrm{ar}\left(∆\mathrm{ABC}\right)=0\phantom{\rule{0ex}{0ex}}⇒\frac{1}{2}\left|{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)\right|=0\phantom{\rule{0ex}{0ex}}⇒{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)=0\phantom{\rule{0ex}{0ex}}$
Also, it is given that
2b + c = 4 .....(2)
Solving (1) and (2), we get
$2\left(7+2c\right)+c=4\phantom{\rule{0ex}{0ex}}⇒14+4c+c=4\phantom{\rule{0ex}{0ex}}⇒5c=-10\phantom{\rule{0ex}{0ex}}⇒c=-2$
Putting c = −2 in (1), we get
$b-2×\left(-2\right)=7\phantom{\rule{0ex}{0ex}}⇒b=7-4=3$
Hence, the respective values of b and c are 3 and −2.
#### Question 32:
If the points A(−2, 1), B(a, b) and C(4, −1) ae collinear and ab = 1, find the values of a and b. [CBSE 2014]
The given points A(−2, 1), B(a, b) and C(4, −1) are collinear.
$\therefore \mathrm{ar}\left(∆\mathrm{ABC}\right)=0\phantom{\rule{0ex}{0ex}}⇒\frac{1}{2}\left|{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)\right|=0\phantom{\rule{0ex}{0ex}}⇒{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)=0\phantom{\rule{0ex}{0ex}}$
Also, it is given that
ab = 1 .....(2)
Solving (1) and (2), we get
$b+1+3b=1\phantom{\rule{0ex}{0ex}}⇒4b=0\phantom{\rule{0ex}{0ex}}⇒b=0$
Putting b = 0 in (1), we get
$a+3×0=1\phantom{\rule{0ex}{0ex}}⇒a=1$
Hence, the respective values of a and b are 1 and 0.
#### Question 33:
If the points form a parallelogram , find the value of and height of the parallelogram taking AB as base .
Since diagonals of a parallelogram bisect eachother.
Coordinates of the midpoint of AC = coordinates of the midpoint of BD.
Since, ABCD is a parallelogram,
Area of ABCD = 2 × area of triangle ABC = 2 × 12 = 24 sq. units
Height of the parallelogram is area of the parallelogram divided by its base.
Base AB is
#### Question 34:
are three vertices of a parallelogram ABCD . If E is the mid-point of DC , find the area of $∆$ ADE.
Three vertices are given, then D can be calulated and it comes out to be (7, 3).
Since, E is midpoint of BD.
Therefore, coordinates of E are $\left(\frac{15}{2},\frac{5}{2}\right)$.
Now, vertices of triangle ABE rae (6, 1), (8, 2) and $\left(\frac{15}{2},\frac{5}{2}\right)$.
#### Question 35:
If are the mid-points of sides of , find the area of .
The midpoint of BC is $D\left(-\frac{1}{2},\frac{5}{2}\right)$,
The midpoint of AB is $F\left(\frac{7}{2},\frac{7}{2}\right)$,
The midpoint of AC is $E\left(7,3\right)$,
Consider the line segment BC,
Solve (i), (ii) and (iii) to get
Let us assume that BC is base of the triangle,
The area of the triangle is
#### Question 1:
Write the distance between the points A (10 cos θ, 0) and B (0, 10 sin θ).
We have to find the distance between A and B.
In general, the distance between A and B is given by,
So,
But according to the trigonometric identity,
Therefore,
#### Question 2:
Write the perimeter of the triangle formed by the points O (0, 0), A (a, 0) and B (0, b).
The distance d between two points and is given by the formula
The perimeter of a triangle is the sum of lengths of its sides.
The three vertices of the given triangle are O(0, 0), A(a, 0) and B(0, b).
Let us now find the lengths of the sides of the triangle.
The perimeter ‘P’ of the triangle is thus,
Thus the perimeter of the triangle with the given vertices is.
#### Question 3:
Write the ratio in which the line segment joining points (2, 3) and (3, −2) is divided by X axis.
Let P be the point of intersection of x-axis with the line segment joining A (2, 3) and B (3,−2) which divides the line segment AB in the ratio.
Now according to the section formula if point a point P divides a line segment joining andin the ratio m: n internally than,
Now we will use section formula as,
Now equate the y component on both the sides,
On further simplification,
So x-axis divides AB in the ratio
#### Question 4:
What is the distance between the points (5 sin 60°, 0) and (0, 5 sin 30°)?
We have to find the distance between A and B.
In general, the distance between A and B is given by,
So,
But according to the trigonometric identity,
And,
Therefore,
#### Question 5:
If A (−1, 3) , B(1, −1) and C (5, 1) are the vertices of a triangle ABC, what is the length of the median through vertex A?
We have a triangle in which the co-ordinates of the vertices are A (−1, 3) B (1,−1) and
C (5, 1). In general to find the mid-point of two pointsand we use section formula as,
Therefore mid-point D of side BC can be written as,
Now equate the individual terms to get,
So co-ordinates of D is (3, 0)
So the length of median from A to the side BC,
#### Question 6:
If the distance between points (x, 0) and (0, 3) is 5, what are the values of x?
We have to find the unknown x using the distance between A and B which is 5.In general, the distance between A and B is given by,
So,
Squaring both the sides we get,
So,
#### Question 7:
What is the area of the triangle formed by the points O (0, 0), A (6, 0) and B (0, 4)?
The given triangle is a right angled triangle, right angled at O. the co-ordinates of the vertices are O (0, 0) A (6, 0) and B (0, 4).
So,
Altitude is 6 units and base is 4 units.
Therefore,
#### Question 8:
Write the coordinates of the point dividing line segment joining points (2, 3) and (3, 4) internally in the ratio 1 : 5.
Let P be the point which divide the line segment joining A (2, 3) and B (3, 4) in the ratio 1: 5.
Now according to the section formula if point a point P divides a line segment joining andin the ratio m: n internally than,
Now we will use section formula as,
So co-ordinate of P is
#### Question 9:
If the centroid of the triangle formed by points P (a, b), Q(b, c) and R (c, a) is at the origin, what is the value of a + b + c?
The co-ordinates of the vertices are (a, b); (b, c) and (c, a)
The co-ordinate of the centroid is (0, 0)
We know that the co-ordinates of the centroid of a triangle whose vertices are is-
So,
Compare individual terms on both the sides-
Therefore,
#### Question 10:
In Q. No. 9, what is the value of $\frac{{a}^{2}}{bc}+\frac{{b}^{2}}{ca}+\frac{{c}^{2}}{ab}$?
The co-ordinates of the vertices are (a, b); (b, c) and (c, a)
The co-ordinate of the centroid is (0, 0)
We know that the co-ordinates of the centroid of a triangle whose vertices are is-
So,
Compare individual terms on both the sides-
Therefore,
We have to find the value of -
Multiply and divide it by to get,
Now as we know that if,
Then,
So,
#### Question 11:
Write the coordinates of a point on X-axis which is equidistant from the points (−3, 4) and (2, 5).
The distance d between two points and is given by the formula
Here we are to find out a point on the x−axis which is equidistant from both the points
A(-3,4) and B(2,5).
Let this point be denoted as C(x, y).
Since the point lies on the x-axis the value of its ordinate will be 0. Or in other words we have.
Now let us find out the distances from ‘A’ and ‘B’ to ‘C’
We know that both these distances are the same. So equating both these we get,
Squaring on both sides we have,
Hence the point on the x-axis which lies at equal distances from the mentioned points is.
#### Question 12:
If the mid-point of the segment joining A (x, y + 1) and B (x + 1, y + 2) is C $\left(\frac{3}{2},\frac{5}{2}\right)$, find x, y.
It is given that mid-point of line segment joining A and B is C
In general to find the mid-point of two pointsand we use section formula as,
So,
Now equate the components separately to get,
So,
Similarly,
So,
#### Question 13:
Two vertices of a triangle have coordinates (−8, 7) and (9, 4) . If the centroid of the triangle is at the origin, what are the coordinates of the third vertex?
We have to find the co-ordinates of the third vertex of the given triangle. Let the co-ordinates of the third vertex be.
The co-ordinates of other two vertices are (−8, 7) and (9, 4)
The co-ordinate of the centroid is (0, 0)
We know that the co-ordinates of the centroid of a triangle whose vertices are is-
So,
Compare individual terms on both the sides-
So,
Similarly,
So,
So the co-ordinate of third vertex
#### Question 14:
Write the coordinates the reflections of points (3, 5) in X and Y -axes.
We have to find the reflection of (3, 5) along x-axis and y-axis.
Reflection of any pointalong x-axis is
So reflection of (3, 5) along x-axis is
Similarly, reflection of any pointalong y-axis is
So, reflection of (3, 5) along y-axis is
#### Question 15:
If points Q and reflections of point P (−3, 4) in X and Y axes respectively, what is QR?
We have to find the reflection of (−3, 4) along x-axis and y-axis.
Reflection of any pointalong x-axis is
So reflection of (−3, 4) along x-axis is
Similarly, reflection of any pointalong y-axis is
So, reflection of (−3, 4) along y-axis is
Therefore,
#### Question 16:
Write the formula for the area of the triangle having its vertices at (x1, y1), (x2, y2) and (x3, y3).
The formula for the area ‘A’ encompassed by three points, and is given by the formula,
The area ‘A’ encompassed by three points, and is also given by the formula,
#### Question 17:
Write the condition of collinearity of points (x1, y1), (x2, y2) and (x3, y3).
The condition for co linearity of three points, and is that the area enclosed by them should be equal to 0.
The formula for the area ‘A’ encompassed by three points, and is given by the formula,
Thus for the three points to be collinear we need to have,
The area ‘A’ encompassed by three points, and is also given by the formula,
Thus for the three points to be collinear we can also have,
#### Question 18:
Find the values of x for which the distance between the point P(2, −3), and Q (x, 5) is 10.
It is given that distance between P (2,−3) and is 10.
In general, the distance between A and B is given by,
So,
On further simplification,
#### Question 19:
Write the ratio in which the line segment doining the points A (3, −6), and B (5, 3) is divided by X-axis.
Let P be the point of intersection of x-axis with the line segment joining A (3,−6) and B (5, 3) which divides the line segment AB in the ratio.
Now according to the section formula if point a point P divides a line segment joining andin the ratio m: n internally than,
Now we will use section formula as,
Now equate the y component on both the sides,
On further simplification,
So x-axis divides AB in the ratio 2:1.
#### Question 20:
Find the distance between the points $\left(-\frac{8}{5},2\right)$ and $\left(\frac{2}{5},2\right)$
We have to find the distance between and .
In general, the distance between A and B is given by,
So,
#### Question 21:
Find the value of a so that the point (3, a) lies on the line represented by 2x − 3y + 5 = 0
If a point is said lie on a line represented by, then the given equation of the line should hold true when the values of the co-ordinates of the points are substituted in it.
Here it is said that the point (3, a) lies on the line represented by the equation.
Substituting the co-ordinates of the values in the equation of the line we have,
Thus the value of ‘a’ satisfying the given conditions is.
#### Question 22:
What is the distance between the points A (c, 0) and B (0, −c)?
We have to find the distance between and .
In general, the distance between A and B is given by,
So,
#### Question 23:
If P (2, 6) is the mid-point of the line segment joining A (6, 5) and B (4, y), find y.
It is given that mid-point of line segment joining A (6, 5) and B (4, y) is P (2, 6)
In general to find the mid-point of two pointsand we use section formula as,
So,
Now equate the y component to get,
So,
#### Question 24:
If the distance between the points (3, 0) and (0, y) is 5 units and y is positive. then what is the value of y?
It is given that distance between P (3, 0) and is 5.
In general, the distance between A and B is given by,
So,
On further simplification,
We will neglect the negative value. So,
#### Question 25:
If P (x, 6) is the mid-point of the line segment joining A (6, 5) and B (4, y), find y.
It is given that mid-point of line segment joining A (6, 5) and B (4, y) is
In general to find the mid-point of two pointsand we use section formula as,
So,
Now equate the y component to get,
So,
#### Question 26:
If P (2, p) is the mid-point of the line segment joining the points A (6, −5) and B (−2, 11). find the value of p.
It is given that mid-point of line segment joining A (6,−5) and B (−2, 11) is
In general to find the mid-point of two pointsand we use section formula as,
So,
Now equate the y component to get,
#### Question 27:
If A (1, 2) B (4, 3) and C (6, 6) are the three vertices of a parallelogram ABCD, find the coordinates of fourth vertex D.
Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (1, 2);
B (4, 3) and C (6, 6). We have to find the co-ordinates of the forth vertex.
Let the forth vertex be
Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.
Now to find the mid-point of two pointsand we use section formula as,
The mid-point of the diagonals of the parallelogram will coincide.
So,
Therefore,
Now equate the individual terms to get the unknown value. So,
Similarly,
So the forth vertex is
#### Question 28:
What is the distance between the points $\mathrm{A}\left(\mathrm{sin}\theta -\mathrm{cos}\theta ,0\right)$ and $\mathrm{B}\left(0,\mathrm{sin}\theta +\mathrm{cos}\theta \right)$? [CBSE 2015]
The given points are $\mathrm{A}\left(\mathrm{sin}\theta -\mathrm{cos}\theta ,0\right)$ and $\mathrm{B}\left(0,\mathrm{sin}\theta +\mathrm{cos}\theta \right)$.
Using distance formula, we have
Thus, the distance between the given points is $\sqrt{2}$ units.
#### Question 29:
Q
Suppose CD be the perpendicular bisector of the line AB.
Suppose CD intersect AB in E.
Coodinates of E are
The coordinates are (0, 13).
#### Question 30:
Find the area of triangle with vertices ( a, b+c) , (b, c+a) and (c, a+b).
The area ‘A’ encompassed by three points, and is given by the formula,
Here, three points, and are .
Area is as follows:
#### Question 31:
If the points A (1,2) , O (0,0) and(a,b) are collinear , then find a : b.
For the three points, and to be collinear we need to have area enclosed between the points equal to zero.
Here, points, and are .
$⇒\frac{1}{2}\left|1\left(0-b\right)+0\left(b-2\right)+a\left(2-0\right)\right|=0\phantom{\rule{0ex}{0ex}}⇒-b+2a=0\phantom{\rule{0ex}{0ex}}⇒2a=b\phantom{\rule{0ex}{0ex}}⇒\frac{a}{b}=\frac{1}{2}\phantom{\rule{0ex}{0ex}}$
#### Question 32:
Find the coordinates of the point which is equidistant from the three vertices $A$( AOB .
It is known that, in a right angled triangle midpoint of the hypotenuse is equidistant from ots vertices.
Suppose D be the midpoint of the hypotenuse AB.
The coordinates of D are$\left(\frac{2x+0}{2},\frac{0+2y}{2}\right)=\left(x,y\right)$.
#### Question 33:
If the distance between the points (4, k) and (1, 0) is 5, then what can be the possible values of k?
Consider the points A(4, k) and B(1, 0).
It is given that the distance AB is 5 units.
By distance formula, disance AB is as follows:
$AB=\sqrt{{\left(4-1\right)}^{2}+{\left(k-0\right)}^{2}}\phantom{\rule{0ex}{0ex}}⇒5=\sqrt{9+{\left(k\right)}^{2}}\phantom{\rule{0ex}{0ex}}⇒25=9+{k}^{2}\phantom{\rule{0ex}{0ex}}⇒16={k}^{2}\phantom{\rule{0ex}{0ex}}⇒±4=k$
Hence, values of k are $±4$.
#### Question 34:
Find the distance of a point P(x, y) from the origin.
The given point is P(xy).
The origin is O(0, 0).
$\mathrm{PO}=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}=\sqrt{{\left(x-0\right)}^{2}+{\left(y-0\right)}^{2}}=\sqrt{{x}^{2}+{y}^{2}}$
Thus, the distance of point P from the origin is PO = .
#### Question 1:
Mark the correct alternative in each of the following:
The distance between the points (cos θ, 0) and (sin θ − cos θ) is
(a) $\sqrt{3}$
(b) $\sqrt{2}$
(c) 2
(d) 1
We have to find the distance between and.
In general, the distance between A and B is given by,
So,
But according to the trigonometric identity,
Therefore,
#### Question 2:
The distance between the points (a cos 25°, 0) and (0, a cos 65°) is
(a) a
(b) 2a
(c) 3a
(d) None of these
We have to find the distance between A(a cos 25°, 0) and.
In general, the distance between A and B is given by,
So,
$AB=\sqrt{{\left(0-a\mathrm{cos}25°\right)}^{2}+{\left(a\mathrm{cos}65°-0\right)}^{2}}\phantom{\rule{0ex}{0ex}}=\sqrt{{\left(a\mathrm{cos}25°\right)}^{2}+{\left(a\mathrm{cos}65°\right)}^{2}}$
But according to the trigonometric identity,
Therefore,
#### Question 3:
If x is a positive integer such that the distance between points P (x, 2) and Q (3, −6) is 10 units, then x =
(a) 3
(b) −3
(c) 9
(d) −9
It is given that distance between P (x, 2) and is 10.
In general, the distance between A and B is given by,
So,
On further simplification,
We will neglect the negative value. So,
#### Question 4:
The distance between the points (a cos θ + b sin θ, 0) and (0, a sin θ − b cos θ) is
(a) a2 + b2
(b) a + b
(c) a2 − b2
(d)
We have to find the distance between and.
In general, the distance between A and B is given by,
So,
But according to the trigonometric identity,
Therefore,
#### Question 5:
If the distance between the points (4, p) and (1, 0) is 5, then p =
(a) ± 4
(b) 4
(c) −4
(d) 0
It is given that distance between P (4, p) and is 5.
In general, the distance between A and B is given by,
So,
On further simplification,
So,
#### Question 6:
A line segment is of length 10 units. If the coordinates of its one end are (2, −3) and the abscissa of the other end is 10, then its ordinate is
(a) 9, 6
(b) 3, −9
(c) −3, 9
(d) 9, −6
It is given that distance between P (2,−3) and is 10.
In general, the distance between A and B is given by,
So,
On further simplification,
We will neglect the negative value. So,
#### Question 7:
The perimeter of the triangle formed by the points (0, 0), (0, 1) and (0, 1) is
(a) 1 ± $\sqrt{2}$
(b) $\sqrt{2}$ + 1
(c) 3
(d) $2+\sqrt{2}$
We have a triangle whose co-ordinates are A (0, 0); B (1, 0); C (0, 1). So clearly the triangle is right angled triangle, right angled at A. So,
Now apply Pythagoras theorem to get the hypotenuse,
So the perimeter of the triangle is,
#### Question 8:
If A (2, 2), B (−4, −4) and C (5, −8) are the vertices of a triangle, than the length of the median through vertex C is
(a) $\sqrt{65}$
(b) $\sqrt{117}$
(c) $\sqrt{85}$
(d) $\sqrt{113}$
We have a triangle in which the co-ordinates of the vertices are A (2, 2) B (−4,−4) and C (5,−8).
In general to find the mid-point of two points and we use section formula as,
Therefore mid-point D of side AB can be written as,
Now equate the individual terms to get,
So co-ordinates of D is (−1,−1)
So the length of median from C to the side AB,
#### Question 9:
If three points (0, 0), and (3, λ) form an equilateral triangle, then λ =
(a) 2
(b) −3
(c) −4
(d) None of these
We have an equilateral triangle whose co-ordinates are A (0, 0); and.
Since the triangle is equilateral. So,
So,
Cancel out the common terms from both the sides,
Therefore,
#### Question 10:
If the points (k, 2k), (3k, 3k) and (3, 1) are collinear, then k
(a) $\frac{1}{3}$
(b) $-\frac{1}{3}$
(c) $\frac{2}{3}$
(d) $-\frac{2}{3}$
We have three collinear points.
In general if are collinear then, area of the triangle is 0.
So,
So,
Take out the common terms,
Therefore,
#### Question 11:
The coordinates of the point on X-axis which are equidistant from the points (−3, 4) and (2, 5) are
(a) (20, 0)
(b) (−23, 0)
(c)
(d) None of these
Let the point be A be equidistant from the two given points P (−3, 4) and Q (2, 5).
So applying distance formula, we get,
Therefore,
Hence the co-ordinates of A are
So the answer is- (D) none of these.
#### Question 12:
If (−1, 2), (2, −1) and (3, 1) are any three vertices of a parallelogram, then
(a) a = 2, b = 0
(b) a= −2, b = 0
(c) a = −2, b = 6
(d) a = 6, b = 2
Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (−1, 2);
B (2,−1) and C(3, 1). We have to find the co-ordinates of the forth vertex.
Let the forth vertex be
Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.
Now to find the mid-point of two points and we use section formula as,
The mid-point of the diagonals of the parallelogram will coincide.
So,
Therefore,
Now equate the individual terms to get the unknown value. So,
Similarly,
So the forth vertex is
#### Question 13:
If A (5, 3), B (11, −5) and P (12, y) are the vertices of a right triangle right angled at P, then y=
(a) −2, 4
(b) −2, −4
(c) 2, −4
(d) 2, 4
Disclaimer: option (b) and (c) are given to be same in the book. So, we are considering option (b) as −2, −4
We have a right angled triangle whose co-ordinates are A (5, 3); B (11,−5);
. So clearly the triangle is, right angled at A. So,
Now apply Pythagoras theorem to get,
So,
On further simplification we get the quadratic equation as,
Now solve this equation using factorization method to get,
Therefore,
#### Question 14:
The area of the triangle formed by (a, b + c), (b, c + a) and (c, a + b)
(a) a + b + c
(b) abc
(c) (a + b + c)2
(d) 0
We have three non-collinear points.
In general if are non-collinear points then are of the triangle formed is given by-
So,
#### Question 15:
If (x , 2), (−3, −4) and (7, −5) are collinear, then x =
(a) 60
(b) 63
(c) −63
(d) −60
We have three collinear points.
In general if are collinear then,
So,
So,
Therefore,
#### Question 16:
If points (t, 2t), (−2, 6) and (3, 1) are collinear, then t =
(a) $\frac{3}{4}$
(b) $\frac{4}{3}$
(c) $\frac{5}{3}$
(d) $\frac{3}{5}$
We have three collinear points.
In general if are collinear then,
So,
So,
So,
Therefore,
#### Question 17:
If the area of the triangle formed by the points (x, 2x), (−2, 6) and (3, 1) is 5 square units , then x =
(a) $\frac{2}{3}$
(b) $\frac{3}{5}$
(c) 3
(d) 5
We have the co-ordinates of the vertices of the triangle aswhich has an area of 5 sq.units.
In general if are non-collinear points then area of the triangle formed is given by-,
So,
Simplify the modulus function to get,
Therefore,
#### Question 18:
If points (a, 0), (0, b) and (1, 1) are collinear, then
(a) 1
(b) 2
(c) 0
(d) −1
We have three collinear points.
In general if are collinear then,
So,
So,
Divide both the sides by,
#### Question 19:
If the centroid of a triangle is (1, 4) and two of its vertices are (4, −3) and (−9, 7), then the area of the triangle is
(a) 183 sq. units
(b) $\frac{183}{2}$ sq. units
(c) 366 sq. units
(d) $\frac{183}{4}$ sq. units
We have to find the co-ordinates of the third vertex of the given triangle. Let the co-ordinates of the third vertex be.
The co-ordinates of other two vertices are (4,−3) and (−9, 7)
The co-ordinate of the centroid is (1, 4)
We know that the co-ordinates of the centroid of a triangle whose vertices are is
So,
Compare individual terms on both the sides-
So,
Similarly,
So,
So the co-ordinate of third vertex is (8, 8)
In general if are non-collinear points then are of the triangle formed is given by-,
So,
#### Question 20:
The line segment joining points (−3, −4), and (1, −2) is divided by y-axis in the ratio
(a) 1 : 3
(b) 2 : 3
(c) 3 : 1
(d) 2 : 3
Let P be the point of intersection of y-axis with the line segment joining A (−3,−4) and B (1,−2) which divides the line segment AB in the ratio.
Now according to the section formula if point a point P divides a line segment joining and in the ratio m:n internally than,
Now we will use section formula as,
Now equate the x component on both the sides,
On further simplification,
So y-axis divides AB in the ratio
#### Question 21:
The ratio in which (4, 5) divides the join of (2, 3) and (7, 8) is
(a) −2 : 3
(b) −3 : 2
(c) 3 : 2
(d) 2 : 3
The co-ordinates of a point which divided two points and internally in the ratio is given by the formula,
Here it is said that the point (4, 5) divides the points A(2,3) and B(7,8). Substituting these values in the above formula we have,
Equating the individual components we have,
Hence the correct choice is option (d).
#### Question 22:
The ratio in which the x-axis divides the segment joining (3, 6) and (12, −3) is
(a) 2: 1
(b) 1 : 2
(c) −2 : 1
(d) 1 : −2
Let P be the point of intersection of x-axis with the line segment joining A (3, 6) and B (12, −3) which divides the line segment AB in the ratio.
Now according to the section formula if point a point P divides a line segment joining andin the ratio m: n internally than,
Now we will use section formula as,
Now equate the y component on both the sides,
On further simplification,
So x-axis divides AB in the ratio
#### Question 23:
If the centroid of the triangle formed by the points (a, b), (b, c) and (c, a) is at the origin, then a3 + b3 + c3 =
(a) abc
(b) 0
(c) a + b + c
(d) 3 abc
The co-ordinates of the vertices are (a, b); (b, c) and (c, a)
The co-ordinate of the centroid is (0, 0)
We know that the co-ordinates of the centroid of a triangle whose vertices are is
So,
Compare individual terms on both the sides-
Therefore,
We have to find the value of -
Now as we know that if,
Then,
#### Question 24:
If Points (1, 2) (−5, 6) and (a, −2) are collinear, then a =
(a) −3
(b) 7
(c) 2
(d) −2
We have three collinear points.
In general if are collinear then,
So,
So,
Therefore,
#### Question 25:
If the centroid of the triangle formed by (7, x) (y, −6) and (9, 10) is at (6, 3), then (x, y) =
(a) (4, 5)
(b) (5, 4)
(c) (−5, −2)
(d) (5, 2)
We have to find the unknown co-ordinates.
The co-ordinates of vertices are
The co-ordinate of the centroid is (6, 3)
We know that the co-ordinates of the centroid of a triangle whose vertices are is
So,
Compare individual terms on both the sides-
So,
Similarly,
So,
#### Question 26:
The distance of the point (4, 7) from the x-axis is
(a) 4
(b) 7
(c) 11
(d) $\sqrt{65}$
The ordinate of a point gives its distance from the x-axis.
So, the distance of (4, 7) from x-axis is
#### Question 27:
The distance of the point (4, 7) from the y-axis is
(a) 4
(b) 7
(c) 11
(d) $\sqrt{65}$
The distance of a point from y-axis is given by abscissa of that point.
So, distance of (4, 7) from y-axis is.
#### Question 28:
If P is a point on x-axis such that its distance from the origin is 3 units, then the coordinates of a point Q on OY such that OP = OQ, are
(a) (0, 3)
(b) (3, 0)
(c) (0, 0)
(d) (0, −3)
GIVEN: If P is a point on x axis such that its distance from the origin is 3 units.
TO FIND: The coordinates of a point Q on OY such that OP= OQ.
On x axis y coordinates is 0. Hence the coordinates of point P will be (3, 0) as it is given that the distance from origin is 3 units.
Now then the coordinates of Q on OY such that OP = OQ
On y axis x coordinates is 0. Hence the coordinates of point Q will be (0, 3)
Hence correct option is (a)
#### Question 29:
If the points(x, 4) lies on a circle whose centre is at the origin and radius is 5, then x =
(a) ±5
(b) ±3
(c) 0
(d) ±4
It is given that the point A(x, 4) is at a distance of 5 units from origin O.
So, apply the distance formula to get,
Therefore,
So,
#### Question 30:
If the points P (x, y) is equidistant from A (5, 1) and B (−1, 5), then
(a) 5x = y
(b) x = 5y
(c) 3x = 2y
(d) 2x = 3y
It is given that is equidistant to the point
So,
So apply distance formula to get the co-ordinates of the unknown value as,
On further simplification we get,
So,
Thus,
#### Question 31:
If points A (5, p) B (1, 5), C (2, 1) and D (6, 2) form a square ABCD, then p =
(a) 7
(b) 3
(c) 6
(d) 8
The distance d between two points and is given by the formula
In a square all the sides are equal to each other.
Here the four points are A(5,p), B(1,5), C(2,1) and D(6,2).
The vertex ‘A’ should be equidistant from ‘B’ as well as D’
Let us now find out the distances ‘AB’ and ‘AD’.
These two need to be equal.
Equating the above two equations we have,
Squaring on both sides we have,
Hence the correct choice is option (c).
#### Question 32:
The coordinates of the circumcentre of the triangle formed by the points O (0, 0), A (a, 0 and B (0, b) are
(a) (a, b)
(b) $\left(\frac{a}{2},\frac{b}{2}\right)$
(c)
(d) (b, a)
The distance d between two points and is given by the formula
The circumcentre of a triangle is the point which is equidistant from each of the three vertices of the triangle.
Here the three vertices of the triangle are given to be O(0,0), A(a,0) and B(0,b).
Let the circumcentre of the triangle be represented by the point R(x, y).
So we have
Equating the first pair of these equations we have,
Squaring on both sides of the equation we have,
Equating another pair of the equations we have,
Squaring on both sides of the equation we have,
Hence the correct choice is option (b).
#### Question 33:
The coordinates of a point on x-axis which lies on the perpendicular bisector of the line segment joining the points (7, 6) and (−3, 4) are
(a) (0, 2)
(b) (3, 0)
(c) (0, 3)
(d) (2, 0)
TO FIND: The coordinates of a point on x axis which lies on perpendicular bisector of line segment joining points (7, 6) and (−3, 4).
Let P(x, y) be any point on the perpendicular bisector of AB. Then,
PA=PB
On x-axis y is 0, so substituting y=0 we get x= 3
Hence the coordinates of point is i.e. option (b) is correct
#### Question 34:
If the centroid of the triangle formed by the points (3, −5), (−7, 4), (10, −k) is at the point (k −1), then k =
(a) 3
(b) 1
(c) 2
(d) 4
We have to find the unknown co-ordinates.
The co-ordinates of vertices are
The co-ordinate of the centroid is
We know that the co-ordinates of the centroid of a triangle whose vertices are is-
So,
Compare individual terms on both the sides-
#### Question 35:
If (−2, 1) is the centroid of the triangle having its vertices at (x , 0) (5, −2), (−8, y), then x, y satisfy the relation
(a) 3x + 8y = 0
(b) 3x − 8y = 0
(c) 8x + 3y = 0
(d) 8x = 3y
We have to find the unknown co-ordinates.
The co-ordinates of vertices are
The co-ordinate of the centroid is (−2, 1)
We know that the co-ordinates of the centroid of a triangle whose vertices are is-
So,
Compare individual terms on both the sides-
So,
Similarly,
So,
It can be observed that (x, y) = (−3, 5) does not satisfy any of the relations 3x + 8y = 0, 3x − 8y = 0, 8x + 3y = 0 or 8x = 3y.
#### Question 36:
The coordinates of the fourth vertex of the rectangle formed by the points (0, 0), (2, 0), (0, 3) are
(a) (3, 0)
(b) (0, 2)
(c) (2, 3)
(d) (3, 2)
We have to find the co-ordinates of forth vertex of the rectangle ABCD.
We the co-ordinates of the vertices as (0, 0); (2, 0); (0, 3)
Rectangle has opposite pair of sides equal.
When we plot the given co-ordinates of the vertices on a Cartesian plane, we observe that the length and width of the rectangle is 2 and 3 units respectively.
So the co-ordinate of the forth vertex is
#### Question 37:
The length of a line segment joining A (2, −3) and B is 10 units. If the abscissa of B is 10 units, then its ordinates can be
(a) 3 or −9
(b) −3 or 9
(c) 6 or 27
(d) −6 or −27
It is given that distance between P (2,−3) and is 10.
In general, the distance between A and B is given by,
So,
On further simplification,
We will neglect the negative value. So,
#### Question 38:
The ratio in which the line segment joining P (x1, y1) and Q (x2, y2) is divided by x-axis is
(a) y1 : y2
(b) −y1 : y2
(c) x1 : x2
(d) −x1 : x2
Let C be the point of intersection of x-axis with the line segment joining and which divides the line segment PQ in the ratio.
Now according to the section formula if point a point P divides a line segment joining andin the ratio m:n internally than,
Now we will use section formula as,
Now equate the y component on both the sides,
On further simplification,
#### Question 39:
The ratio in which the line segment joining points A (a1, b1) and B (a2, b2) is divided by y-axis is
(a) −a1 : a2
(b) a1 : a2
(c) b1 : b2
(d) −b1 : b2
Let P be the point of intersection of y-axis with the line segment joiningandwhich divides the line segment AB in the ratio.
Now according to the section formula if point a point P divides a line segment joining andin the ratio m:n internally than,
Now we will use section formula as,
Now equate the x component on both the sides,
On further simplification,
#### Question 40:
If the line segment joining the points (3, −4), and (1, 2) is trisected at points P (a, −2) and Q , Then,
(a)
(b)
(c)
(d)
We have two points A (3,−4) and B (1, 2). There are two points P (a,−2) and Q which trisect the line segment joining A and B.
Now according to the section formula if any point P divides a line segment joining andin the ratio m: n internally than,
The point P is the point of trisection of the line segment AB. So, P divides AB in the ratio 1: 2
Now we will use section formula to find the co-ordinates of unknown point A as,
Equate the individual terms on both the sides. We get,
Similarly, the point Q is the point of trisection of the line segment AB. So, Q divides AB in the ratio 2: 1
Now we will use section formula to find the co-ordinates of unknown point A as,
Equate the individual terms on both the sides. We get,
#### Question 41:
If the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its centre are (−2, 5), then the coordinates of the other end of the diameter are
(a) (−6, 7) (b) (6, −7) (c) (6, 7) (d) (−6,−7) [CBSE 2012]
Let O(−2, 5) be the centre of the given circle and A(2, 3) and B(x, y) be the end points of a diameter of the circle.
Then, O is the mid-point of AB.
Using mid-point formula, we have
$\therefore \frac{2+x}{2}=-2$ and $\frac{3+y}{2}=5$
$⇒2+x=-4$ and $3+y=10$
$⇒x=-6$ and $y=7$
Thus, the coordinates of the other end of the diameter are (−6, 7).
Hence, the correct answer is option A.
#### Question 42:
The coordinates of the point P dividing the line segment joining the points A (1, 3) and B (4, 6) in the ratio 2 : 1 are
(a) (2, 4) (b) (3, 5) (c) (4, 2) (d) (5, 3) [CBSE 2012]
It is given that P divides the line segment joining the points A(1, 3) and B(4, 6) in the ratio 2 : 1.
Using section formula, we get
Coordinates of P
Thus, the coordinates of P are (3, 5).
Hence, the correct answer is option B.
#### Question 43:
In Fig. 14.46, the area of ΔABC (in square units) is
(a) 15 (b) 10 (c) 7.5 (d) 2.5 [CBSE 2013]
The coordinates of A are (1, 3).
∴ Distance of A from the x-axis, AD = y-coordinate of A = 3 units
The number of units between B and C on the x-axis are 5.
∴ BC = 5 units
Now,
Area of ∆ABC = $\frac{1}{2}×\mathrm{BC}×\mathrm{AD}=\frac{1}{2}×5×3=\frac{15}{2}=7.5$ square units
Thus, the area of ∆ABC is 7.5 square units.
Hence, the correct answer is option C.
#### Question 44:
The point on the x-axis which is equidistant from points (−1, 0) and (5, 0) is
(a) (0, 2) (b) (2, 0) (c) (3, 0) (d) (0, 3) [CBSE 2013]
Let A(−1, 0) and B(5, 0) be the given points. Suppose the required point on the x-axis be P(x, 0).
It is given that P(x, 0) is equidistant from A(−1, 0) and B(5, 0).
∴ PA = PB
⇒ PA2 = PB2
$⇒{\left[x-\left(-1\right)\right]}^{2}+{\left(0-0\right)}^{2}={\left(x-5\right)}^{2}+{\left(0-0\right)}^{2}$ (Using distance formula)
$⇒{\left(x+1\right)}^{2}={\left(x-5\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒{x}^{2}+2x+1={x}^{2}-10x+25\phantom{\rule{0ex}{0ex}}⇒12x=24\phantom{\rule{0ex}{0ex}}⇒x=2$
Thus, the required point is (2, 0).
Hence, the correct answer is option B.
#### Question 45:
If A(4, 9), B(2, 3) and C(6, 5) are the vertices of ∆ABC, then the length of median through C is
(a) 5 units (b) $\sqrt{10}$ units (c) 25 units (d) 10 units [CBSE 2014]
It is given that A(4, 9), B(2, 3) and C(6, 5) are the vertices of ∆ABC.
Let CD be the median of ∆ABC through C. Then, D is the mid-point of AB.
Using mid-point formula, we get
Coordinates of D =
∴ Length of the median, AD
Thus, the length of the required median is $\sqrt{10}$ units.
Hence, the correct answer is option B.
#### Question 46:
If P(2, 4), Q(0, 3), R(3, 6) and S(5, y) are the vertices of a parallelogram PQRS, then the value of y is
(a) 7 (b) 5 (c) −7 (d) −8 [CBSE 2014]
It is given that P(2, 4), Q(0, 3), R(3, 6) and S(5, y) are the vertices of a parallelogram PQRS.
Join PR and QS, intersecting each other at O.
We know that the diagonals of the parallelogram bisect each other. So, O is the mid-point of PR and QS.
Coordinates of mid-point of PR = $\left(\frac{2+3}{2},\frac{4+6}{2}\right)=\left(\frac{5}{2},\frac{10}{2}\right)=\left(\frac{5}{2},5\right)$
Coordinates of mid-point of QS = $\left(\frac{0+5}{2},\frac{3+y}{2}\right)=\left(\frac{5}{2},\frac{3+y}{2}\right)$
Now, these points coincides at the point O.
$\therefore \left(\frac{5}{2},\frac{3+y}{2}\right)=\left(\frac{5}{2},5\right)\phantom{\rule{0ex}{0ex}}⇒\frac{3+y}{2}=5\phantom{\rule{0ex}{0ex}}⇒3+y=10\phantom{\rule{0ex}{0ex}}⇒y=7$
Thus, the value of y is 7.
Hence, the correct answer is option A.
#### Question 47:
If A(x, 2), B(−3, −4) and C(7, −5) are collinear, then the value of x is
(a) −63 (b) 63 (c) 60 (b) −60 [CBSE 2014]
The given points A(x, 2), B(−3, −4) and C(7, −5) are collinear.
$\therefore \mathrm{ar}\left(∆\mathrm{ABC}\right)=0\phantom{\rule{0ex}{0ex}}⇒\frac{1}{2}\left|{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)\right|=0\phantom{\rule{0ex}{0ex}}⇒{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}-{y}_{2}\right)=0$
$⇒x\left[-4-\left(-5\right)\right]+\left(-3\right)\left(-5-2\right)+7\left[2-\left(-4\right)\right]=0\phantom{\rule{0ex}{0ex}}⇒x+21+42=0\phantom{\rule{0ex}{0ex}}⇒x+63=0\phantom{\rule{0ex}{0ex}}⇒x=-63$
Thus, the value of x is −63.
Hence, the correct answer is option A.
#### Question 48:
The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is
(a) 7 + $\sqrt{5}$ (b) 5 (c) 10 (d) 12 [CBSE 2014]
Let A(0, 4), O(0, 0) and B(3, 0) be the vertices of ∆AOB.
Using distance formula, we get
OA = $\sqrt{{\left(0-0\right)}^{2}+{\left(4-0\right)}^{2}}=\sqrt{16}=4$ units
OB = $\sqrt{{\left(3-0\right)}^{2}+{\left(0-0\right)}^{2}}=\sqrt{9}=3$ units
AB = $\sqrt{{\left(3-0\right)}^{2}+{\left(0-4\right)}^{2}}=\sqrt{9+16}=\sqrt{25}=5$ units
∴ Perimeter of ∆AOB = OA + OB + AB = 4 + 3 + 5 = 12 units
Thus, the required perimeter of the triangle is 12 units.
Hence, the correct answer is option D.
#### Question 49:
If the point P (2, 1 ) lies on the line segment joining points A (4,20 and B (8, 4) , then
(a) (b) AP = BP (C) PB$\frac{1}{3}AB$ (D)
Use section formula for finding out the ratio in which P divided the line segment AB.
#### Question 50:
A line intersects the y-axis and x-axis at P and Q , respectively. If (2, $-$5) is the mid-point of PQ, then the coordinates of P and Q are, respectively
(a) (0, $-$5) and (2, 0) (b) (0, 10) and ($-$4, 0)
(c) (0, 4) and ($-$10, 0 ) (d) (0, $-$10) and (4 , 0) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 252, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8313623070716858, "perplexity": 740.2500878162286}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250599718.13/warc/CC-MAIN-20200120165335-20200120194335-00482.warc.gz"} |
http://physics.stackexchange.com/questions/60705/bells-theorem-graph | # Bell's Theorem graph
My friends and I got into an argument about determinism, and I brought up that quantum events are random. But I couldn't prove it.
I found the Wikipedia page on Bell's theorem, which seems to imply what I'm trying to show, because it disqualifying local hidden variable models. But I don't understand how the experiment works. I think I understand the steps taken:
1. An electron-positron pair is produced, with opposite spins.
2. Alice measures the spin of the electron along the x-axis.
3. Bob measures the spin of the positron along some axis, which could be the x-axis.
4. Alice and Bob compare their results, recording a +1 if their spins match, and a -1 if they do not.
5. A graph of "angle between Alice and Bob's axes" vs. "sum of many trials" is created.
The part I don't get is: Why would local hidden variable theories predict a triangular pattern for the graph, and likewise, why would entanglement predict a cosine?
-
You don't need entanglement and Bell's theorem to see that QM contains randomness. Consider a single spin 1/2 particle polarized along the x-direction and measure the component of the spin along z. You (randomly) get $\pm\hbar/2$ with equal probability. – Thomas Apr 10 '13 at 23:35
But how do we know that the spin axis isn't determined when the particles are created? You'd still get half spin-up and half spin-down in that case. – Henry Swanson Apr 11 '13 at 0:01
Which of the two questions you pose are you most interested in? Understanding Bell's theorem or randomness in quantum mechanics? It appears that your question is really two questions! – Juan Miguel Arrazola Apr 11 '13 at 2:54
I'm trying to show that the spin of the electrons is not predetermined. Is Bell's theorem not the right way to go about that? – Henry Swanson Apr 11 '13 at 3:10
Bell's theorem is basically a logical argument (as in mathematical logic) in probability theory. You do not have to use Quantum prepositions to understand it. This paper : arxiv.org/abs/1212.5214, has a figure which explains it well. – daaxix Apr 11 '13 at 6:50
Bell's theorem basically states that some predictions of quantum mechanics cannot be obtained from a local hidden variable model of the theory. Some people (like Nielsen and Chuang) refer to this as the fact that there cannot exist a local realist theory that has the same predictions as quantum mechanics.
Roughly speaking, a local theory is one in which systems that are space-like separated cannot influence each other. A realist theory is one in which the properties of systems have definite values, independent of measurements of them. Within this terminology, what you are trying to show to your friends is that quantum mechanics is not a realist theory, there is inherent uncertainty about the value of physical properties before they are measured.
But you see, Bell's theorem only formally tells us that we cannot have both realism and locality. However, it says nothing about keeping one but dropping the other. So, can there be a non-local realist model that makes the same predictions as quantum mechanics? Well yes there can!
An example is the Brohm-deBroglie interpretation of quantum mechanics, which you can learn more about if you are interested. The bottom line is that we cannot prove that the predictions of quantum mechanics imply that the properties of physical systems, like spin, are not determined before measurement, because we know that there is a theory in which they are determined that makes the same experimental predictions!
-
As a note on the randomeness of quantum mechanics (though this might not be what you're directly asking in your question).
Time evolution of a state/system is perfectly deterministic in quantum mechanics. It's only measurements that give "random" results. In a certain perspective, that's an effective model for our ignorance of how measurements work (for eg: Steve Weinberg has been grappling with that for a while now). One of the ideas is that any measuring device is typically a macroscopic classical system and (roughly) decohorence turns a pure quantum state density matrix into a mixed state which gives a classical probability distribution over the possible outcomes of the measurement.
Note: Some people try to make stochastic models of quantum mechanics, where QM is augmented with random variables, which makes those models non-deterministic. But that is beyond mainstream "core" quantum mechanics and is still to be tested.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8083229064941406, "perplexity": 410.88564896879456}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802775085.124/warc/CC-MAIN-20141217075255-00056-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/3180247/solving-int-sint2-dt-without-trigonometric-identities | # Solving $\int \sin(t)^2\, dt$ without trigonometric identities
Can you solve $$\int \sin(t)^2\, dt$$ without trigonometric identities?
I wanted to solve this integral and I actually had a rough time with it... First I tried a normal product integration, with $$u'=\sin(t)$$ and $$v=\sin(t)$$, which lead nowhere.
Then I tried $$u'=1$$ and $$v=\sin(t)^2$$ which was actually do able, with the identity $$\sin(t)\cos(t)=2\sin(2t)$$, which I had to look up...
But the easiest way seems to be, that one uses $$\sin(t)^2=\frac12-\frac12\cos(2t)$$
The problem is, that you have to know these identities. Which I barely do.
Is there an elementary way to solve this integral, which uses as less knowledge about these identities as possible.
• Don't begin by trying to improve the world. Instead, improve yourself: learn the identities.:)
– avs
Apr 8 '19 at 21:53
• But improving yourself is selfish. :( But yeah, you are right. I should learn them, to use them once in a while. Apr 8 '19 at 21:55
• With Euler's formula: $\sin t=\dfrac{\mathrm e^{it}-\mathrm e^{-it}}{2i}$. Apr 8 '19 at 22:04
• That's a trigonometric identity :) Apr 8 '19 at 22:21
The easiest one requires you to know that $$\sin^2x+\cos^2x=1$$. You can also derive other identities from De Moivre's formula namely $$\exp i\theta=\cos\theta+i\sin\theta$$.
$$\int\sin^2t\mathrm dt=\int \sqrt{1-\cos^2t}\sin t\mathrm dt=-\int\sqrt{1-u^2}\mathrm du$$
This form can be handled using Integration by Parts. Can you proceed?
Aliter:
Using Taylor series for $$\sin^2 x$$ and integrate term-by-term. This gives you an answer in the form of an infinite series. $$\int\sin^2t\mathrm dt=\int\left(\sum_{n=0}^{\infty}(-1)^n\dfrac{x^{2n+1}}{(2n+1)!}\right)^2\mathrm dt$$
Now expand this expression namely the integrand using the Multinomial Theorem and integrate term-by-term. However, that does not look very good. Also you have to be sure if it holds for infinitely many terms.
• I'll leave the OP to decide whether $\sin^2 t+\cos^2 t=1$ is also a disallowed identity (but really, who studies these functions without learning the Pythagorean theorem?), but $\sin t$ can differ from $\sqrt{1-\cos^2t}$ by a sign, so it's better to Integrate by parts with $u=\sin t,\,v=-\cos t$, viz. $$\int\sin^2 tdt=-\sin t\cos t+\int (1-\sin^2 t)dt\implies\int\sin^2 tdt=\frac{1}{2}(t-\sin t\cos t)+C.$$
– J.G.
Apr 9 '19 at 6:34
• Thanks for the insight @J.G. Also I was wondering if that expression I've written as infinite series indeed holds. It would be great if you could check the validity of what I've written as an alternative answer. Apr 9 '19 at 6:38
• A better approach there is to show the series squares to $(1-\cos 2x)/2$ as expected (this is an exercise in combinatorial identities, which is a good learning exercise of you haven't done it), but whether the OP is OK with using "banned" results of we first prove them isn't clear.
– J.G.
Apr 9 '19 at 6:49
• @J.G I accept the Pythagorean theorem. This I know by heart. Apr 9 '19 at 7:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 12, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9368540048599243, "perplexity": 310.12074255794636}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301730.31/warc/CC-MAIN-20220120065949-20220120095949-00008.warc.gz"} |
https://astronomy.stackexchange.com/questions/38257/why-are-stars-still-distant-from-one-another/38260 | # Why are stars still distant from one another? [duplicate]
Why are stars so far from each other? Shouldn't gravity pull them closer over time? And if the effects of gravity are negligible is there an explanation why stars have to be so distant from one another?
The closest star is Alpha Centauri (I think) and it is 4.4 light years away.
• Do you think stars in a galaxy behave differently than planets in a solar system? Planets are attracted to each other (and their suns) through gravity too, and yet they do not collapse into one another. Of course the effects of gravity aren't negligible - if they were, the stars/planets would shoot out of the galaxy/solar system rather than keeping their orbits :) Also, stars don't always form far from one another - but then they're gravitationally bound. See binary/multiple stars :) – Luaan Aug 3 '20 at 9:26
The initial star formation regions were regions that have a high enough mass density to form a star. The density of the early universe was not constant at different locations. Some regions had high enough density to form a star, and some didn't.
When a star forms it draws in matter from a large distance away. This forms an accretion disk and leaves a temporary relatively empty space around the star at a large distance away. There is no way a star can form in this low mass region. Once you get further away from this region the matter density may return to a level where a star can form. But this region is far from a star.
Yes, gravity from a particular star is pulling at other stars but there are many stars pulling on the particular star of interest. So the net force on the star may be very small.
• Stars often form in dense clusters and often as multiple systems. – ProfRob Aug 3 '20 at 8:29
• It's not the accretion that causes the relative depletion IIRC - it's the very harsh solar wind of the young star that pushes ridiculous amounts of materials away. Even today, the heliopause, which is essentially the area where the Sun's solar wind pressure equalises with the "ambient" pressure of interstellar space is pretty far away (though still just a tiny fraction of the distances between stars). – Luaan Aug 3 '20 at 9:30
As jmh has answered, stars naturally form at large distances from each other. To add to the answer, what is the reason for this particular distance scale?
If we imagine a very large, homogeneous cloud of gas it will be unstable to gravitational collapse over the Jeans length scale $$\lambda_J=c_s/\sqrt{G\rho}$$ where $$c_s$$ is the speed of sound in the gas, $$G$$ the gravitational constant, and $$\rho$$ the density. For typical values this about a light-year, giving a sense of the distance scale.
But why are those values what they are? $$G$$ is a fundamental constant and since it is small we get long distances. The speed of sound depends on temperature and molecular mass; since molecules are very light it is high. Why are molecules light? This is because another fundamental constant, the ratio between proton and electron mass, is large. The density of gas in the universe is low, since the universe has a density close to the critical density (had it not been that, it would either have recollapsed or expanded so fast there would not have been many stars).
A universe where stars form much closer to each other than in ours needs to have strong gravity (making stars burn much hotter and be short lived, beside lots of gravitational interactions disrupting planetary orbits), have heavy molecules (making chemistry weird), or have a high density (likely collapsing rapidly). So it is likely that there would not be any life and observers there.
The question also asks why gravity does not pull them closer to each other. Note that if the cloud turns into stars with typical mass $$M\approx \rho \lambda_J^3$$ separated by distance $$r\approx \lambda_J$$ then the acceleration between them will be $$a=GM/r^2=G\rho \lambda_J^3/\lambda_J^2=G\rho \lambda_J =\sqrt{G\rho}c_s$$. So the accelerations will be very small, again for the same reasons as discussed above.
(Further, due to the ratio between the electromagnetic force strength and the gravitational force strength, objects like stars have equilibrium sizes that are small compared to $$\lambda_J$$, so they rarely collide with each other on this timescale. They just miss, and fly past. )
• Argument about acceleration is dimensionally incorrect. – ProfRob Aug 3 '20 at 8:25
• The typical initial Jeans length is orders of magnitude bigger than a light year. – ProfRob Aug 3 '20 at 8:28
• @RobJeffries - Ah, slipped up on the acceleration formula. Correcting. – Anders Sandberg Aug 4 '20 at 15:02
All these stars orbit the galaxy center of mass. The galaxy outweights any possible star by many orders of magnitude and dominates the orbital motion, unless some stars come pretty close together.
The orbits of the stars are more or less stable in the same sense that orbits of planets are stable in the solar system. They are not absolutely stable, but stable enough on a tens of bilions of years timescale.
Even if two stars come closer, they rarely collide. There is too much of space and too few stars. In most cases they just miss each other in an open hyperbolic orbits.
It is estimated that if the Milky way galaxy collides with the Andromeda galaxy, there will be less than 10 actual star collisions out of bilions and bilions of stars in both galaxies.
bonus: a pretty graph of nearest stars approaching and passing by the Sun, from Wikipedia: https://upload.wikimedia.org/wikipedia/commons/e/e9/Near-stars-past-future-en.svg
• Yeah, I wanted to point that perspective out too. Stars move in relation to each other. As long as two stars aren't gravitationally bound (i.e. binary+ stars), they will inevitably drift apart even if the difference in their orbits is tiny. That's just how orbital mechanics work. – Luaan Aug 3 '20 at 9:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 9, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.895378589630127, "perplexity": 640.6026847792084}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487636559.57/warc/CC-MAIN-20210618104405-20210618134405-00337.warc.gz"} |
http://mathhelpforum.com/pre-calculus/207267-trigonometric-equation.html | # Math Help - Trigonometric Equation
1. ## Trigonometric Equation
How do you solve the equation 5cos(2x)=2 ?
2. ## Re: Trigonometric Equation
I would divide through by 5 first. Then I would solve the 2 cases (where $k\in\mathbb{Z}$):
a) $\cos(2x)=\cos(2x+2k\pi)=\frac{2}{5}$
b) $\cos(2x)=\cos(2k\pi-2x)=\frac{2}{5}$
Recall, if $\cos(f(x))=a$ then $f(x)=\cos^{-1}(a)$, where $a$ is an arbitrary constant. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9959091544151306, "perplexity": 1675.4750916812143}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375100481.40/warc/CC-MAIN-20150627031820-00029-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://research.ipmu.jp/seminar/?seminar_id=487 | # ACP Seminar (Astronomy - Cosmology - Particle Physics)
Speaker: Morimitsu Tanimoto (Niigata U.) Impact of Large \theta_{13} on Flavor Physics Thu, Jun 30, 2011, 13:30 - 15:00 Seminar Room A 487.pdf seminar: 2pm--3pm. Prior to the seminar, 1:30pm--2pm, a pedagogical introduction to very elementary things in neutrino oscillation and flavor mixing will be given by one of IPMU members. Seminar Abstract: The long baseline neutrino experiment T2K has presented new data, which indicates the relatively large neutrino mixing angle $\theta_{13}$. Now we should not persist in the paradigm of the tri-bimaximal mixing. In this seminar, we discuss the impact on the model building of flavors. We also discuss the CP violation in neutrino oscillations and neutrinoless double beta decays. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9468086957931519, "perplexity": 3146.4043301374545}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572033.91/warc/CC-MAIN-20220814113403-20220814143403-00147.warc.gz"} |
http://physics.stackexchange.com/questions/29902/why-is-the-center-of-mass-of-2-bodies-at-the-focus-of-their-elliptical-orbits | # Why is the center-of-mass of 2 bodies at the focus of their elliptical orbits?
Why is the center-of-mass of 2 bodies (which interact only via Newtonian gravity) located at a focus of each of the elliptical orbits?
I know that when there are no external forces, the center of mass moves at a constant speed, but that doesn't explain it.
-
"they orbit around their center of mass" That's a bit of an oversimplified statement. You do know that orbits are, in general, not circular with a well-defined center point? – leftaroundabout Jun 11 '12 at 13:53
The important thing here is that the center of mass is an invariant in a frame where it is initially at rest. This actually follows from the definition of the thing and Newton's laws, but I've become convinced that the point is subtle and important enough (if basic) to merit a careful answer. – dmckee Jun 11 '12 at 14:00
If the orbit is elliptic, the center of mass will be at one of the ellipse's focuses. But why? – Omer Jun 11 '12 at 14:55
Wait. Is the question about the invariance of the CoM or about the shape of two body orbits? – dmckee Jun 11 '12 at 16:22
The question is about the shape of two body orbits - why the CoM is one of the two ellipses' focuses. – Omer Jun 11 '12 at 17:13
I'm going to assume that Omer is specifically asking why the centre of mass is at the focus (well, one of the foci) of the orbits. Omer, if this isn't what you meant please ignore what follows because it's completely irrelevant!
If you have a body moving in a central field (i.e. the force is always pointing towards the centre), and the field is inversely proportional to the square of the distance from body to the centre, then the orbit is an ellipse with the centre at one of the foci. For now let's just assume this and we can come back to prove it later.
So if we can show that both of the bodies feel a central inverse square force, with the COM at the centre, this guarantees the orbits will be ellipses with a focus at the COM. Given that the force is due to the two bodies attracting each other, and that both bodies are orbiting around, it may seem a bit odd that each body just feels a central inverse square force, but actually this is easy to show.
The picture shows the two masses and the COM. I haven't shown the velocities because it doesn't matter what they are. For now let's just consider $m_1$ and calculate the force on it. By Newton's law this is simply:
$$F_1 = \frac{Gm_1m_2}{(r_1 + r_2)^2}$$
First is this force central? We know the centre of mass doesn't move. For two bodies this seems obvious to me, but in any case dmckee proved it in his answer. If the COM doesn't move it must lie on the line joining the two mases, otherwise there'd be a net force on it. So the force $F_1$ must always point towards the COM i.e. the force is central.
Second is this an inverse square law force i.e. is $F_1 \propto 1/r_1^2$? Well the definition of the centre of mass is that:
$$m_1r_1 = m_2r_2$$
or
$$r_2 = r_1 \frac{m_1}{m_2}$$
If we substitute for $r_2$ in the expression for $F_1$ we get:
$$F_1 = \frac{Gm_1m_2}{(r_1 + r_1(m_1/m_2))^2}$$
or with a quick rearrangement:
$$F_1 = \frac{1}{(1 + m_1/m_2)^2} \frac{Gm_1m_2}{r_1^2}$$
and this shows that $F_1$ is inversely proportional to $r_1^2$. I won't work through it, but it should be fairly obvious that exactly the same reasoning applies to $F_2$ so:
$$F_2 = \frac{1}{(1 + m_2/m_1)^2} \frac{Gm_1m_2}{r_2^2}$$
This is the key result. Even though the two bodies are whizzing around each other, each body just behaves as if it were in a static gravity field, but the strength of the field is reduced by a factor of $(1 + m_1/m_2)^2$ for $m_1$ or $(1 + m_2/m_1)^2$ for $m_2$. This applies to all two body systems, even such unequal ones as the Sun and the Earth (ignoring perturbations from Jupiter etc).
I did start by assuming that a body in a central gravity field orbits in an ellipse with the foci at the centre, but I'm going to wimp out of proving this since it would double the length of this answer and you'd all go to sleep. The proof is easily Googled.
NB this only applies to two body systems. For three or more body systems the orbits are generally not ellipses with the centre of mass at the focus.
-
Thanks! It was exactly what I searched for. – Omer Jun 11 '12 at 18:25
This actually follows from Newton's laws (and it only holds true for an isolated system).
For simplicity we'll consider an isolated system of two bodies on a line. Call their masses $m_1$ and $m_2$ and put them at $x_1$ and $x_2$. Now computer their center of mass: $$X = \frac{\sum_i m_i x_i}{\sum_i m_i} = \frac{1}{M} \sum_i m_i x_i$$
Assume that there is some force, $F$, between them. Newton's second law tells us that the force on body one due to body two $F_1$ is equal and opposite that on body two due to body one $F_2 = -F_1$.
This is enough information to compute the acceleration of the CoM: $$A = X'' = \frac{1}{M} \sum_i m_i a_i = \frac{1}{M} \sum_i m_i \frac{F_i}{m_i},$$ cancel the masses inside the sum and take note of the relation between the forces and you get $$A = 0 .$$
For more bodies and more dimensions the math gets more complicated but the result is the same.
-
I think Omer is specifically asking why the centre of mass is at the focus of the orbits – John Rennie Jun 11 '12 at 15:11
@JohnRennie: This is the reason they move "around" the CoM. Any movement away on the mart of mass one must be mirrored by a mass-ratio weighted movement away on the part of mass two (as long as only internal forces are acting). And likewise for movements toward, and in all relevant dimensions. I know it seems like a different question, but it's not. That is the subtle point that makes this question interesting and pedagogically important. – dmckee Jun 11 '12 at 16:19
Ah...seeing the OP's comment above maybe I have answered the wrong question. – dmckee Jun 11 '12 at 16:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9034016132354736, "perplexity": 171.70190188063293}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464053209501.44/warc/CC-MAIN-20160524012649-00169-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/27037/greens-formula-for-nonorientable-manifolds/27067 | # Green's formula for nonorientable manifolds
Usually in differential geometry one proves the Stokes theorem and then obtains divergence theorem and Green's formulas as corollaries. However, divergence theorem is also valid for nonorientable riemannian manifolds when one replaces forms with densities. But then the Green's formulas should also be valid. Am I missing something? I haven't found a discussion about where precisely the orientability is needed.
Further supposing Green formulas in nonorientable manifolds one could define variational formulation for example for elliptic PDEs there. I wonder if the (non)orientability has some effect on the solutions of such PDEs?
-
Similar answer, a friend recently asked about twisted forms for non-orientable manifolds, what I found was pages 79-88 in Bott and Tu, "Differential forms in algebraic topology." I also think a twisted Stokes' Theorem is possible, as they present an entire twisted de Rham complex. Anyway, take a look:
I googled "twisted Stokes theorem." My friend Dmitry asked originally based on some physics inquiries. It appears that these physics people give a pretty direct discussion, maybe it is enough. "Foundations of classical electrodynamics: charge, flux, and metric" By Friedrich W. Hehl, Yuri N. Obukhov, (2003) Birkhauser
-
A discussion of the kind you want indeed seems to be difficult to locate. I am not an expert but I guess one could prove your claim (and more broadly, some version of the Stokes theorem) for non-orientable manifolds by passing to the two-sheeted covering oriented manifold, as suggested, in a slightly different setting, in the book Geometry VI: Riemannian geometry by Postnikov (here is the link to the relevant page on Google preview).
EDIT: As explained in the Bott--Tu book (see the link in Will's answer and also these two pages), rescuing the Stokes theorem in the non-orientable case requires passing from differential forms to densities.
-
So it seems that Bott & Tu are saying that orientability doesn't really matter: Stokes theorem remains valid if one replaces forms by densities. But then it's curious why this fact is not clearly stated in differential geometry books. For example some projective spaces are nonorientable and obviously projective spaces are fundamental in all mathematics. So it's funny why there is no explicit discussion of these matters in projective spaces.
- | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9291641712188721, "perplexity": 580.0538486719408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246661733.69/warc/CC-MAIN-20150417045741-00179-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/forced-vibration-on-a-cantilever-beam.65726/ | Forced Vibration on a Cantilever Beam
1. Mar 3, 2005
phiska
If a cantilever beam is subject to forced vibration (from a shaker at fixed end), what will the effect be of placing a mass at a)nodes and b)between nodes?
I presume that as there is no displacement from the normal at the nodes, the mass will have little effect, but what about between the nodes?
Any textbook, or website suggestions also greatfully received!
2. Mar 3, 2005 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8074942231178284, "perplexity": 2591.1731887189876}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719136.58/warc/CC-MAIN-20161020183839-00456-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://www.coursehero.com/file/5863137/differentialgeometryofcurvesandsurfaces-1909/ | 498 Pages
#### differential_geometry_of_curves_and_surfaces-1909
Course: PHILOSOPHY 650, Spring 2010
School: Ohio State
Word Count: 143184
Rating:
###### Document Preview
TREATISE mmm Wf / I xSm*^i?c?^n A ON THE DIFFERENTIAL GEOMETRY OF PURVES AND SURFACES BY LUTHER PFAHLER EISENHART PROFESSOR OF MATHEMATICS IN PRINCETON UNIVERSITY GINN AND COMPANY BOSTON NEW YORK CHICAGO LONDON CvA COPYRIGHT, 1909, BY LUTHER PFAHLER EJSENHART ALL RIGHTS RESERVED 89-8 gftc SUftengum PKOU.S.A. GINN AND COMPANY PRILTORS BOSTON 6 A-.ATH.. STAT. LIBRARY PEEFACE This book feel that...
##### Unformatted Document Excerpt
Coursehero >> Ohio >> Ohio State >> PHILOSOPHY 650
Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
TREATISE mmm Wf / I xSm*^i?c?^n A ON THE DIFFERENTIAL GEOMETRY OF PURVES AND SURFACES BY LUTHER PFAHLER EISENHART PROFESSOR OF MATHEMATICS IN PRINCETON UNIVERSITY GINN AND COMPANY BOSTON NEW YORK CHICAGO LONDON CvA COPYRIGHT, 1909, BY LUTHER PFAHLER EJSENHART ALL RIGHTS RESERVED 89-8 gftc SUftengum PKOU.S.A. GINN AND COMPANY PRILTORS BOSTON 6 A-.ATH.. STAT. LIBRARY PEEFACE This book feel that is a development from courses which I have given in number of years. During this time I have come to more would be accomplished by my students if they had an otherwise adapted to the introductory treatise written in English and Princeton for a use of men beginning their graduate work. the method Chapter I is devoted to the theory of twisted curves, in general being that which is usually followed in discussions of this I have introduced the idea of moving axes, subject. But in addition and have derived the formulas pertaining thereto from the previously In this way the student is made familiar with a method which is similar to that used by Darboux in the first volume of his Lemons, and to that of Cesaro in his Geometria obtained Frenet-Serret formulas. not only of great advantage in the treat ment of certain topics and in the solution of problems, but it is valu Intrinseca. This method is able in developing geometrical thinking. book may be divided into three parts. The deals with the geometry of a sur first, consisting of Chapters II-YI, face in the neighborhood of a point and the developments therefrom, The remainder of the such as curves and systems tions. of curves defined by differential equa by large extent the method from the discussion of two quad properties of a surface are derived ratic differential forms. However, little or no space is given to the To a is that of Gauss, which the and their invariants. In algebraic treatment of differential forms as defined in the first chapter, addition, the method of moving axes, has been extended so as to be applicable to an investigation of the properties of surfaces and groups of is surfaces. The extent of the no attempt has theory concerning ordinary points consider the exceptional problems. For a discussion been made to of such questions as the existence of integrals of differential equa tions and boundary conditions the reader must consult the treatises which deal particularly with these subjects. In Chapters VII and VIII the theory previously developed is as the quadrics, ruled applied to several groups of surfaces, such minimal surfaces, surfaces of constant total curvature, and surfaces, so great that surfaces with plane and spherical lines of curvature. iii iv PEEFACE The idea of applicability of surfaces is introduced in Chapter III as a particular case of conformal representation, and throughout the book attention called to examples of applicable surfaces. However, the general problems concerned with the applicability of surfaces are discussed in Chapters IX and X, the latter of which deals entirely is with the recent method of Weingarten and its developments. The remaining four chapters are devoted to a discussion of infinitesimal deformation of surfaces, congruences of straight lines and of circles, and triply orthogonal systems of surfaces. It will be noticed that the book contains many them examples, and the are merely direct student will find that applications of w hereas r certain of formulas, theory which might properly be included as portions of a more ex tensive treatise. At first I felt constrained to give such references as the others constitute extensions of the would enable the reader to consult the journals and treatises from which some of these problems were taken, but finally it seemed best to furnish no such key, only to remark that the Encyklopadie der mathematisclicn Wissenschaften may be of assistance. And the same may the book. be said about references to the sources of the subject-matter of Many important citations have been made, but there has not been an attempt to give every reference. However, I desire to acknowledge my indebtedness to the treatises of Darboux, Bianchi, and Scheffers. But the difficulty is that for many years I have con sulted these authors so freely that now it is impossible for except in certain cases, what specific debts I owe to each. me to say, In its present form, the material of the first eight chapters has been given to beginning classes in each of the last two years; and the remainder of the book, with certain enlargements, has constituted an advanced course which has been followed several times. It is im suitable credit for the suggestions made and for me to give possible the assistance rendered by my students during these years, but I am conscious of helpful suggestions made by my colleagues, Professors Veblen, Maclnnes, and Swift, and by my former colleague, Professor Bliss of Chicago. I wish also to thank Mr. A. K. Krause for making the drawings for the figures. It remains for me to express my appreciation of the courtesy shown by Ginn and Company, and of the assistance given by them during the printing of this book. LUTHER PFAHLER E1SENHART CONTENTS CHAPTEE SECTION 1. I CURVES IN SPACE PAGE 1 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. PARAMETRIC EQUATIONS OF A CURVE OTHER FORMS OF THE EQUATIONS OF A CURVE LINEAR ELEMENT TANGENT TO A CURVE ORDER OF CONTACT. NORMAL PLANK CURVATURE. RADIUS OF FIRST CURVATURE OSCULATING PLANE PRINCIPAL NORMAL AND BINORMAL OSCULATING CIRCLE. CENTER OF FIRST CURVATURE TORSION. FRENET-SERRET FORMULAS FORM OF CURVE IN THE NEIGHBORHOOD OF A POINT. THE SIGN OF TORSION CYLINDRICAL HELICES 3 4 6 8 9 10 12 14 .... 16 18 20 22 13. INTRINSIC EQUATIONS. FUNDAMENTAL THEOREM 14. 15. RICCATI EQUATIONS 25 THE DETERMINATION OF THE COORDINATES OF A CURVE DEFINED BY ITS INTRINSIC EQUATIONS 27 30 16. 17. MOVING TRIHEDRAL ILLUSTRATIVE EXAMPLES OSCULATING SPHERE .33 37 39 41 18. 19. 20. 21. 22. BERTRAND CURVES TANGENT SURFACE OF A CURVE INVOLUTES AND EVOLUTES OF A CURVE MINIMAL CURVES 43 . 47 CHAPTER II CURVILINEAR COORDINATES ON A SURFACE. ENVELOPES 23. 24. 25. 26. PARAMETRIC EQUATIONS OF A SURFACE PARAMETRIC CURVES TANGENT PLANE ONE-PARAMETER FAMILIES OF SURFACES. ENVELOPES v 52 54 56 .... 59 vi SECTION 27. 28. 29. CONTENTS PAGE DEVELOPABLE SURFACES. RECTIFYING DEVELOPABLE APPLICATIONS OF THE MOVING TRIHEDRAL ENVELOPE OF SPHERES. CANAL SURFACES .... 61 04 66 CHAPTER III LINEAR ELEMENT OF A SURFACE. DIFFERENTIAL PARAME TERS. CONFORMAL REPRESENTATION 30. LINEAR ELEMENT ISOTROPIC DEVELOPABLE 70 72 72 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. TRANSFORMATION OF COORDINATES ANGLES BETWEEN CURVES. THE ELEMENT OF AREA FAMILIES OF CURVES MINIMAL CURVES ON A SURFACE VARIATION OF A FUNCTION DIFFERENTIAL PARAMETERS OF THE FIRST ORDER DIFFERENTIAL PARAMETERS OF THE SECOND ORDER SYMMETRIC COORDINATES ISOTHERMIC AND ISOMETRIC PARAMETERS ISOTHERMIC ORTHOGONAL SYSTEMS CONFORMAL REPRESENTATION ISOMETRIC REPRESENTATION. APPLICABLE SURFACES CONFORMAL REPRESENTATION OF A SURFACE UPON ITSELF CONFORMAL REPRESENTATION OF THE PLANE SURFACES OF REVOLUTION CONFORMAL REPRESENTATIONS OF THE SPHERE . .... 74 78 81 82 84 .... 88 91 93 95 98 .... . . 100 101 104 107 109 CHAPTER IV GEOMETRY OF A SURFACE IN THE NEIGHBORHOOD OF A POINT 48. 49. 50. 51. 52. 53. 54. 55. 56. FUNDAMENTAL COEFFICIENTS OF THE SECOND ORDER RADIUS OF NORMAL CURVATURE PRINCIPAL RADII OF NORMAL CURVATURE LINES OF CURVATURE. .... 114 117 118 121 EQUATIONS OF RODRIGUES TOTAL AND MEAN CURVATURE EQUATION OF EULER. DUPIN INDICATRIX CONJUGATE DIRECTIONS AT A POINT. CONJUGATE SYSTEMS ASYMPTOTIC LINES. CHARACTERISTIC LINES CORRESPONDING SYSTEMS ON Two SURFACES GEODESIC CURVATURE. GEODESICS 123 124 . 126 128 ....... ... 130 131 133 57. 58. FUNDAMENTAL FORMULAS . CONTENTS SECTION 59. GO. 61. vii PAGE 137 141 62. GEODESIC TORSION SPHERICAL REPRESENTATION RELATIONS BETWEEN A SURFACE AND ITS SPHERICAL REPRE SENTATION HELICOIDS 143 146 CHAPTEE V FUNDAMENTAL EQUATIONS. THE MOVING TRIHEDRAL 63. ClIRISTOFFEL SYMBOLS 152 64. 65. 66. 67. 68. THE EQUATIONS OF GAUSS AND OF CODAZZI FUNDAMENTAL THEOREM FUNDAMENTAL EQUATIONS IN ANOTHER FORM TANGENTIAL COORDINATES. MEAN EVOLUTE THE MOVING TRIHEDRAL FUNDAMENTAL EQUATIONS OF CONDITION LINEAR ELEMENT. LINES OF CURVATURE CONJUGATE DIRECTIONS AND ASYMPTOTIC DIRECTONS. SPHER ICAL REPRESENTATION FUNDAMENTAL RELATIONS AND FORMULAS PARALLEL SURFACES SURFACES OF CENTER FUNDAMENTAL QUANTITIES FOR SURFACES OF CENTER SURFACES COMPLEMENTARY TO A GIVEN SURFACE . . 153 157 160 162 166 69. 168 171 172 70. 71. 72. 73. 74. 75. 76. 174 177 179 . 181 184 CHAPTER VI SYSTEMS OF CURVES. GEODESICS 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. ASYMPTOTIC LINES SPHERICAL REPRESENTATION OF ASYMPTOTIC LINES FORMULAS OF LELIEUVRE. TANGENTIAL EQUATIONS CONJUGATE SYSTEMS OF PARAMETRIC LINES. INVERSIONS SURFACES OF TRANSLATION ISOTHERMAL-CONJUGATE SYSTEMS SPHERICAL REPRESENTATION OF CONJUGATE SYSTEMS TANGENTIAL COORDINATES. PROJECTIVE TRANSFORMATIONS EQUATIONS OF GEODESIC LINES GEODESIC PARALLELS. GEODESIC PARAMETERS GEODESIC POLAR COORDINATES AREA OF A GEODESIC TRIANGLE LINES OF SHORTEST LENGTH. GEODESIC CURVATURE GEODESIC ELLIPSES AND HYPERBOLAS . 189 191 .... .... . . 193 195 197 198 . . . 200 201 204 206 207 209 .... . . 212 213 viii CONTENTS PAGE 214 . . . SECTION 91. 92. 93. 94. SURFACES OF LIOUVILLE INTEGRATION OF THE EQUATION OF GEODESIC LINES GEODESICS ON SURFACES OF LIOUVILLE LINES OF SHORTEST LENGTH. ENVELOPE OF GEODESICS 215 218 220 CHAPTEK VII QUADRICS. RULED SURFACES. MINIMAL SURFACES 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. CONFOCAL QUADRICS. ELLIPTIC COORDINATES FUNDAMENTAL QUANTITIES FOR CENTRAL QUADRICS FUNDAMENTAL QUANTITIES FOR THE PARABOLOIDS LINES OF CURVATURE AND ASYMPTOTIC LINES ox QUADRICS GEODESICS ON QUADRICS GEODESICS THROUGH THE UMBILICAL POINTS ELLIPSOID REFERRED TO A POLAR GEODESIC SYSTEM PROPERTIES OF QUADRICS EQUATIONS OF A RULED SURFACE LINE OF STRICTION. DEVELOPABLE SURFACES CENTRAL PLANE. PARAMETER OF DISTRIBUTION PARTICULAR FORM OF THE LINEAR ELEMENT ASYMPTOTIC LINES. ORTHOGONAL PARAMETRIC SYSTEMS MINIMAL SURFACES LINES OF CURVATURE AND ASYMPTOTIC LINES. ADJOINT MINI MAL SURFACES MINIMAL CURVES ON A MINIMAL SURFACE DOUBLE MINIMAL SURFACES ALGEBRAIC MINIMAL SURFACES ASSOCIATE SURFACES FORMULAS OF SCHWARZ . . . . 226 . 229 .... . 230 232 231 236 . 236 239 241 242 244 247 . . 248 250 253 254 258 260 263 264 . r CHAPTER VIII SURFACES OF CONSTANT TOTAL CURVATURE. TF-SURFACES. SURFACES WITH PLANE OR SPHERICAL LINES OF CUR VATURE 115. 116. 117. 118. 119. 120. 121. SPHERICAL SURFACES OF REVOLUTION PSEUDOSPHERICAL SURFACES OF REVOLUTION GEODESIC PARAMETRIC SYSTEMS. APPLICABILITY TRANSFORMATION OF HAZZIDAKIS TRANSFORMATION OF BIANCHI TRANSFORMATION OF BACKLUND THEOREM OF PERMUTABILITY 270 272 275 278 280 284 286 CONTENTS SECTION 122. 123. ix PAGE . . TRANSFORMATION OF LIE JF-SURFACES. 289 291 FUNDAMENTAL QUANTITIES . 124. 125. 126. 127. 128. 129. EVOLUTE OF A TF-SuitFACE SURFACES OF CONSTANT MEAN CURVATURE RULED IF-SuRFACES SPHERICAL REPRESENTATION OF SURFACES WITH PLANE LINES OF CURVATURE IN BOTH SYSTEMS SURFACES WITH PLANE LINES OF CURVATURE IN BOTH SYSTEMS SURFACES WITH PLANE LINES OF CURVATURE IN ONE SYSTEM. 294 ....... 296 299 300 302 SURFACES OF MONGE 130. 305 307 308 131. 132. 133. 134. MOLDING SURFACES SURFACES OF JOACHIMSTHAL SURFACES WITH CIRCULAR LINES OF CURVATURE CYCLIDES OF DUPIN SURFACES WITH SPHERICAL LINES OF CURVATURE SYSTEM 310 312 IN ONE 314 CHAPTER IX DEFORMATION OF SURFACES 135. 136. 137. 138. 139. 140. 141. 142. PROBLEM OF MINDING. SURFACES OF CONSTANT CURVATURE SOLUTION OF THE PROBLEM OF MINDING DEFORMATION OF MINIMAL SURFACES SECOND GENERAL PROBLEM OF DEFORMATION DEFORMATIONS WHICH CHANGE A CURVE ON THE SURFACE INTO A GIVEN CURVE IN SPACE LINES OF CURVATURE IN CORRESPONDENCE CONJUGATE SYSTEMS IN CORRESPONDENCE ASYMPTOTIC LINES IN CORRESPONDENCE. DEFORMATION OF A . 321 323 327 331 333 336 338 143. RULED SURFACE METHOD OF MINDING PARTICULAR DEFORMATIONS OF RULED SURFACES 342 344 144. 345 CHAPTER X DEFORMATION OF SURFACES. THE METHOD OF WEINGARTEN 145. 146. 147. 148. REDUCED FORM OF THE LINEAR ELEMENT GENERAL FORMULAS THE THEOREM OF WEINGARTEN OTHER FORMS OF THE THEOREM OF WEINGARTEN 351 .... . 149. SURFACES APPLICABLE TO A SURFACE OF REVOLUTION . 353 355 357 362 x SECTION 150. 151. CONTENTS PAGE 364 MINIMAL LINES ON THE SPHERE PARAMETRIC SURFACES OF GOURSAT. SURFACES APPLICABLE TO CERTAIN PARABOLOIDS 366 CHAPTER XI INFINITESIMAL DEFORMATION OF SURFACES 152. 153. 154. GENERAL PROBLEM CHARACTERISTIC FUNCTION ASYMPTOTIC LINES PARAMETRIC ASSOCIATE SURFACES .... 373 374 376 155. 156. 157. 158. 159. 378 379 PARTICULAR PARAMETRIC CURVES RELATIONS BETWEEN THREE SURFACES S, S v S SURFACES RESULTING FROM AN INFINITESIMAL DEFORMATION ISOTHERMIC SURFACES 382 385 387 CHAPTER XII RECTILINEAR CONGRUENCES 160. 161. 162. DEFINITION OF A CONGRUENCE. LIMIT POINTS. SPHERICAL REPRESENTATION 392 393 395 398 NORMAL CONGRUENCES. RULED SURFACES OF A CONGRUENCE FOCAL SURFACES PRINCIPAL SURFACES 163. DEVELOPABLE SURFACES OF A CONGRUENCE. 164. ASSOCIATE NORMAL CONGRUENCES 165. 166. 167. 401 168. 169. 170. 171. 172. 173. DERIVED CONGRUENCES FUNDAMENTAL EQUATIONS OF CONDITION SPHERICAL REPRESENTATION OF PRINCIPAL SURFACES AND OF DEVELOPABLES FUNDAMENTAL QUANTITIES FOR THE FOCAL SURFACES ISOTROPIC CONGRUENCES CONGRUENCES OF GUICIIARD PSEUDOSPHERICAL CONGRUENCES IT-CONGRUENCES CONGRUENCES OF RIBAUCOUR ........ ...... . . . 403 406 407 409 412 414 415 417 420 CHAPTER XIII CYCLIC SYSTEMS 174. 175. 176. GENERAL EQUATIONS OF CYCLIC SYSTEMS CYCLIC CONGRUENCES SPHERICAL REPRESENTATION OF CYCLIC CONGRUENCES 426 431 . 432 CONTENTS SECTION 177. 178. XI PAGE 436 437 439 179. 180. SURFACES ORTHOGONAL TO A CYCLIC SYSTEM NORMAL CYCLIC CONGRUENCES CYCLIC SYSTEMS FOR WHICH THE ENVELOPE OF THE PLANES OF THE CIRCLES is A CURVE CYCLIC SYSTEMS FOR WHICH THE PLANES OF THE CIRCLES PASS THROUGH A POINT 440 CHAPTEE XIV TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 181. TRIPLE SYSTEM OF SURFACES ASSOCIATED WITH A CYCLIC 446 447 449 451 SYSTEM 182. 183. 184. GENERAL EQUATIONS. THEOREM OF DUPIN EQUATIONS OF LAME TRIPLE SYSTEMS CONTAINING ONE FAMILY OF SURFACES OF REVOLUTION TRIPLE SYSTEMS OF BIANCHI AND OF WEINGARTEN THEOREM OF RIBAUCOUR 185. .... 452 186. 187. 188. THEOREMS OF DARBOUX TRANSFORMATION OF COMBESCURE . 457 458 461 INDEX 467 DIFFERENTIAL GEOMETRY CHAPTER I CURVES IN SPACE 1. Parametric equations of a curve. Consider space referred to fixed rectangular axes, and let (x, y, z) denote as usual the coordi nates of a point with respect to these axes. In the plane 2 = draw a circle of radius r and center (a, b). The coordinates of a point P on the circle can be expressed in the form (1) x a -{- r cos u, y = b H- r sin u, 2 = 0, where u denotes the angle which the radius to P makes with the to 360, the point P describes As u varies from positive o&gt;axis. the circle. The quantities a, 5, r determine the position and size it. of the circle, whereas u determines the position of a point upon a variable or parameter for the equations (1) are called parametric it is is In this sense circle. And equations of the circle. straight line in space A determined by a direction-cosines point on a, /3, it, P (a, Q 6, c), and its 7. The latter fix also the sense of the line. Let P distance be another point on the line, and let the P be denoted by u, which is positive Q P or negative. The rectangular coordinates of are then expressible in the form (2) P x = a + ua, of y = b + u(B, z c + wy. FIG. 1 To each value u there corresponds a point on the line, and the coordinates of any point on the line are expressible as in (2). These equations are consequently parametric equations of the straight line. When, as in fig. 1, a line segment PD, of constant length , per pendicular to a line OZ at D, revolves uniformly about OZ as axis, 2 CURVES IN SPACE and at the same time locus of D moves along it P is called a circular helix. If with uniform velocity, the the line OZ be taken for the a&gt;axis, 2-axis, the initial position of PD for the positive and the angle between the u, latter and a subsequent position of PD be denoted by the equations of the helix can be written in the parametric form (3) x =a cos u, y a sin u, z = bu, where the constant PD and of translation of D. radian, D determined by the velocity of rotation of Thus, as the line PD describes a moves the distance b along OZ. b is the above equations u is the variable or parameter. Hence, with reference to the locus under consideration, the coordi indicate this by writing these nates are functions of u alone. In all of We equations The functions / /2 / have x, , definite circle, straight line or circular helix. eral case and consider equations (4), forms when the locus is a But we proceed to the gen when /r /2 / are any func , tions whatever, analytic for all values of u, or at least for a certain domain.* The locus of the point whose coordinates are given by (4), as u takes all values in the domain considered, is a curve. Equa tions (4) are said to be the equations of the curve in the parametric all the points of the curve do not lie in the same plane form. When it is called a space curve or a twisted curve ; otherwise, a plane curve. It is evident that a necessary and sufficient condition that a curve, defined linear relation (5) by equations (4), be plane, between the functions, such ofi + 5f2 + c/3 is that there exist a as +d= 0, where dition If (6) a, b, c, is satisfied d denote constants not all equal to zero. This con by equations (1) and (2), but not by (3). by any function of *fc*^(*0, v, u in (4) be replaced say equations (4) assume a new form, * E.g. in case it is u is when complex, it lies two fixed values; supposed to be real, it lies on a segment between within a closed region in the plane of the complex variable. EQUATIONS OF A CURVE It is evident that the values of x, y, z, 3 (7) for a given by value are equal to those given by i u obtained from (6). Consequently equations (4) and (7) define the same curve, u and v being the respective parameters. Since of of , (4) for the corresponding value there is no restriction upon the function &lt;/&gt;, except that it be ana lytic, it follows that a curve can be given parametric representation in an infinity of ways. 2. Other forms of the equations of a curve. If the first of equa tions (4) be solved for w, giving parameter, equations (7) are (8) u $(#), then, in terms of x as x = x, is y = F (x), 2 z = F (x). 8 In this form the curve or, if it really defined by the its last two equations, is be a plane curve in the o?y-plane, equation in the customary form (9) y =/(*) points in space whose coordinates satisfy the equation lie on the cylinder whose elements are parallel to the The z y = F (x) 2-axis and whose cross section by the xy-pl&ne is the curve y = F2 (x). In like manner, the equation z = F3 (x) defines a cylinder whose Hence the curve with the common to two cylinders equations (8) with perpendicular axes. Conversely, if lines are drawn through the points of a space curve normal to two planes perpendicular to one another, we obtain two such cylinders whose intersection is the given curve. Hence equations (8) furnish a perfectly gen elements are parallel to the #-axis. is the locus of points eral definition of a space curve. in In general, the parameter u can be eliminated from equations (4) such a way that there result two equations, each of which in all volves (10) three rectangular coordinates. Thus, y, z) Qfa if y, z) = 0, &lt;S&gt; a (a;, = 0. Moreover, the form two equations x, of this kind be solved for y and z as functions of get equations of the form (8), and, in turn, of of u. Hence (4), by replacing x by an arbitrary function It will also are the general equations of a curve. equations (10) we be seen later that each of these equations defines a surface. 4 It CURVES IN SPACE should be remarked, however, that when a curve is defined (10), it as the intersection of two cylinders (8), or of two surfaces may happen that these curves of intersection consist of several parts, so that the new equations define more than the original ones. For example, the curve defined by the parametric equations (i) x = w, y = w2 , z = w3 , is a twisted cubic, for every plane meets the curve in three points. Thus, the plane ax + by -f cz + + d = meets the curve in the three points whose parametric values are the roots of the e(l uation CM* lies + &n" + an d = 0. This cubic upon the three cylinders y = x2 , z = x3 , y3 = z2 . of the first and second cylinders is a curve of the sixth degree, of the sixth degree, whereas the last two intersect in a curve of the ninth degree. Hence in every case the given cubic is only a part of The intersection of the first and third it is the curve of intersection Again, (ii) we may eliminate that part which lies on all three cylinders. u from equations (i), thus xy = z, y* = xz, and the second a hyperboliclies on both of these surfaces, parabolic cone. The straight line y = 0, z = but not on the cylinder y = x 2 Hence the intersection of the surfaces (ii) consists of which the first defines a hyperbolic paraboloid . of this line and the cubic. The generators of the paraboloid are defined by x = a, z = ay 6. ; y 6, z = bx ; for all values of the constants a and From (i) we see that the cubic meets each generator of the 3. is first family in one point and of the second family in two points. Linear element. By the limit, when it exists, definition the length of an arc of a curve toward which the perimeter of an inscribed polygon tends as the number of sides increases lengths uniformly approach zero. Curves for does not exist will be excluded from the subsequent discussion. and their which such a limit Consider the arc of a curve whose end points m mined by the parametric values U Q and # and let intermediate points with parametric values u^ w 2 , , , ma are mv m , deter , 2, be . The length l k of the chord mkmk+l is =V2,r/;.^, L1 )-/v(oi 2 . = i, 2, 3 LINEAR ELEMENT By the 5 is mean value theorem of the differential calculus this equal to where f . t = wt + 0&lt;( w*+i ~ %) differentiation. &lt; ^ &lt; * and the primes indicate , As denned, the length of the arc m a is the limit of 2Z4 , as the lengths in k k+l tend to zero. From the definition of a definite m m integral this limit is equal to ra n ) Hence, if s denotes the length of the arc from a fixed point (u to a variable point (u), we have This equation gives (12) s as a function of w. We write it =(w), (11) it follows that and from which we may write (14) in the form i ds ds 2 =dx As thus expressed is called the element of length, or linear element, of the curve. In the preceding discussion we have tacitly assumed that u real. is When it is complex we take equation (11) as the definition of the length of the arc. If equation (12) be solved for u in terms of s, and the result be substituted in (4), the resulting equations also define the curve, and s is the parameter. From (11) follows the theorem : necessary and sufficient condition that the parameter arc measured from the point U = U Q is 2 A u be the (15) /, +/r+/s = l2 2 An exceptional case should be noted here, namely, /r+/ "+/ a 2 =o. 6 CURVES IN SPACE // 2, 3 Unless//, be zero and the curve reduce to a point, at least one of the coordinates must be imaginary. For this case s is zero. Hence these imaginary curves are called curves of length zero, or minimal curves. For the present they will be excluded from the discussion. Let the arc be the parameter of a given curve and s and s + e its values for two points M(x, y, z) and M^(x^ y^ z^. By Taylor s theorem we have (17) ^ =z -f z e where an accent indicates Unless x 1 differentiation with respect to is, s. , y , z ! are all zero, that unless the locus is a point z is zl and not a curve, one at least of the lengths x l x, If these lengths be denoted by of the order of magnitude of e. &u, y^y, %, Sz, and e by 8, then we have denotes the aggregate of terms of the second and higher the ratio of the lengths orders in 8s. Hence, as l approaches limit of the chord and the arc approaches unity and in the where 1 2 M M MM . l ; we have 4. ds 2 = 2 dx2 + dy 2 + dz is tangent to a curve at a point of a point and l the limiting position of the secant through as a limit. the curve as the latter approaches In order to find the equation of the tangent we take s for par in the 1 ameter and write the expressions for the coordinates of are and The equations of the secant through l form Tangent to a curve. The M M M M (17). M M M of these equations be multiplied by e and the denominators be replaced by their values from (17), we have in If each member the limit as M 1 approaches M y TANGENT TO A CURVE If #, /3, 7 7 denote the direction-cosines of the tangent in conse quence of (15), we may take u * any whatever, these equations are f! /2 . When * (20) 9 the parameter ft Jl / is a= nt *i ^ * o =, . /wo 0= / /*o . /*-&gt;. = = &gt; /!*) y * = -= // ft 3 They may /oi\ also be written thus : 21) a dx = :T ds = dy ds / dz T" V ds From these equations it follows that, if the convention be made that the positive direction on the curve is that in which the par ameter increases, the positive direction upon the tangent is the same as upon the curve. fundamental property of the tangent is discovered by con sidering the expression for the distance from the point M^ with the coordinates (17), to any line through M. tion of such a line in the form A We write the equa (22) *= a a, 5, c = !=* = b =, c where are the direction-cosines. The (23) distance from {[(bx&gt;- M l to this line is equal to $$bx"- ay")e* ay )e + + 1 2 ] 2 . bz )e + 2 -] + [(az - cx )e + -] }*. Hence, if MM l be considered an infinitesimal of the first order, this distance also is of the first order unless in which case it is of the second order at least. But when these equations are satisfied, equations (22) define the tangent at M. Therefore, of all the lines through a point of a curve the tangent is nearest to the curve. , , * Whenever the functions x y z appear in a formula it is understood that the arc s is the parameter otherwise we use /{, /2 /3 indicating by accents derivatives with respect to the argument u. ; , , 8 5. CURVES IN SPACE Order of contact. Normal plane. When the curve is such that there are points for which (24) ^=4 x y z the distance from to the tangent is of the third order at least. l In this case the tangent is said to have contact of the second order, whereas, ordinarily, the contact is of the first order. And, in gen eral, the tangent to a curve has contact of the wth order at a point, M if the following conditions are satisfied for n = 2, , n 1, and n : &lt; 25 ) ^=^=^rr Jl fl ~jTf xw i(0 jyV V 5&gt;V"y the parameter of the curve is any whatever, equations (24), (25) are reducible to the respective equations f (rt-1) 2, When J% J% // J\ -f(.n-$$ J Ja f(n-l) The plane normal contact (26) is to the tangent to a curve at the point of normal plane at the point. Its equation is called the (X a, /3, x) a + (Y y} ft + (Z z) 7 = 0, where 7 have the values (20). EXAMPLES 1. Put the equations of the circular helix (3) in the form (8). Express the equations of the circular helix in terms of the arc measured from a point of the curve, and show that the tangents to the curve meet the elements of the circular cylinder under constant angle. 2. 3. Show is that if at every point of a curve the tangency line. is of the second order, the curve 4. a straight sufficient condition that at the point (x 2/o) of = / "(BO) f(x) the tangent has contact of the nth order is/"(x ) the tangent crosses the = also, that according as n is even or odd r=/()(z ) = curve at the point or does not. , Prove that a necessary and the plane curve y . . . ; Prove the following properties of the twisted cubic the cubic one and only one meets the (a) Of all the planes through a point of 3 cubic in three coincident points its equation is 3 u*x - 3 uy + z - w = 0. on a plane has a the orthogonal projection (6) There are no double points, but 5. : ; double point. a variable chord of the cubic and by each of (c) Four planes determined by four fixed points of the curve are in constant cross-ratio. FIKST 6. CUKVATUKE first 9 Let Jf, be two l points of a curve, As the length of the arc between these points, and A0 the angle between the tangents. The limiting value of Curvature. Radius of curvature. M A0/As as M l approaches Jf, namely dd/ds, measures the rate of as the point of con change of the direction of the tangent at tact moves along the curve. This limiting value is called the its reciprocal the radius of the latter will be denoted by p. first In order to find an expression for p in terms of the quantities defining the curve, we introduce the idea of spherical representa first M curvature of the curve at M, and ; curvature take the sphere * of unit radius with center at the origin and draw radii parallel to the positive directions ofthe tangents to the curve, or such a portion of it that no two tion as follows. We tangents are parallel. The locus of the extremities is a curve upon the sphere, which is in one-to-one correspondence with the given curve. In this sense we have a spherical representation, or spherical indicatrix, of the curve. The angle A# between the tangents to the curve at the points M, M^ is measured by the arc of the great circle between their on the sphere. If ACT denotes the representative points m, l of the arc of the spherical indicatrix between and m^ length then by the result at the close of 3, m m dO =v lim ! p A&lt;9 = 1. da Ao- Hence we have (27) = ds , where da- is The the linear element of the spherical indicatrix. coordinates of are the direction-cosines a, /3, 7 of the m tangent at M\ consequently When (28) the arc s is the parameter, this formula becomes ^.jji+yw+gw * Hereafter we refer to this as the unit sphere. 10 CURVES IN SPACE is However, when the parameter from (12), (13), (20), any whatever, u, we have and W-fifl+JSJf*flfr find Hence we (30) by substitution which sometimes is written thus : \ _(d*x}* fmake sign of p is not determined by these formulas. the convention that it is always positive and thus fix the sense of The We a displacement on the spherical indicatrix. 7. Osculating plane. and through a point M^ of the curve. The a curve at a point is called the of this plane as approaches l limiting position and thus establish osculating plane at M. In deriving its equation M Consider the plane through the tangent to M M ing its existence we assume that the arc s is the parameter, and in the form (17). take the coordinates of l M The equation (32) of a plane through M (x, y, z) is of the form X, Y", Z being (X- x)a + (Y- y)l + (Z-z)c = Q, the current coordinates. When the Jf, through the tangent at (33) If the values (17) for the coefficients a, c &gt;, plane passes are such that xa a; , +yb+ zc= y^ z^ 0. be substituted in (32) for X, F, Z, e* , and the resulting equation be divided by we get where limit (34) 77 represents the aggregate of the terms of first orders in As e. we have M 1 approaches Jf, 77 approaches zero, and higher and in the x"a + y"b + z"c = 0. OSCULATING PLANE Eliminating a, 11 c &gt;, from equations (32), (33), (34) we obtain, as the equation of the osculating plane, X-x Y-y Z-z (35) x 1 y y" = 0. x" From plane (36) this we find that when the curve w, in terms of a general is parameter is defined by equations (4) the equation of the osculating x_ x Y // y /," z_ z fi The plane defined by either of these equations is unique except when the tangent at the point has contact of an order higher than the In the latter case equations (33), (34) are not independent, as follows from (24); and if the contact of the tangent is of the wth order, the equations = first. ^+^+^ ~\., G 0? for all values of r one another. up to and including n are not independent of this equation and (33) are inde But for r = n + pendent, and we have as the equation of the osculating plane at this singular point, X-x Yx Z-z = 0. y plane, and its plane is taken for the rry-plane, reduces to Z = 0. Hence the osculating plane the equation (35) of a plane curve is the plane of the latter, and consequently is the same for all points of the curve. Conversely, when the osculating When a curve is plane of a curve is the same for all its points, the curve is plane, for all the points of the curve lie in the fixed osculating plane. The equation reducible to of the osculating plane of the twisted cubic (2) is readily where JT, F, Z are current coordinates. From the definition of the osculating plane and the fact that the curve is a cubic, it follows that the osculating plane meets the curve only at the point of osculation. As equation (i) is a cubic in w, it follows that through a point (o, 2/o, ZQ) not on tne cnrve there pass three planes which osculate the cubic. Let MI, w 2 u 3 denote the parameter values of these points. , Then from (i) we have = 3 XG, 3 ?/o, 2t i \ n 12 By means CURVES IN SPACE of these relations the equation of the plane through the corresponding is reducible to three points on the cubic (X - XQ) 3 7/0 - Y( y , ) 3x + ZQ) 5 (Z - z ) = 0. : This plane passes through the point (x 2/0, hence we have the theorems The points of contact of the three osculating planes of a twisted cubic through a point not on the curve lie in a plane through the point. The osculating planes at three points of a twisted cubic meet in a point which lies in the plane of the three points. By means of these theorems we can establish a dual relation in space by mak ing a point correspond to the plane through the points of osculation of the three osculating planes through the point, and a plane to the point of intersection of the three planes which osculate the cubic at the points where it is met by the plane. In particular, to a point on the cubic corresponds the osculating plane at the point, and vice versa. 8. Principal normal and binormal. Evidently there are an in finity of normals to a curve at a point. : Two of these are of par ticular interest the normal, which lies in the osculating plane at the point, called the principal normal; and the normal, which is perpendicular to this plane, called the binormal. If the direction-cosines of the binormal be denoted by X, /&gt;t, z&gt;, we have from X : (35) : / v = (y z"- z y") : (z x"~ z : z") (* / -y x"). In consequence of the identity the value of the (28) to common ratio is reducible by means of (19) and take the positive direction of the binormal to p.* be such that this ratio shall" be -f- p then ; We (37) \ = p(y z"-z f y"), ^ = P (z is x"~ x z"), v = p(x y When (38) the parameter u general, these formulas are x= 282 : or in other form /oof ~ P dycPz *For / dzcfy ~~df~ 0, ^~ P ~ dzd*xdxd?z ds 3 ~ P dxd y 1 dyd^x ds* SV = as is seen by differentiating 2x"*= with respect to s. PRINCIPAL NORMAL AND BINORMAL By definition the principal 13 normal tangent and binormal. We make direction is such that the positive directions of the tangent, prin cipal normal and binormal at a point have the same mutual ori entation as the positive directions of the x-, y-, z-axes respectively. perpendicular to both the the convention that its positive is These directions are represented in fig. 2 by the lines if Z, MT, MC, MB. Hence, m, n, denote the direc tion-cosines of the principal normal, we have* (39) m /JL =4-1, FIG. 2 from which it follows that I = mv = \ = ftn a I fjij , nu, vft^ 777*, ft n\ va yl Iv, 7 m= /Ji \7, #n, n i^ = = Z/i = Xp = am m\, yuo:, ftl. Substituting the values of #, /3, 7; X, /u-, i^ from (19) and (37) in the m, w, the resulting equations are reducible to expressions for , Hence, when the parameter u (42) is general, we have l=-(W2 or in other form, _ In consequence of (29) equations (42) 2 dzd 2 s may be written: da (43) dft dj ds ds (27), ds or by means of da m dft do- dy da- Hence the tangent to the principal to the spherical indicatrix of a curve is parallel normal to the curve and has the same sense. *C. Smith, Solid Geometry, llth ed., p. 31. 14 9. CURVES IN SPACE Osculating circle. Center of first curvature. We have defined the osculating plane to a curve at a point to be the limiting of the plane determined by the tangent at and by a position of the curve, as the latter approaches the curve. point l along M M M M We consider now gent at to the tion of this circle, as M as the curve, and passes through M The limiting posi M approaches called the osculating { . the circle in this plane which has the same tan l 7I/, is circle curve at M. It is evident that its center C is on the prin normal at M. Hence, with reference F the coordinates of (7 denoted by Q cipal , to X , , Z Q, any fixed axes in space, are of the form rn, X =x + rl, Y y -f rm, Z^=z + where the absolute value of of the circle, r is the radius of the osculating circle. r, In order to find the value of we return to the consideration its when M l does not have limiting position, and we let X, F, Z\ Zj, m^ n^ r^ denote respectively coordinates of the cen ter of the circle, the direction-cosines of the diameter through and the radius. If x v y v z l be the coordinates of M^ they have the M values (17), and since rl If M l is on the circle, we have i 2 e*x". = 2(A - xtf = 2(7-^- ex 2 -) . we divide ^x ^ = 0, and through by e we have notice that , after reducing the above equation 1-r^Z/ where 77 +*? = (), involves terms of the limi-t r l becomes r, ^x\ and higher orders in e. In the becomes 2z 7, that is - and this equation first , reduces to so that r is equal to the radius of curvature. circle On this account the of curvature and its center the osculating circle is called the center of first curvature for the point. Since r is positive the center of curvature is on the positive half of the principal normal, and coordinates are pl, consequently (44) its X =x + Y =y + pm, Z = z + pn. Q CENTER OF CURVATURE The ture is 15 line normal to the osculating plane at the center of curva called the polar line or polar of the curve for the corre sponding point. Its equations are /45\ X-x-pl = Y-y-pm = Zz \ fig. JJL pn ^ v In 2 C represents the center of curvature and CP the polar line for M. A curve may be looked upon as the path of a point moving under the action of a system of forces. From this point of view it is convenient to take for parameter the time which has elapsed since the point passed a given position. Let t denote this parameter. As t is a function of 8, we have dx _dx ~ ds _ ds C dy ds~dt~ *dt Hence the _ dt~ ds ~dt dz _ ds ~dt~ y ~dt rate of change of the position of the point with the time, or its velocity, laid off dt may be represented by the length (41), on the tangent to the curve. In like manner, by means of we have d^ _ ~ From this it is seen that the rate of d?s n /ds\ 2 7 change of the velocity at a point, or the acceleration, be represented by a vector in the osculating plane at the point, through the latter and whose components on the tangent and principal normal may d*s - , and - 1 /dY I ) df* P \dtj EXAMPLES 1. Prove that the curvature of a plane curve defined by the equation M (x, y)dx cy p 2. ex (J/ 2 + N z Show that the normal planes to the curve, x a sin 2 it, y = a sin u cos w, = a cos M, pass through the origin, and find the spherical indicatrix of the curve. 3. The straight line is the only real curve of zero curvature at every point. : 4. (a) Derive the following properties of the twisted cubic is In any plane there planes can be drawn. (6) one line, and only one, through which two osculating fixed osculating planes are cut by the line of intersection of any two osculating planes in four points whose cross-ratio is constant. and four fixed points of the curve (c) Four planes through a variable tangent Four are in constant cross-ratio. (d) What is the dual of (c) by the results of 7? 16 5. CURVES IN SPACE Determine the form of the function curve x 6. = w, y = sin w, z &lt;f&gt; so that the principal normals to the (u) are parallel to the yz-plane. first Find the osculating plane and radius of x a cos u -f 6 sin w, y = a sin u curvature of z + 6 cos w, = c sin 2 u. 10. Torsion. Frenet-Serret formulas. less a It has been seen that, un curve be plane, the osculating plane varies as the point moves along the curve. The change in the direction depends evidently upon the form of the curve. The ratio of the angle A^ between the binormals at two points of the curve and their curvi linear distance As expresses our idea of the mean change in the direction of the osculating plane. this ratio, as And so we take the limit of one point approaches the other, as the measure of rate of this change at the latter point. This limit is called the the second curvature, or torsion, of the curve, and its inverse the radius of second curvature, or the radius of torsion. will be denoted by r. The latter In order to establish the existence of this limit and to find an expression for it in terms of the functions defining the curve, we draw radii of the unit sphere parallel to the positive binormals of the curve and take the locus of the end points of these radii as a second spherical representation of the curve. The coordinates of points of this representative curve on the sphere are X, /*, v. Pro ceeding in a manner similar to that in 6, we obtain the equation (46) i_ r dcr l is 2 ds* where the linear element of the spherical indicatrix of the binormals. In order that a real curve have zero torsion at every point, the cosines X, /*, v must be constant. By a change of the fixed axes, which evidently has no effect upon the form of the curve, the cosines can be given the values X = 1, /* = v = 0. const. Hence a necessary It follows from (40) that a = 0, and consequently x and sufficient condition that the torsion of a real curve be zero at every point is that the curve be plane. In the subsequent discussion we shall need the derivatives with 7; I, respect to s of the direction-cosines a, & m, w; X, p, v. We deduce them now. (4T) From a (41) /3 we have =i, =, y-. FRENET-SERRET FORMULAS In order to find the values of X respect to s , 17 /* , i/, we differentiate with the identities, X 2 +At2+z 2 = 1? , a\ + /A + 7* = 0, r and, in consequence of (47), obtain XX + 1 fjLfj, + vv 1 = 0, f : \ + /V + yv = 0. From and these, by the proportion (40), follows X fjL :v r =l:m:n, is the factor of proportionality algebraic sign of r its is 1/r, as is seen from (46). The We fix (48) not determined by thus sign by writing the above proportion the latter equation. : V= A T I ,: &gt; = T t V = T s l*&gt; If the identity = ^7 vfi be differentiated with respect to the result (49) is reducible by (40), (47), and (48) to I and n Similar expressions can be found for Gathering to fundamental in gether these results, we have the following formulas . m the theory of twisted curves, and called the Frenet-Serret formulas : (50) \^ v v ^-/l+2), Y As an example, we If the equation derive another expression for the torsion. \ = p(y z " z y") be differentiated with respect to s, the result may be written and similar ones for WI/T, rz/r be multiplied by ?, wz, n respectively and added, we have, in consequence of (50) and (41), If this equation x (51) y y" " " z z" x y z " 18 CUKVES IN SPACE The last three of equations (50) give the rate of change of the direction-cosines of the osculating plane of a curve as the point of osculation moves along the curve. From these equations it follows that a necessary and sufficient condition that this rate of change at a point be zero is that the values of s for the point make the determinant in equation (51) vanish. At such a point the osculat ing plane 11. torsion. is said to be stationary. of curve in the neighborhood of a point. The sign of have made the convention that the positive directions of the tangent, principal normal, and binormal shall have the same Form We relative orientation as the fixed take these lines at a point for axes, the equations of the curve Q can be put in a very convenient form. If the coordinates be ex M x-, y-, 2-axes respectively. When we and pressed in terms of the arc measured from = (41) that for s M^ we P have from (19) When the values of I and X from (41) and (37) are substituted in the fourth of equations (50), (5 2) we r obtain x this " = ---- (y p z" -z y") " x". p for f From for and similar expressions -i y and -i 1 z" we find that s= 2 P* p PT #, y, z Hence, by Maclauriri pressed in the form s / theorem, the coordinates -i can be ex (^%\ I \&lt;JO &lt; v v z = Ct 2p bp s J 2 s 3 -f---, -f- , 6 pr where p and r are the radii of first and second curvature at the = 0, and the unwritten terms are of the fourth and higher point s powers in s. the last of these equations it is seen that for sufficiently small values of 8 the sign of z changes with the sign of s unless From THE SIGN OF TORSION I/T 19 Hence, unless the osculating plane is stationary at the curve crosses the plane at the point.* Furthermore, a point, when a point moves along a curve in the positive direction, it side of the osculating passes from the positive to the negative at a point, or vice versa, according as the torsion at the at . = M plane or negative. In the former case the curve said to be sinistrorsum, in the latter dextrorsum. latter is positive is As another consequence variable point to the osculating plane at M on the curve approaches Jf the distance from M M of the third order of magnitude in , of this equation, we remark that as a Q is comparison with MM Q . By means any find that the distance to of the other equations (53) we is of the other plane through M : second order at most. Hence we have the theorem to The osculating plane crossed by the curve, a twisted curve at an ordinary point all the is and of planes through the point it lies nearest to the curve. it is positive for suffi or negative. Hence, in the positive neighborhood of an ordinary point, the curve lies entirely on one on side of the plane determined by the tangent and binomial From the second of (53) , is seen that y ciently small values of the side of the positive direction of the principal normal. These properties of a twisted curve are discovered, likewise, from a consideration of the projections upon the coordinate planes of the approximate curve, whose equations consist of the first terms in to the (53). The y= parabola x the cubic = *, x curve. On s, the projection on the osculating plane is whose axis is the principal normal /2 p, the plane of the tangent and binomial it is s 2 = z = s 8 /6 pr, which has the curve for an inflectional tangent. And the plane of the binormal and principal normal into the semi3 2 with the latter for s /Q pr, cubical parabola y = s /2 /o, z= cuspidal tangent. These results are represented by the following figures, which picture the pro upon the osculating plane, normal plane, and the plane of the In the third figure the heavy line corresponds to the case line to the case is tangent to the the curve projects upon jection of the curve tangent and binormal. where r positive and the dotted where r is negative. *This result can be derived readily by geometrical considerations. 20 CUEVES IN SPACE The preceding results serve also to give a means of determining; ue variation in the osculating plane as the point moves along the curve. By r ns of (50) the direction-cosines X, /u,, v can be given the form where the subscript null indicates the value of a function for s = and the un written terms are of the second and higher terms in s. If the coordinate axes are those which lead to (53), the values of X, p, v for the point of parameter 5s are X = . =-, TO v = \ to within terms of higher order, and consequently the equation of this osculating plane at this point MI is Y- + Z = TO ; 0. M we put = po, we get the z-coordinate of the point in which this plane is cut by it is the polar line for the point s = po5s/T Hence, according as T O is positive or negative at Jf, the osculating plane at the near-by point MI cuts the polar line for on the negative or positive side of the osculating plane at If Y . M . 12. Cylindrical helices. of the use of formulas (50) we derive several properties of cylindrical helices. By definition, a cylindrical helix is a curve which lies upon a cylinder and cuts the elements of the cylinder under constant angle. If the axis of As another example z be taken parallel to the elements of the cylinder, we have 7 = const. Hence, from (50), from which it follows that the cylindrical helices have the following properties : The principal normal is perpendicular to the element of the cylinder and consequently coincides with the normal to the cylinder at the point ( The radii offirst and second curvature are in constant ratio. : at the point, 22). Bertrand has established the converse theorem Every curve whose radii of first and second curvature are in constant ratio is a cylindrical helix. In order to prove = icp, and remark from (50) that it, we put T da_d\ ds ds K\ dp _ */* dfj. dy ds _ = KV dv ds ds~ /3 ds from which we get where a2 a + C2 a, = + 6, 7 + cy c, a, 6, c are constants. From these equations we find + 52 + _ i + K^ aa + bp + = CYLINDRICAL HELICES Hence the lines ta 21 -ents to the curve make V * the constant angle cos- * =; is with the whose lin 3n-cosines are ts Vl + helix, Consequently the curve K2 a cylindrical and t .u e of the helix have the above direction. EXAMPLES sin w), y = a cos w, between the points 1. Find the length of the curve x = a (u show that the locus of the center of curva TT and IT for which u has the values ture is of the same form as the given curve. ; 2. Find the coordinates of the center of curvature of x = a cos M, y = y a sin w, z a cos 2 u. 3. Find the radii of curvature and torsion of x 4. is = a (u sinw), = a (1 cosw), z = bu. If the principal normals of a curve are parallel to a fixed plane, the curve a cylindrical helix. , V% u is a cylindrical helix and that e u y = er M , z 5. Show that the curve x the right section of the cylinder is a catenary also that the curve lies upon a cylin der whose right section is an equilateral hyperbola. Express the coordinates in terms of the arc and find the radii of first and second curvature. ; 6. Show that if 6 and curve 7. first make with a fixed line in space, then denote the angles which the tangent and binormal to a sin 6 dd =r - sin &lt;/&gt; d&lt;p p When two curves are symmetric with respect to the origin, their radii of curvature are equal and their radii of torsion differ only in sign. 8. The osculating circle at an ordinary point of a curve has contact of the sec ond order with the latter and all other circles which lie in the osculating plane and are tangent to the curve at the point have contact of the first order. ; 9. A necessary and sufficient condition that the osculating circle at a point have contact of the third order is that p = and I/T = at the point at such a point ; the circle 10. is said to superosculate the curve. Show that any twisted curve may be defined by equations of the form where 11. p and r are the radii of first and second curvature at the point (4), s is 0. When the equations of a curve are in the form /I f the torsion given by /2 fff /3 f/f f where has the significance of equation (12). 22 12. CURVES IN SPACE The locus is of the centers of curvature of a twisted curve of constant first curvature 13. a curve of the same kind. all When is the osculating planes of a curve pass through a fixed point, the curve 14. plane. plane. Determine f(u) so that the curve x What is the form of the curve ? = a cosw, y = a sin w, z =f(u) shall be two curves defined Fundamental theorem. Let C^ and Cz be s, and let points each with the same values of s correspond. We assume, upon 13. Intrinsic equations. in terms of their respective arcs ture have the same value, furthermore, that at corresponding points the radii of first curva and also the radii of second curvature. We shall show that Cl and Cz are congruent. By a motion in space the points of the two curves for which = can be made to coincide in such a way that the tangents, s principal normals, and binomials to them at the point coincide also. Hence s indicate use the notation of the preceding sections and by subscripts 1 and 2 the functions of Cl and C2 we have, if , we when (54) = 0, xl =x z, al =a z, ^ = Z 2, \=\ z, and other similar equations. The Frenet-Serret formulas for the two curves are ds ds r ds -=ds = = p &gt; = ds ds ( -| -- I = r -&gt; \p T/ the functions without subscripts being the same for both curves. If the equations of the first row be multiplied by 2 Z 2 X 2 respec tively, and of the second row by a^ l^ X : and all added, we have , , , (55) and consequently This constant is ^(,.H*,+ i\)=0. a^ + IJ + \\ = const. 2 x equal to unity for s = 0, as is seen from (54), and hence for all values of s we have INTRINSIC EQUATIONS Combining this equation 23 with the identities we obtain (a t - a + ft 2 2) 2 &gt;) + (X, - X = 0. 2 2) Hence a &=& = a# ^ = X = X = 7 we nave 7i l Z 2, t 2 . Moreover, since in like manner 2 1(^-^=0, But ^-^=0, I , &lt;*,-,) = o. Consequently the differences 2^ #2 y^y^ z l z 2 are constant. for s = they are zero, and so we have the theorem : curves whose radii of first and second curvature are the same functions of the arc are congruent. Two From this it follows that a curve is determined, to within its position in space, by the expressions for the radii of first and second curvature in terms of the arc. And so the equations of a curve may be (56) written in the form /&gt;=/,&lt;), T =/,(.). They are called its intrinsic equations. inquire, conversely, whether two equations (56), in which f^ are any functions whatever of a parameter s, are intrinsic We and/ 2 equations of a curve for which s is the length of arc. In answering this question we show, in the first place, that the equations /trrv (pi) du ds =v -, p dv ds = /u /--| w\ , dw ds v __ \p T/ : r admit of three sets of solutions, namely (58) u = a, ; v = 1, w= \; u = fi, v = m, w = /JL ; u y^ v = n, w = v; which are such that for each value of s the quantities a, fi, 7; v are the direction-cosines of three /, 7?z, n X, mutually perpen dicular lines. In fact, we know * that a system (57) admits of a set of solutions whose values for s = are given arbitra unique /-i, rily. Consequently these equations admit of three sets of solutions II, p. * Vol. Picard, Tralte d Analyse, Vol. II, p. 356. 313; Goursat, Cours d Analyse Mathematique, 24 CURVES IN SPACE s whose values for = s are 1, 0, ; 0, 1, ; 0, 0, 1 respectively. By (59) an argument similar to that applied to equation (55) we prove the solutions (58) satisfy the conditions 0, it that for all values of aft + Im + \p = 7 + mn + pv = 0, dv ya + nl + v\ = 0. In like manner, since follows from (57) that u-r+v as + w as we prove ( du dw = as A (), that these solutions satisfy the conditions a 60) o +Z a +X = l, a 2 /3 +m 2 +/x 2 = l, y+n +i/^l. (59), (60) are equivalent to (40), and conse 7; Z, TH, w; X, /z, v are quently the three sets of functions a, the direction-cosines of three mutually perpendicular lines for all values of s. But the conditions & Suppose we have such a set of solutions. For the curve (61) x I ads, y= , I fids, z=* I yd*, the functions since ds* (61) = a, /3, 7 are the direction-cosines of the tangent, and dx 2 + dy* + dz 2 s measures the arc of the curve. From first and the of (57) we get d?x_l_ ofy^m ~ ds 2 ds*~p p* d^z^n. df~~p /d*x\* /^&gt;\ 2 W/ W/ W/ /^\ = 1 p* 2 Hence if p be positive for all values of s, it is the radius of curva ture of the curve (61), and Z, m, n are the direction-cosines of the principal normal in the positive sense. In consequence of (40) the functions X, yu., v are the direction-cosines of the binomial; hence of (57) it follows that r is from (50) and the third the radius of torsion of the curve. Therefore we have the following theorem in the theory of curves fundamental : is Given any two analytic functions, f^s), f2 (s), of which the former a positive for all values of s within a certain domain ; there exists and s is the arc, for values of s in curve for which p =/j(s), r 2 (), the given domain. The determination of the curve reduces to the find ing of three sets of solutions of equations (57), satisfying the conditions (59), (60), =/ and to quadratures. KICCATI EQUATIONS 25 We proceed now to the integration of set of integrals of the desired kind (62) must equations (57). Since each satisfy the relation u2 cr we * two functions introduce with Darboux and &), defined by 1 (63) w iv u 1 iv u \ w u cr +w + iv and ft) It is evident that the functions v, - are conjugate imaginaries. Solving for u, (64) w, we 1 get v u =1 o-ft) = ,1 i + o-ft) i w= cr + co If these values be substituted in equations (57), it is found that the functions cr and co are solutions of the equation (65) Miieip. ds 2r p 2r when substi conversely, any two different solutions of (65), tuted in (64), lead to a set of solutions of equations (57) satisfying the relation (62). Our problem reduces then to the integration of And equation (65). 14. Riccati equations. Equation (65) 2 may be written (66) ^=L+ N for him. MO + NP, are functions of s. This equation is a generalized where L, M, form of an equation first studied by Riccati, f and consequently is named As theory of curves properties. and Riccati equations occur frequently in the surfaces, we shall establish several of their a particular integral of a Riccati equation known, the general integral can be obtained by two quadratures. Theorem. When is * treatise frequently, t . Lemons sur la Thdorie Generate des Surfaces, Vol. I, p. 22. We shall refer to this and for brevity give our references the form Darboux, I, 22. Cf Forsyth, Differential Equations, chap, v also Cohen, Differential Equations, ; pp. 173-177. 26 CURVES IN SPACE we put Let O l be a particular integral of (66). If is the equation for the determination of &lt;/&gt; = l/c -f 0^ - (67) + 2(M+2WJ&lt;l&gt;+N**Q. and of the first As this equation is linear order, it can be solved by two quadratures. &lt;=/1 (s)+ 0/2(3), general integral of equation (66) Since the general integral of (67) is of the form where a denotes the constant of integration, the is of the form -Sriwhere P, (), R, S are functions of s. Theorem. When two particular integrals of a Riccati equation are known, the general integral can be found by one quadrature. Let l and 2 be two solutions of equation (66). If we effect the substitution 6 --\-6 , the equation in ^ is respec equation and (67) be multiplied by 1/^r and and subtracted, the resulting equation is reducible to tively, If this !/&lt; ty/&lt;l&gt;)=N(0 1 2 )^/&lt;t&gt;. Consequently the general integral of cts (66) is given by 69 &lt; &gt; 00, -v|r fv(0 t -0c/ where a is the constant of integration. Since equation (68) may be looked upon as a linear fractional corre substitution upon a, four particular solutions V 2 # 3 4 , , , sponding to four values a v a z , a 3 a , of a, are in the : same cross-ratio as these constants. Hence we have the theorem The cross-ratio is of any four particular integrals of a Riccati equation constant. From this it follows that if three particular integrals are known, the general integral can be obtained without quadrature. DETERMINATION OF COORDINATES 15. 27 The determination equations. of the coordinates of a curve defined by its intrinsic We return to (o i the consideration of equation (65) (70) and indicate by o\ = a.P+O -^ -^i = bP+Q -v T&gt; (i = l,2,3) we obtain six particular integrals of this equation. three sets of solutions of equations (57), *1 I From namely these (71) for /8, m, ft; and similar expressions in cr 2 3 respectively 3 2 These expressions satisfy the conditions (60). In order 7, n, ZA that (59) also may be satisfied we must have , &&gt; ; &lt;7 , &&gt; CT ft),, ft), ft), which (72) is reducible to = -1. of the three pairs of constants z. A/ _1 ; JL 9 ^ ^ q O * ^ , Hence each two 5X 2, J2 ; a3 form a harmonic range. When the for a, /3, 7, values (70) for it is found that &) &lt;T., . t are substituted in the expressions (73) 7 = ,1+a, a where, for the sake of brevity, we have put (74) RS PQ PS-QR 28 CURVES IN SPACE The coefficients of U, F, and W in , (73) are of the z&gt;. same form as the expressions (71) for cr, Z, X ; ra, /* ; 7, n, Moreover, the of condition (59) are equivalent to (72). Hence these equations coefficients are the direction-cosines of three fixed directions in space mutually perpendicular to one another. If lines through the origin of coordinates parallel to these three lines be taken for a new set of axes, the expressions for #, reduce to U, V, 7 with reference to these axes ^respectively.* These results may be stated thus /3, : If the general solution of equation (65) (68) the curve " be = STV o aii -f- whose radii of is first and second curvature are p and T respectively given by f\ T)v w-K ) ci I/ RS-/~r&gt; PS- QR must be remarked that the new axes of coordinates are not necessarily real, so that when it is important to know whether the It curves are real (73). it will be advisable to consider the general formulas this will be given later. An shall example of We When the curve apply the preceding results to several problems. is plane the torsion is zero, and conversely. For this case equa tion (65) reduces to = ds - of which the general ae integral is p = where a is -if J 1 P = ae~ &lt;r i&lt;r , an arbitrary constant, and by (27) is the measure of the arc of the of the -I spherical indicatrix of the tangent. This solution is form (08), with O Therefore the coordinates are given by (75) x=Ccos&lt;rds, y=Csir\&lt;rds, 2 = Hence the coordinates * This is of any plane curve can be put as taking &i in this form. the same thing ai = l, c*2= b^ i, 3 oo, &3=0. Rechnung auf Geometrie, t Scheffers, Anwendung Leipsic, 1902. der Differential und Integral Vol. I, p. 219. DETERMINATION OF COORDINATES We (65) 29 radii of first have seon that cylindrical helices are characterized by the property that the und second curvature are in constant ratio. If we put T = pc, equation bj written may ^ = _l(l_2c0-02). ds 2T V Two 2 + 2 cd - 1 = 0. particular integrals are the roots of the equation we consider only this case, and put if c is real roots are real and unequal ; These (76) el = -c -Vc 2 + 1, 2 = - c + Vc2 + l, 01&2 = - 1. is From (77) solution of the (69) it follows that the general above equation where we have put (78) Since &lt;r and -- in (63) are conjugate imaginary, if we take then a and 6 must be such that aeit _ i &oe - my _ 6. 0j where (76) (7 9 , 6 denotes the conjugate imaginary of 4- This reduces, in consequence of to , ; n M * = -|=-* &lt;x&gt; One 63 solution of this If these = 0. is given by taking values be substituted in (72), and for a 4- and 0, 6; we put a s = i GO, we get a kr where = 1, 2. So that equation (79) becomes && = 0^, where = e0 2 are 61 = 1? 6 2 = - i0i- From (77) i P , = 1, 2. The solutions of this equation Q = - 0i, R = e S = - 1, so that , W==Vc + 2 l When a, /3, the foregoing values are substituted in (73), and the resulting values of 7 in (61), we C get (80) xthe la Vc2 + From a constant ang c - Ccoslds, J 1 y= - Vc 2 + 1 Jfstafdt, g = Vc 2 + = 1 -v\ ie expressions we find that the tangent to the curve makes the direction of the elements of the cylinder. the z-axis is And the cross-section of the cylinder Xi defined by y\ = fcos t dsi, = J sin t dsi, where Si denotes the arc of this section measured from a point of it. If pi !) denotes the radius of curvature of the right section, we find that pc 2 = 2 pi(c + 30 CURVES IN SPACE EXAMPLES 1. Find the coordinates of the cylindrical helix whose intrinsic equations are p = T = S. 2. Show upon a cylinder whose cross-section 3. that the helix whose intrinsic equations are p is a catenary. = T = 2 (s + 4)/V2 lies p = as, r Establish the following properties for the curve with the intrinsic equations = 6s, where a and b are constants : to (a) the Cartesian coordinates are reducible where J_, B, h are functions of a and 6 ; x=Ae ht cos, y = Aeht sm t, z Behi , (6) the curve lies upon a circular cone whose axis coincides with the z-axis and cuts the elements of the cone under constant angle. 16. Moving trihedral. In 11 we took for fixed axes of refer ence the tangent, principal normal, and binormal to a curve at a of it, and expressed the coordinates of any other point of Q point the curve with respect to these axes as power series in the arc s is any point of the of the curve between the two points. Since M M curve, there is a set of such axes for each of its points. Hence, instead of considering only the points whose locus is the curve, we may look upon the moving point as the intersection of three mutually perpendicular lines which move along with the point, the whole figure rotating so that in each position the lines coin cide with the tangent, principal normal, and binormal at the point. We shall refer to such a configuration as the moving trihedral. In the solution of certain problems it is of advantage to refer the curve to this moving trihedral as axes. proceed to the con We sideration of this idea. With cosines of reference to the trihedral at a point Jf, the directionthe tangent, principal normal, and binormal at M have the values a=l, As /3 = 7 = 0; I = 0, m=l, n = 0; X =p= 0, i/ = l. functions with the trihedral begins to move, the rates of change of these s are found from the Frenet formulas (50) to have the values da ds _ = = ft o d{$ _ = 1 , dy ds _ ft o dl _ j 1 5 dm _ ds ~ u, ds 1 j p ds dfJL p dn ds d\ r~ = n "i r ds ~T~ 1 =~ dv ds 7~ = ft U * ds r MOVING TRIHEDRAL Let f, 77 , 31 f, f f denote coordinates referring to the axes at Jf, and those with reference to the axes at Jf , and let JfJf = As 77, (see fig. 4). Since the rate of change of a is zero and a =1 at Jf, the cosine of the angle between the - and f -axes is 1 to within terms of higher than the first order in As. Likewise the cosine of the angle between the f- and Tj -axes cosines of the angles between all is As/p. We calculate the the axes, and the results tabulated as follows: f may be v S As (81) ., _ As FIG. 4 Let at P be f, M are a point whose coordinates with respect to the trihedral 77, f. Suppose that as Jf describes the given curve (7, P describes a path T. It may happen that in this motion P is fixed P in the relatively to the moving trihedral, but in general the change will be due not only to the motion of the trihedral position of P but also to a motion relative to In the latter general case, if it. on (7, the coordinates the point on T corresponding to denotes and of relative to the axes at may be written M P M M ?4Af Thus A 2 # moving Tj+A^, f+A^; and f +A 2 f, 17 +A^, f+A 2 ?. indicates the variation of a function relative to the trihedral, A^ the variation due to the latter and to the motion of the trihedral. are within terms of higher order the coordinates of with respect to the axes at Jf, and with the aid of (81) the equations of the transformation of coordinates with respect (As, 0, 0) To M to the two axes are expressible thus . : 32 CURVES IN SPACE These reduce to As As . o As . . As T As In the limit as Jtf As H /a approaches M these equations become ds ds c?s p ds p T ds ds 0, T 80 d0 denotes the absolute rate of change of thus -T- and -=- that relative to the trihedral.* If t denotes the distance between 2 P and 2 , a point P^ (f^ find rj^ fj), that is ^ = (? -f) 1 2 2 +(7? 1 -7;) + (? -?) we 1 by means of the formulas (82) that denote the direction-cosines of PP^ with respect to the axes at Jf, then If a, 5, &lt;? express the condition that f 1? 77 L ft as well as f, ?/, f satisfy equations (82), we are brought to the following fundamental relations between the variations of a, 6, c: , When we &a (83) ~T~ c?s b da ~ ~7--- &b == T" db ~T a --I 1 c * ds p ds ds p T $c __ dc ~r == ds ds "T b * r If the point P remains fixed in space as M moves are along the zero curve, the left-hand members of equations (82) and the equations reduce to (84) ds --l, p = ds \p T/ c?s T Moreover, the direction-cosines of a line fixed in space satisfy the equations da or . (85) b =/) db c?s = (a c?s (\p c\ +T) )i b dc _ = _. ds r Naples, 1896. * Cf. Cesaro, Lezioni di Geometria Intrinseca, pp. 122-128. MOVING TRIHEDRAL These are the Frenet-Serret formulas, expected. as 33 might have been We it shall show that the X 1, the solution of (84). I, ; solution of these equations carries with Suppose we have three sets of solutions 7, n, v, 0, of (85), a, (86) & m, 0, /JL ; whose values 1, for s 0, = are 0; 0; 0, 1. They are the direction-cosines, with respect to the moving trihedral with vertex M, of three fixed directions in space mutually perpen be a fixed point, and through it Let with the directions just found. Take these lines for coordinate axes and let #, y, z denote the coordinates of with respect to them. If f, ??, f denote the coordinates of dicular to one another. lines draw the M with respect to the moving trihedral, then f are the ?;, f with respect to the trihedral with vertex at coordinates of , M and edges parallel to the corresponding edges of the trihedral at M. Consequently we have (87) = - (ax -f &y + rj=-(lx+my + m), f 7*)&gt; If these values be substituted in (84) and we take account of are identically satisfied. (50) (85), we find that the equations for s = 0, it follows from If fo ^o ?o Denote the values of f, ?;, (86) and (87) that they differ only in sign from the initial values and of x, y, z. Hence if we write, in conformity with (21), (88) x and substitute these values in (87), they become the general solu tion of equations (84). We have seen that the solution of equa tions (85) reduces to the integration of the Riccati equation (65). 17. Illustrative examples. As an example of the foregoing method we consider which is the locus of a point on the tangent to a twisted curve C at a the curve constant distance a from the point of contact. The coordinates of the point MI of the curve with reference to the axes at are a, 0, (i) M 0. In this case equations (82) reduce to ^-l^ds~ ds~ a P ^ds~ 34 Hence if Si CURVES IN SPACE denotes the length of arc of C\ from the point corresponding to s = on C, we have and the direction-cosines iven by are given b 7 of the tangent to Ci with reference to the moving axes a2 is + p 2 Va2 4- p2 Hence the tangent point of C. to Ci parallel to the osculating plane at the corresponding By means ds of (83) we find p Sa:\ d / \_ + p 2 a 2 p Va + ds y V(i 2 p 2 ( 2 + P 2 ) P Va 2 + (ii), p 2 Proceeding in like manner with 0i and d(*i 71, and making use 5/3i of we have & 2 pp (a 2 d 22 ) _ ~ (a 2 2 ftp p 22 ) p + p a2 + p 2 5Si + p a2 + p2 871 _ ap these expressions and (21 of the first curvature of C\ From } we obtain the following expression for the square : app/ Pi 2 a2 + p 2 \a 2 + -II -t The direction-cosines of the principal normal of C\ are 5/?i 571 By means of (40) : we derive the following expressions for the direction-cosines of the binormal r (a 2 + p )^ 2 r (a 2 + p )^ 2 ft2 &gt; + ? Va 2 + p 2 In order to find the expression for TI, the radius of torsion of Ci, substitute the above values in the equation we have only to _ ~ S\i _ p /d\i \ _ MI\ P 5i Va2 + p 2 ds We leave this calculation to the reader and proceed to an application of the preceding results. inquire whether there is a curve necessary and sufficient condition is that We C I/pi, such that Ci is a straight line. The be zero (Ex. 3, p. 15). From (iii) it follows that we must have ILLUSTRATIVE EXAMPLES From the second of these equations it 35 be plane, and from follows that C must the former we get, by integration, log (a* + ,*) = + , where take c c is = a constant of integration. If the point s 2 log a , this equation reduces to = be chosen so that we may P If 6 = a \e - 1. denotes the angle which the line C\ makes with the -axis, tan 6 we have, from (i), = 8rj a =-= e 1 - 1 Differentiating this equation with respect to s, we can put the result in the form dd__ ds 1 p consequently (89) When in the these values are substituted in equations (75), we obtain the coordinates of C form x = = i \1 e as, y = ae = a sin 0. or, in terms of 0, (90) x a log tan - + cos 6 , y The curve, with these equations, is called the tractrix. As just seen, it possesses the property that there is associated with it a straight line such that the segments of the tangents between the points of tangency and points of intersection with the given line are of constant length. Theorem. The orthogonal trajectories of the osculating plane of a twisted curve can be found by quadratures. plane are (, reference to the moving axes the coordinates of a point in the osculating The necessary and sufficient condition that this point describe 77, 0). an orthogonal trajectory of the osculating plane as moves along the given curve With M is that and ds ds the equations in (82) be zero. Hence we have for the determination of and 77 ?*_+ da- ^+ da, = 0, where a is given by (89). Eliminating d?-t] we have " ^+ = - Hence 77 can be found by quadratures as a function of and then is given directly. &lt;r, and consequently of S, 36 CURVES IK SPACE Problem. Find a necessary and sufficient condition that a curve If , lie upon a sphere. f denote the coordinates of the center, and R the radius of the sphere, we have 2 -f if* + f 2 = R 2 Since the center is fixed, the derivatives of , 17, f are given by (84). Consequently, when we differentiate the above equation, the result = 0, which shows that the normal plane to the curve ing equation reduces to ??, . at each point passes through the center of the sphere. If this equation be differen tiated, we get 77 = p hence the center of the sphere is on the polar line for each point. Another differentiation gives, together with the preceding, the following ; coordinates of the center of the sphere : When (92) the last of these equations is differentiated we obtain the desired condition - -f (rp Y = 0. Conversely, lies when this condition is satisfied, the point with the coordinates (91) is fixed in space and at constant distance from points of the curve. A curve which upon a sphere is called a spherical curve. Hence equation (92) is a necessary sufficient condition that and a curve be spherical. EXAMPLES 1. Show 2. Let C be a plane curve and Ci an orthogonal trajectory of the normals to C. that the segments of these normals between C and Ci are of the same length. Let in C and Ci be two curves in the same plane, and say that the points corre which the curves are met by a line through a fixed point P. Show that if the tangents at corresponding points are parallel, the two curves are similar and P is the center of similitude. spond of the point of projection of a fixed point upon the tangent to called the pedal curve of C with respect to P. Show that if r is the makes on (7, and 6 the angle which the line to a point distance from with the tangent to C at M, the arc Si and radius of curvature pi of the pedal 3. The locus P a curve C is P M PM curve are given by where s and p are the arc and the radius of curvature of C. 4. Find the intrinsic and parametric equations of a plane curve which is such that the segment on any tangent between the point of contact and the projection of a fixed point is of constant length. 5. Find the all angle 6. intrinsic equation of the plane curve which meets under constant the lines passing through a fixed point. is The plane curve which of the such that the locus of the mid-point of the seg . ment normal between a point of the curve and the center of curvature is 2 2 2 line is the cycloid whose intrinsic equation is p -f- s = a a straight 7. Investigate the curve which is the locus of the point on the principal normal of a given curve and at constant distance from the latter. OSCULATING SPHERE 18. Osculating sphere. 37 to its moving trihedral. The point whose Consider any curve whatever referred coordinates have the values (91) lies on the normal to the osculating plane at the center of curvature, that is, on the polar line. Consequently the moving sphere whose center is is at this point, and whose radius cuts the osculating plane in the osculating This sphere is called the osculating sphere to the curve at circle. the point. shall derive the property of this sphere which -f, Vp 2 r // 2 2 We accounts for its name. the tangent to a curve at a point is tangent likewise to a sphere at this point, the center of the sphere lies in the normal When M denotes its radius and the curve is plane to the curve at M. If referred to the trihedral at M, the coordinates of the center C of the 2 Let P(x, y, z) sphere are of the form (0, y v z t ) and yl + z* = ^ be a point of the curve near M, and Q the point in which the line CP cuts the sphere. If PQ be denoted by 8, we have, from (53), . R 6/r which reduces to Hence ?/ 1 8 is of the second order, in comparison with JMTP, unless is is =/3, that is, unless the center is on the polar line; then it of the third order unless z l = p r, in which case the sphere the osculating sphere. Hence we have the theorem : The osculating sphere to a curve at a point has contact with the curve of the third order ; oilier spheres with their centers on the polar line, and tangent to the curve, have contact with the curve of the second order ; all other spheres tangent to the curve at a point have contact of the first order. The (93) radius of the osculating sphere JS* is given by =,! + TV, in space, are and the coordinates of the center, referred to fixed axes (94) xl = x + pi f p T\, y^ = y + pm p rfji, zl = z + pn p rv. 88 CURVES IN SPACE Hence when p is and the osculating circle coincide. constant the centers of the osculating sphere Then the radius of the sphere is necessarily constant. Conversely, it follows from the equation (93) that a necessary and sufficient condition that be con IP E stant [P, is that is is, either the curvature is constant, or the curve spherical. If equations (94) be differentiated with respect to s, we get (96)- #1 = From sions it is these expres seen that of the center the is osculating sphere fixed only in case of spherical curves. Also, the tangent to the locus of the cen ter is parallel to the binormal. Combin with ing this result FIG. 5 a previous one, we have the theorem: is The polar line for a point on a curve tangent to the locus of the center of the osculating sphere to the curve at the corresponding point. represented in fig. 5, in which the curve is the are the correspond locus of the points M\ the points (7, C^ C2 are normal ing centers of curvature the planes MCN, M^C^N^ and the are the polar lines the lines CP, C^P^ to the curve This result is , ; ; ; points P, Pj, P 2 , are the centers of the osculating spheres. BEETEAND CUKVES 19. 39 : Bertrand curves. the To determine the Bertrand proposed the following problem curves whose principal normals are the principal normals of another curve. moving trihedral. We generate a curve C^ whose principal normal coincides with the ?;-axis of the moving remains on the moving ?/-axis, we trihedral. Since the point 1 condition that the point l( M In solving this problem we make use of must find the necessary and sufficient = 0, TJ = k, ? = 0) M have d% this = d% = 0. And since M tends to move at axis, Brj = 0. Now equations (82) reduce to l right angles to (96) }Ll-, ds p the second ds is vO, &gt; 5--*. ds r Moreover, if co From we see that k a constant. denotes makes with the tangent at M, the angle which the tangent at l first and third of these equations, we have, from the tan or sin co M co 8? = -^r = Sf cos co kp T (k p) sin co (97) k have seen ( 11) that according as r is positive or negative, near the osculating plane to a curve at a point cuts the below or above the osculating plane at M. From polar line for We M M M these considerations fourth, or first &lt; it follows that when r &gt; &gt; 0, co is in &lt; &lt; the third, &lt; and when r accordingly. k ^, p, or k quadrants according as k co is in the second, first, or fourth quadrant, 0, ; It is readily (97). is found that these results are consistent with equation By means of (97) it found from (96) that the negative sign being taken so that the left-hand member may be positive. Thus far we have expressed only the condition that the locus of orthogonally, but not that this axis For this we shall be the principal normal to the curve Cl also. consider the moving trihedral for Cl and let a x b^ c^ denote the M^ cut the moving T^-axis , 40 CUKVES IN SPACE as direction-cosines with respect to it of a fixed direction in space, M^D in fig. 6. They satisfy equations similar to (85), namely l (99 ) M If a, 6, c are the direction-cosines of the same direction, with respect to the &&gt; mov ing trihedral at M, we must have a l a cos -f- c sin &), b l = 6, ^ = a sin eo + cos for all possible cases, &lt;? a&gt;, as enumerated above. When these values are sub stituted in the above equations, of (98), sin tw _l_ we get, by means _ p P sin I T G) &) sm sin &)1 \d -\- cos Tsin I I ft) a sin &) ( &)) = _ as c 0, cos COS PI ft) &lt;w to sin PI cos TI &)1 I ~ [T */&gt; TI J L ~| , * , u, J dco ~ &) /3 [sin cos &) k . 6 -h , (&lt;? sm . &) T r t r sin + a cos &))- = x 0. &)J as Since these equations must be true for every fixed line, the cients of a, 6, c in each of these equations must be zero. resulting equations of condition reduce to &) coeffi The = const., &) (100) sm cos &) 1 sm ;; &lt;w = ~ " Since the &) is first a constant, equation (97) is a linear relation between and second curvatures of the curve C. And the last of for the curve Cr equations (100) shows that a similar relation holds a curve C whose first and second curvatures Conversely, given satisfy the relation (ioi) + p -4, 7 = c&gt; where B, C are constants different from zero ; if we take k =A , COt ft) = B ;&gt; TANGENT SURFACE OF A CURVE and for p l isfied 41 and r l the values given by (100), equations (99) are sat identically, and the point (0, k, 0) on the principal normal gather these results generates the curve Cv conjugate to C. about the curves of Bertrand into the following theorem: necessary and sufficient condition that the principal normals one curve be the principal normals of a second is that a linear of relation exist between the first and second curvatures; the distance We A between corresponding points of the two curves is constant, the oscu lating planes at these points cut under constant angle, and the torsions of the two curves have the same sign. We consider, finally, several particular cases, which we have excluded in the consideration of equation (101). When C = the curve that is, and A=Q, its the ratio of p and T is constant. Hence 0, is a helix and conjugate is at infinity. When A = the curve has constant torsion, the conjugate curve coincides with the original. When A = C = 0, k is indeterminate ; when hence plane curves admit of an infinity of conjugates, they are the curves parallel to the given curve. The only other curve which has more than one conjugate is a circular helix, for since p and T are constant, A/C can be given any value whatever both the given helix and the circular helices conjugate to it are traced on circular cylinders with the same axis. ; 20. Tangent surface of a curve. For the further discussion of the properties of curves it is necessary to introduce certain curves and surfaces which can be associated with them. However, in con sidering these surfaces we limit our discussion to those properties which have to do with the associated curves, and leave other con siderations to their proper places in later chapters. The totality of all the points on the tangents to a twisted curve C constitute the tangent surface of the curve. As thus defined, the sur face consists of an infinity of straight lines, which are called the on this surface lies on one generators of the surface. Any point of these lines, and is determined by this line and the distance t from P P to arc , fig. 7. the point where the line touches the curve, as is shown in If the coordinates x, y, z of are expressed in terms of the M M the coordinates of P are given by (102) f 42 CURVES IN SPACE s. where the accents denote differentiation with respect to the equations of the curve have the general form When the coordinates of (103) P can be expressed thus : =./+/, v where = -(104) ?=/,() +./, f -/,() From is this it is seen that v is equal to the distance MP only when s the parameter. As given by equations (102) or (103), the coordinates of a point on the tangent surface are functions of two parameters. A rela tion between these parameters, such as f(s, t) = 0, upon this defines a curve which lies the surface. FIG. 7 For, when t equation is solved for in terms sion of s and the resulting expres substituted in (102), the coordinates f, ?;, f are functions of a single parameter, and consequently the is locus of the point (f, 77, f) is a curve (1). By definition, the element of arc of this curve da2 = di; 2 -f drf + 2 c?f . This is is given by by means of (102) and expressible (41) in the form z d&lt;r (105) = l +- 2 ds 2 + 2dsdt + dt\ where t is supposed to be the expression in s obtained from (104), and p is the radius of curvature of the curve (7, of which the sur face is the tangent surface. This result is true whatever be the relation (104). Hence equation (105) gives the element of length of any curve on the surface, and do- is called the linear element of the surface. in equations (102) has a positive or negative value, the point lies on the portion of the tangent drawn in the According as t TANGENT SUBFACE OF A CUKVE 43 in the opposite direction. It positive direction from the curve or is now our purpose to get an idea of the form of the surface in the neighborhood of the curve. In consequence of (53) equations (102) can be written 1 \ L -^r+.-.u, 1 6 pr The for f plane f s it is at which = = 0, cuts the surface in a curve F. is also a point of F. From The point Q of (7, the above expression M M t seen that for points of s F near only in s Q the parameters and t differ sign. Hence, neglecting powers of and of higher orders, the equations of the neighborhood of J/ are F in f=0 By we , ,=-., r=_ 2/o t &lt; 3 pr eliminating from the last two equa find that in the neighborhood of tions, the curve F has the form of a semiQ M cubical parabola with the T^-axis, that is the principal normal to (7, for cuspidal tangent. Since any point of the curve C can be taken for Jf we have the theorem , : The tangent surface of a curve consists of two sheets, corresponding which are tangent to respectively to positive and negative values of t, one another along the curve, and thus form a sharp edge. On this account the curve is called the edge of regression of the surface. An idea of the form of the surface may be had from fig. 8. 21. Involutes and evolutes of a curve. When the tangents of a curve C are normal to a curve Cv the latter is called an involute of (7, and C is called an evolute of Cr As of a twisted curve lie upon its thus defined, the involutes of a tangent surface, and those 44 plane curve in CURVES IN SPACE its plane. The latter is only a particular case of the former, so that the problem of finding the involutes of a curve is that of finding the curves upon the tangent surface which cut the generators orthogonally. write the equations of the tangent surface in the form We Assuming that s is to the determination of a relation the parameter of the curve, the problem reduces between t and s such that ds c of (50) this reduces to dt 0, so that t s, where c is an arbitrary constant. Hence the coordinates x^ y v z l of an involute are expressible in the form (106) By means + = 2^= x + a(c s), #! =#+ (&lt;?), zx =z+ ; ?(&lt;? s). Corresponding to each value of c there is an involute consequently a curve has an infinity of involutes. If two involutes correspond to values c^ and c 2 of c, curves is of length c l c2 . the segment of each tangent between the Hence the involutes are said to form a system of parallel curves on the tangent surface. When s is known the involutes by equations the complete de termination of the involutes of a (106). are given directly Hence given curve requires one quad rature at most. FIG. 9 its From the definition of t and above value, an involute can be generated mechanically in the following manner, as represented in fig. 9. Take a string of length c and bring it into coincidence call the other with the curve, with one end at the point s = end A. If the former point be fixed and the string be unwound trace out gradually from the curve beginning at A, this point will ; an involute on the tangent surface. By differentiating equations (106), we j ds, get dz l , dx l 7 = I (c s} j ds, , dy l = m (c s) n (c - s) , ds. INVOLUTES AND EVOLUTES Hence the tangent 45 to an involute is parallel to the principal nor mal of the curve at the corresponding point, and consequently the tangents at these points are perpendicular to one another. As an example of the foregoing theory, cular helix, whose equations are x we determine the involutes of the cir = a cos te, y = a sin M, z = au cot 0, where a is the curve makes with s the radius of the cylinder and 6 the constant angle which the tangent to axis of the cylinder. Now a cosec 6 - u, a, /3, 7 = - -sin u, cos M, cot : cosec Hence the equations Xi of the involutes are c sin 0)sin M, = a cos u + (au yi it = a sin u (aw c sin 0) cos u, zi = c cos 0. From the last of these equations follows that the involutes are plane curves whose planes are normal to the axis of the cylinder, and from the expressions for x\ and yi it is seen that these curves are the involutes of the circular sections of the cylinder. We proceed to the inverse problem C, to find its evolutes. : Given a curve normals to The problem reduces to the determination of a succession of C which are tangent to a curve G If Q be the point on (7, it lies in the normal plane to C at on C corresponding to . M M If, and consequently its coordinates are of the form where p and q are the distances from Q to the binomial and prin cipal normal respectively. These quantities p and q mi^t be such that the line is Q tangent to the locus of Jf at tiis point, that M MM is, we must have ,, where /c denotes a factor of proportionality. values are substituted in these equations, we "When the above et P and two other equations obtained by replacing a, Z, X by /3, m, ft and 7, n, v. Hence the expressions in parentheses vanish. From 46 the first it CURVES IN SPACE follows that p the polar line of , written ds C at M. lies on equal to /o; consequently Q The other equations of condition can be is M dp q J: + 1 + p r da * a, , ds _p+ T 0&gt; Eliminating /c, we get p For the sake of convenience we put integration &lt;o = I - &gt; and obtain by P = tan (o&gt; + c), where c is the constant of integration. As c is arbitrary, there is an infinity of evolutes of the curve C\ they are defined by the following equations, in which c is constant for an evolute but changes with it: xQ =x+lp + \p tan(o) + c), y =y + mp + fip Z = z -f np + vp tan c). Q tan(o&gt; -f c), Q -f- (o&gt; From the definition of q it of the angle which MM : follows that q/p is equal to the tangent to Q makes with the principal normal 6 C at M. Calling this angle 0, we have = &&gt; + c. The foregoing results give the following theorem A curve f" C (7, normals admits of an infinity of evolutes; when each of the which are tangent to one of its evolutes, is turned to C, these through the sa^g angle in the corresponding normal plane new normals are tangent to another evolute of C. In fig. 5 the locus of the points o; E is an evolute of the given curve. Each system normals to C which are tangent to an evolute C constitute a tangen t surface of which C is the edge of regression. Hence the evolves of C are the edges of regression of an infinity of tangent svff aces? all o f which pass through follows that w is C.. From is tbe definition of w it constant only when the curve C we have plane. j n this case we may take w equal to zero. Then when c the evol\te C in the plane of the curve. The other evolutes lie upon the right MINIMAL CUEVES 47 cylinder formed of the normals to the plane at points of Co, and cut the elements of the cylinder under the constant angle 00 c, and consequently are helices. Hence we have the theorem : The evolutes of a plane curve are the helices traced on the right cylinder whose base is the plane evolute. Conversely, every cylindrical helix is the evolute of an infinity of plane curves. EXAMPLES 1. Find the coordinates of the center of the osculating sphere of the twisted cubic. 2. The angle between the radius of the osculating sphere for any curve and the locus of the center of the sphere is equal to the angle between the radius of the osculating circle and the locus of the center of curvature. 3. The locus is of the center of curvature of a curve is an evolute only when the curve 4. plane. first Find the radii of y = a cos 2 it, z = construction. 5. asinw. Show that the curve Find its evolutes. and second curvature of the curve x = a sin u cos w, is spherical, and give a geometrical without the use of the moving Derive the properties of Bertrand curves ( 10) trihedral. 6. Find the involutes and evolutes of the twisted cubic. Determine whether there Derive the results of is 7. a curve whose bmormals are the binormals of a second curve. 8. 21 by means of the moving trihedral. Minimal curves. In the preceding discussion we have made exception of the curves, defined by 22. z =/! (w), y =/ a (u), z =/ 8 (u), when these functions satisfy the condition As these imaginary curves are of interest in certain parts of the theory of surfaces, we devote this closing section to their discussion. The equation of condition may be written in the form _ J3 f where equivalent to the following (108) : Jl f __ if V2 These equations are v is a constant or a function of u. ^ ^lz* : SI.&lt;l+*&gt;:.. 48 CURVES IN SPACE At most, the common ratio is a function of M, say f(u). And so we disregard additive constants of integration, as they can be removed by a translation of the curve in space, we can replace if the above equations by (1 09) x =.f(u)du, first y = if( 2 n)du, z = call it a. We consider the case when v is constant and I If we change the parameter of the curve by replacing new parameter which we call w, we have, without loss (110) f(u) du by a of generality, 1-a x= 2 t ^u y = .1+a i^--u, z = au. For each value of a these are the equations of an imaginary straight line through the origin. Eliminating #, we find that the envelope of these lines origin, is is the imaginary cone, with vertex at the whose equation (111) z2 +2/ 2 +z 2 =0. Every point on the cone is at zero distance from the vertex, and from the equations of the lines it is seen that the distance between any two points on a line is zero. We call these generators of the cone minimal straight lines. Through any point in space there are an infinity of them ; their direction-cosines are proportional to &gt; where a vertex is is arbitrary. The locus of these lines is the cone whose and whose generators pass through the circle at infinity. For, the equation in homogeneous coordinates of the 2 2 = w2 sphere of unit radius and center at the origin is 3? + y + z at the point , so that the equations of the circle at infinity are Hence the cone of u. If (111) passes through the circle at infinity. this function of it, We consider now the case where v in equations (109) is a function we take u for a new convenience (112) call it equations (109) may parameter, and for be written in the form g _ll~p &gt;()&lt;**, y = i^^F(u)du, z= CuF(u)du, MINIMAL CUEVES where, as is 49 seen from (108), F(u) can be any function of u different third derivative of a function f(u), from If zero. we replace F(u) by the thus F(u)=f form "(u], equations (112) can be integrated by parts and put in the uf (u)-f(u), (113) 1y-4 Since F must be c l u*+ than c 2 u -f c8 , different from zero, f(u) can have any form other where ^, c 2 , c3 are arbitrary constants. EXAMPLES that the tangents to a minimal curve are minimal lines, and that a curve whose tangents are minimal lines is minimal. 1. 2. Show Show that the osculating plane of a minimal curve can be written Q, + (Y-y)B + (Z-z)C = of this sort 3. is where A + B2 + C = 0. A plane 2 2 (X x) A is whose equation called an isotropic plane. Show lie that through each point of a plane two minimal straight lines pass which 4. in the latter. Determine the order of the minimal curves for which the function /in (113) condition satisfies the 5. 4/ v "/ - 5/ iv2 = , 0. Show that the equations of a minimal v condition 4/ 5/ iv2 = a/ ///3 where a is "/ curve, for which /in (113) satisfies the a constant, can be put in the form . x = 8 - cos , y 8 - sin , z = 8i, t. GENERAL EXAMPLES 1. Show that the equations of any plane curve can be put in the form x=J*cos0/(0)d0, 2. y J 0. sin 0/(0) d0, and determine the geometrical significance of Prove that the necessary and sufficient condition that the parameter u in the in Ex. 1 is equations x =fi(u), y =f2 (u) have the significance of 3. Prove that the general projective transformation transforms an osculating plane of a curve into an osculating plane of the transform. 4. The principal normal to a curve is is normal to the locus of the centers of curvature at the points where p a maximum or minimum. 50 . CURVES IN SPACE A 5 certain plane curve possesses the property that if C be its center of curva ture for a point P, Q the projection of on the x-axis, and T the point where the meets this axis, the area of the triangle is constant. Find the tangent at equations of the curve in terms of the angle which the tangent forms with the x-axis. P P CQT 6. The binormal at a point Mot a curve is the limiting position of the as and approaches M. perpendicular to the tangents at common M M , N 7. The tangents to the spherical indicatrices of the tangent and binormal of a twisted curve at corresponding points are parallel. 8. Any curve upon the unit sphere serves for the spherical indicatrix of the binormal of a curve of constant torsion. Find the coordinates of the curve. 9. The equations r Idk - kdl x a J #2 + 2 + 12 I - i y I a r hdl I fc2 Idh z J hZ d r kdh I ~ hdk + + 1-2 J h2 + where a is constant and h, k, whose radius of torsion is a. 10. If, in Ex. 9, are functions of a single parameter, define a curve we have k = sm/i0 + -%sinX0 is I = 2 \/ cos 2 commensurable, the integrands are expressible as linear homogeneous functions of sines and cosines of multiples of 0, /x where X and are constants whose ratio and consequently the curve is algebraic. t. 11. Equations (1) define a family of circles, if a, &, r are functions of a parameter Show that the determination of their orthogonal trajectories requires the solution of the Riccati equation, *! dt = l*?,__L* r dt where 0=tanw/2. 12. 8rdr i-"), ( Find the vector representing the rate of change of the acceleration of a point. moving 13. When a curve is of the perpendicular upon the osculating plane spherical, the center of curvature for the point is the foot at the point from the center of the sphere. 14. The radii of first and second curvature of a curve which lies upon a sphere 2r = 0, and cuts the meridians under constant angle are in the relation 1 + ar + b are constants. where a and fy&gt; An epitrochoidal curve is generated by a point in the plane of a circle which without slipping, on another circle, whose plane meets the plane of the first rolls, circle under constant angle. Find its equations and show that it is a spherical curve. 15. 16. If two curves are in a one-to-one correspondence with the tangents at are parallel corresponding points parallel, the principal normals at these points and likewise the binormals two curves so related are said to be deducible from one another by a transformation of Combescure. ; 17. If two curves are in a one-to-one correspondence and the osculating planes at corresponding points are parallel, either curve can be obtained from the other by a transformation of Combescure. GENERAL EXAMPLES 18. 51 Show [x"" E2 T2p4 2 that the radius of the osculating sphere of a curve is given by + /z + z ///2 ] r 2 where the prime denotes differentiation with y" , respect to the arc. its At corresponding points of a twisted curve and the locus of the center of osculating sphere the principal normals are parallel, and the tangent to one curve is parallel to the binorinal to the other also the product? of the radii of torsion of the two curves is equal to the product of the radii of first curvature, 19. ; or to within the sign, according as the positive directions of the principal normals are the same or different. 20. Determine the twisted curves which are such that the centers of the spheres osculating the curve of centers of the osculating spheres of the given curve are points of the latter. 21. Show that the binormals to a curve do not constitute the tangent surface of another curve. 22. Determine the directions of the principal of a given curve. normal and binormal to an involute 23. Show x that the equations (u) sin = a C&lt;f&gt; u du, y = ^ 2 a ) \$ (u) cos u du, and \f/ z a f 4&gt;(u)\l/(u)du, where (u) (1 4- ^2 4- ^ /2 )- (1 4- * (u) is any function whatever, define a curve of constant curvature. 24. Prove that when ^ (u) = tan w, in example 23, the curve is algebraic. 25. Prove that in order that the principal normals of a curve be the binor- mals of another, the relation a I stants. h = ) - must hold, where a and of 6 are con Show that such curves are defined by equations (1 _|_ example 23 when . = 26. Let \i, /ii, *i (1 4. ^2 _|_ ^/2)3 2 i// _|_ ^2)3(^" _j_ ^,\2 (1 4. )^(l 4- 2 1// 4- 1// )^ 2 be the coordinates of a point on the unit sphere expressed as &lt;TI functions of the arc of the curve. Show that the equations / x = ek I \idffi k cot w k cot w k cot w (MI^I y z = = ek j I mdai vida-i \ (v\\{ v{\\] d&lt;?i, ek | (\i/4 where k and w are constant, e = 1, and the primes indicate differentiation with respect to o-i, define a Bertrand curve for which p and T satisfy the relation (97) show also that X 1? /t 1? v\ are the direction-cosines of the binormal to the conjugate ; curve. CHAPTER II CURVILINEAR COORDINATES ON A SURFACE* ENVELOPES In the preceding chapter seen that the coordinates of a point on the tangent surface of a curve are expressible in the form 23. Parametric equations of a surface. (1) we have x where f l (u), ?=/(*), are the equations of the curve, and v is proportional to the distance between the points (f 77, f ), (x, y, z) on the same generator. Since , the coordinates of the surface are expressed by (1) as functions of two independent parameters equations of the surface be written w, v, the may Consider also a sphere of radius a whose center (fig. is at the origin 10). If v denotes the angle, measured in the positive sense, which the plane through the z-axis FIG. 10 of the sphere makes and a point with the #z-plane, and u denotes the angle between the radius OM and the positive z-axis, the coordinates of may be written M M (3) x = a sin u cos v, y = a sin u sin v, z = a cos u. the Here, again, the coordinates of any point on the sphere are ex pressible as functions of two parameter^, and the equations of sphere are of the form (2)*. is * Notice that in this case /^ a function of u alone. PARAMETRIC EQUATIONS OF A SURFACE , 53 In the two preceding cases the functions fv /2 /3 have par consider the general case where /1? /2 /3 are ticular forms. any functions of two independent parameters w, v, analytic for all We v, , values of u and or at least for values within a certain domain. The locus of the point whose coordinates are given by (2) for all values of u and v in the domain is called a surface. And equa tions (2) are called parametric equations of the surface. It is to be understood that one or more of the functions / may be involve a single parameter. For instance, any cylinder defined by equations of the form may x =fi If M =F l y =/ M 2 z =/a u ( ^ v )- u and v in (2) by independent functions other parameters u v v v thus replace (4) we of two u (u v v,), v =F z (u l9 vj, the resulting equations (5) may y be written x = fa (u^ VJ, = fa K, vj, z = fa (u t, vj. If particular values of ing values of of #, y, z u and v l be substituted in (4) and the result v be substituted in (2), we obtain the values ^ and given by ticular values. face, and t^ have been given the par (5), when u^ Hence equations (2) and (5) define the same sur are of such a form that fa, fa, fa s. Hence the satisfy the general conditions imposed upon the equations of a surface may be expressed in parametric form in provided that F l and F 2 F the number of ways y, Suppose the terms of x and solutions. first two arbitrary functions. two of equations (2) solved for u and v in and let u = Ft (x, y), v = F2 (x, y) be a set of of the generality of When (5) these equations are taken as equations (4), equations become x = x, y = y, =/(*, z =f(x, y}, which may be replaced by the single (6) relation, 2 y). first If there is only one set of solutions of the two of equations (2), equation (6) defines the surface as completely as (2). If, however, there are n sets of solutions, the surface would be defined by n equations, z =f (x^ t y). 54 It CURVILINEAR COORDINATES ON A SURFACE may be said that equation (6) is obtained from equations (2) by eliminating u and v. This is a particular form of elimination, the more general giving an implicit relation between x, y, z, as (7) F(x,y,z)=0. If we have form of the a locus of points whose coordinates satisfy a relation For, if we take (6), it is a surface in the above sense. v, / x and y equal to any analytic functions of u and and substitute in (6), we obtain z =/8 (w, v). 2 , namely f^ and In like manner equation (7) may be solved for z, and one or more equations of the form (6) obtained, unless z does not appear in (7). In the latter case there is a relation between x and y alone, so that the surface and its a cylinder whose elements are parallel to the z-axis, parametric equations are of the form is x =/i W y =/ 2 M 2 =/ (w, 3 v). Hence (6), or a surface can be denned analytically by equations (2), Of these forms the last is the oldest. It was used (7). exclusively until the time of Monge, who proposed the form (6); the latter has the advantage that many of the equations, which define properties of the surface, are simpler in form than when equation (7) is used. The parametric method of definition is due to Gauss. In It many will respects it is methods. treatment. be used almost superior to both of the other entirely in the following 24. Parametric curves. is When the parameter u in equations (2) put equal to a constant, the resulting equations define a curve on the surface for which v is the parameter. If we let u vary continu ously, we get a continuous array of curves whose totality consti tutes the surface. Hence a surface may be considered as generated by the motion of a curve. Thus the tangent surface of a curve is described by the tangent as the point of contact moves along the curve and a sphere results from the revolution of a circle about ; a diameter. have just seen that upon a surface (2) there of curves whose equations are given by equations constant, each constant value of We lie (2), an infinity is when u u determining a curve. We call them the curves u = const, on the surface. In a similar way, PARAMETRIC CURVES there 55 The curves of is an infinite family of curves v = const.* two families are called the parametric curves for the given these equations of the surface, and u and v are the curvilinear coordinates say that the positive direction upon the surface.f of a parametric curve is that in which the parameter increases. If we replace v in equations (2) by a function of w, say of a point (8) We v #, y, z are = &lt;t&gt;(u), functions of a single parameter w, and consequently the locus of the point (#, y, z) is a curve. Hence equation (8) defines a curve on the surface (2). For example, the coordinates the equation v = au defines x = a cos w, (8) is a helix on the cylinder y o&gt; sin u, z = v. Frequently equation (9) written in the implicit form, v) F(u, = 0. of this form. Conversely, any curve upon the surface is defined by an equation For, if t be the parameter of the curve, both u and v in equations (2) are functions of t\ thus w =^ 1 (Q, v = (j&gt; z (t). Elimi t between these equations, we get a relation such as (9). return to the consideration of the change of parameters, defined by equations (4). To a pair of values of u^ and v l there nating We correspond unique values of u and v. On the contrary, it may happen that another pair of values of u^ and v l give the same values of u and v. But the values of x, y, z given by (5) will be the same in both cases ; this follows from the manner in which these equations were derived. On this account when equations (4) are solved for u^ and v l in terms of u and v, and there is more than one set of solutions, be used. (10) we must specify which solution will We write the solution u^ = &lt;$&gt;! (w, v), v^ = 4&gt; 2 (u, v). In terms of the original parameters, the parametric lines u^= const. and v l = const, have the equations, * On the sphere defined by equations (3) the curves v const, are meridians and u t const, parallels. When a plane two families is referred to rectangular coordinates, the parametric lines are the of straight lines parallel to the coordinate axes. 56 CURVILINEAR COORDINATES ON A SURFACE b where a and denote constants. Unless u or v is absent from either of these equations the curves are necessarily distinct from the parametric curves u const, and v const. Suppose, now, that = = v does not appear in and vice versa. then u^ is constant when u is constant, Consequently a curve u^ = const, is a member of ^j the family of curves parameters is Hence, when a transformation of made by means of equations of the form u = const. or ^=(2,, ^(M), the two systems of parametric curves are the same, the difference being in the value of the parameter which is constant along a curve. EXAMPLES 1 . A surface which is ; straight line orthogonally right conoid its (v) the locus of a family of straight lines, which meet another and are arranged according to a given law, is called a that 2. when = equations are of the form x = u cos v, y = u sin u, z a cot v + b the conoid is a hyperbolic paraboloid. is = &lt;j&gt; (v). Show passes through the ellipse x 3. Find the equations of the right conoid whose axis z2 V 2 -" the axis of z, and which a, -\ 1. When a sphere of radius a is defined by (3), find the relation between -f u and a4 . v along the curve of intersection of the sphere and the surface x4 Show that the curves of intersection are four great circles. 4. y* + z4 = Upon the surface x v w2 + -J- cos t&gt;, y Ma 4- sin v, z = w, determine the that two and second curves whose tangents make with the z-axis the angle tan- 1 \/2. of these curves pass through every point, and find their radii of curvature. Show first 25. is Tangent plane. A tangent line to a curve upon a surface called a tangent line to the surface at the point of contact. It is evident that there are an infinity of tangent lines to a surface at a shall show that all of these lines lie in a plane, which point. We is called the tangent plane to the surface at the point. To this end we consider a curve C upon a surface y, z) and let M(XJ be the point at which the tangent ( is drawn. The equations of the tangent are 4) f-s = t)-y _ ?-g = ^ dx ds dy ds dz ds TANGENT PLANE where f, 77, 57 for their values f are the coordinates of a point on the line, depending upon the parameter X. If the equation in curvi linear coordinates of the curve C is v = * &lt;f&gt;(u), the above equations may be written . ( ^ , , \ dx\ j du dvl ds =^ \ \du -f- ^^ 4&gt; cv/ ds )-r In order to obtain the where the prime indicates equations. differentiation. locus of these tangent lines, we eliminate$ arid X from these This gives (U) = 0, which evidently is the equation of a plane through the point M. The normal to this plane at the point of contact is called the normal to the surface at the point. As an example, we of a curve at find the equation of the tangent plane to the tangent surface If the values any point. is from (1) be substituted in equation (11), the resulting equation reducible to (12) /i fi fi fi is fi fs Hence the equation upon u. of the tangent plane (I, In consequence of to the 36) * we have independent of the theorem : u, and depends only The tangent plane generator; touches the curve. it is the tangent surface of a curve is the same at all points of a osculating plane of the curve at the point where the generator When the surface is defined by an equation of the form F(x, y, z) = 0, we to imagine that x, y. z are functions of u and v, and differentiate with respect the latter. This gives ~ Hx du dy du dz du dx dv dy dv dz dv * In references of this sort the Roman numerals refer to the chapter. 58 CURVILINEAR COORDINATES ON A SURFACE of these equations the equation (11) of the tangent plane can be given By means the form (l-^+fo-jO^+tf-*)?^. ex cz cy When it is ( the Monge form of the equation of a surface, dz namely z =/(x, y\ t is used, customary to put rdx 14 = -P cz = is 9- cy Consequently the equation of the tangent plane (15) (* - x)p + (77 - y)q -(f-z) = is 0. In the (16) first chapter we found that a curve defined by two equations of the form F l (x,y,z) = J F2 (x, y, z) = 0. Hence a curve is the locus of the points tions of the tangent to the curve are common to two surfaces. The equa g-X^q -y _- Z dx ^ dy dz where cfcc, dy, dz satisfy the relations 5*1* + dx ^dy + ?*& = cz cy z 0, ?**, + dx ^dy + ?** = dz cy $0. Consequently the equations of the tangent can be put in the form (17) 77 - y dx dz - z dy . dz dz dy (13), dz dx dx dy dy dx Comparing is this result with we the intersection of the tangent planes at the curve. see that the tangent line to a curve at a point to two surfaces which intersect along M M *" EXAMPLES 1. Show that the volume of the tetrahedron formed by the coordinate planes and y the tangent plane at any point of the surface x 2. = w, = u, z = a s/uv is constant. Show that the sum of the squares of the intercepts of the axes by the tan gent plane to the surface z at = w 3 sin 3 u, y = M 3 cos 3 v, z = (a 2 - it 2 )*, any point 3. is constant. Given the right conoid for which 0(u) = a sin 2 u. Show that any tangent plane to the surface cuts it in an ellipse, and that if perpendiculars be drawn to the generators from any point the feet of the perpendiculars lie in a plane ellipse. ENVELOPES 4. 59 Show (u) which 5. = a Vtan u, that the tangent planes, at points of a generator, to the right conoid for in parallel lines. meet the plane z to the curve Find the equations of the tangent ax 2 whose equations are + by* + cz 2 = 1, 6x 2 + cy 2 + az 2 = 1. 6. Find the equations of the tangent z(x to the curve whose equations are a) + z)(x is a) = a 3 , z(y + z)(y = a3 , and show that the curve 7. plane. The distance from a point M is point M is of the second order when through M the distance from M form of a surface to the tangent plane at a near-by is of the first order and for other planes MM ; ordinarily of the first order. 26. One-parameter families of surfaces. of the (18) Envelopes. An equation F(x, y, z,a) = Q value of the parameter defines an infinity of surfaces, each surface being determined by a Such a system is called a one-parameter a. For example, the tangent planes to the tangent surface of a twisted curve form such a family. The two surfaces corresponding to values a and a of the param family of surfaces. eter meet in a curve whose equations may be written &gt; * a) = o. a a As a approaches 1 a, this curve approaches a limiting form whose equations are (19) ^( W ,)=0, is *(**.)-(). The curve thus a. defined called the characteristic of the surface of a family of these characteristics, parameter and their locus, called the envelope of the family of surfaces, is a a varies As we have surface whose equation is equations (19). This elimination the second of (19) for a, thus: a obtained by eliminating a from the two may be accomplished by solving =$ (#, y, z), and substituting in the first with the result 60 ENVELOPES The equation of the tangent plane to this surface is in For a particular value of a, say a equations (19) define the curve which the surface F(x, y, z, a ) = meets the envelope and from , ; the second of (19) it follows that at all points of this curve equa tion (20) of the tangent plane to the envelope reduces to This, however, is F(x, y, z, a ) = 0. the equation of the tangent plane to the surface If we say that two surfaces with the same tan gent plane at a common point are tangent to one another, we have is : to The envelope of a family of surfaces of one parameter each surface along the characteristic of the latter. tangent The equations of the characteristic of the surface of parameter a l are (21) This characteristic meets the characteristic (19) in the point whose coordinates satisfy (19) and (21), tions (19) or, what is the same thing, equa and z, F(x,y, a l )-F(x,^z, a) a l approaches 0, this point of intersection approaches a limiting position whose coordinates satisfy the three equations (22) As F-0, ^=0, da a;, !" da 2 0. If these equations be solved for y, z, we have *=/.(a). is (23) * = /, y=/,(a), These are parametric equations of a curve, which edge of regression of the envelope. called the DEVELOPABLE SURFACES The 61 direction-cosines of the tangent to the edge of regression - are proportional to da -^, -. da da If we imagine that x, y, z in (19) are replaced by the values (23), and we differentiate these tions with respect to a, we get, in consequence of (22), equa dx da dy da tfF dy da dy da | dz da dz == " *r da d# ** , c?# da dz da From these we obtain follows that the minors of the right-hand mem ber are proportional to the direction-cosines of the tangent to the But from curve (17) it (19). Hence we have the theorem: The characteristics of a family of surfaces of one parameter are tangent to the edge of regression. Rectifying developable. simple ex ample of a family of surfaces of one parameter is afforded by a family of planes of one parameter. Their envelope is called a developable surface ; the full significance of this term will be 27. Developable surfaces. A shown later (43). The characteristics are straight lines which are tangent to a curve, the edge of regression. When the edge of regression is a point, the surface is a cone or cylinder, according as the point is at a finite or infinite distance. exclude this We case for the present and assume that the coordinates x, y, z of a point on the edge of regression are expressed in terms of the arc s. may write the equation of the plane We (24) (X- x)a + (Y, where by 5, c also are functions of its s. The characteristics are defined s, this equation and derivative with respect to namely : (25) (X- x)a +(Y- y)V+(Z- z)c - ax - by - cz = 0. 62 ENVELOPES Since these equations define the tangent to the curve, they must be equivalent to the equations X-x _Y-y = Zz x y r z Hence we must have (26) ax +%+ cz = 0, ax + Vy + c z = 0. If the first of these equations the resulting equation of (26), to is be differentiated with respect to s, reducible, in consequence of the second if ax" + E jf by" + , IT cz"= A 0. From this equation and (26) ab : : we : find (z x"- c = (y z"- z y") x z") : (x y"- y x"}. Hence by ( 7) we have the theorem: On the envelope of a one-parameter family of planes the planes osculate the edge of regression. We itself. leave it to the sion of the osculating planes of a twisted curve reader to prove that the edge of regres is the curve normal to the principal normal to a curve at a point of the curve is called the rectifying develop able of the latter. We shall find the equations of its edge of The envelope of the plane regression. The equation (27) If of this plane I is (X- x) + (Y- y) m + (Z - z) n = 0. curve, we differentiate this equation with respect to the arc of the and make use of the Frenet formulas (I, 50), we obtain (28) (I - + we derive the equations of the character From istic in these equations the form RECTIFYING DEVELOPABLE t 63 being the parameter of points on the characteristic. In order to find the value of t corresponding to the point where the character istic touches the edge of regression, we combine these equations with the derivative of (28) with respect to s, namely : and obtain (jL--J\t + - s*Q. P VP PT/ the coordinates of the edge of regression of the rectifying developable are (29 ) Hence t=x p , pr TP pr y, z) Tp pr Problem. Under what conditions does the equation F(x, ? = define a devel opable surface We u assume that x, y, z are = const, are the generators, is functions of two parameters w, u, such that the curves and v = const, are any other lines. The equation of the tangent plane This equation should involve u and be independent of given by (i) u. Its characteristic is and where we have put, for the sake of brevity, .t.^.^w+y, ax 2 _ dxdy dxdz dx Since equation (i) is independent of u, we have (iii) A* + B* + c dv dv (ii) = 0. at) Comparing equations and (iii) with (13), we see that ^-X^=0, 3x B-\^=0, dy C-X^ oz 64 ENVELOPES where X denotes a factor of proportionality. If we eliminate x, and X from these equations and (i), we obtain the desired condition 2 X Y y, Z z, F F d2 F F 2 d^F dxdz dz dF dx z2 2 dxdy d2 F " dF dy ? ~fa dx dy dy dy dz az 2 = 0. dx dz d_F_ dy dz aF dy ** dz dx EXAMPLES 1. Find the envelope and edge of regression of the family of planes normal Find the rectifying developable of a cylindrical helix. is to a given curve. 2. 3. Prove that the rectifying developable of a curve the polar developable of its involutes, 4. and conversely. Find the edge of regression of the envelope of the planes x sin u au = 0. y cos u -f z Determine the envelope of a one-parameter family of planes parallel line. 5. to a given 6. stant angle Given a one-parameter family of planes which cut the xy-plane under con the intersections of these planes with the latter plane envelop a curve C. Show that the edge of regression of the envelope of the planes is an ; evolute of C. 7. When a plane curve lies on a developable surface its plane meets the tangent planes to the surface in the tangent lines to the curve. Determine the developable surface which passes through a parabola and the circle, described in a perpendicular plane, on the latus rectum for diameter, and show that it 4s a cone. y2 Determine the developable surface which passes through the two parabolas z = 0; x2 = 4 ay, z = 6, and show that its edge of regression lies on the surface y*z = x 3 (6 z). 8. = 4 ox, the moving trihedral. Problems concern ing the envelope of a family of surfaces are sometimes more readily solved when the surfaces are referred to the moving trihedral of a curve, which is associated in some manner with 28. Applications of the family of surfaces, the parameter of points on being the parameter of the family. the curve Let (30) F(& 77, , *) = APPLICATIONS OF THE MOVING TRIHEDRAL define such a family of surfaces. Since f, 77, f are functions of the equations of the characteristics are (30) and 65 *, ^-^^ + ^^ + ^^4.^=0 ~ ~ ds d% ds dr) ds 0f ds ds But the characteristics being fixed in space, we have (I, 84) Hence the equations (32) of the characteristics are ,_ , /i If, for the sake of brevity, we let $(, ??, f, *) = denote the second of these equations, the edge of regression is defined by (32) and 8S &lt; &gt; !( - For example, the family of osculating planes of a curve is defined with refer ence to the moving trihedral by f = 0. In this case the second of (32) is rj = 0, and - 4(33) is = 0. Hence the tangents are the ; characteristics, and the edge of regres sion is the curve for, we have = =f= ?? 0. In like of .(32) is 17 manner p=0 ; the family of normal planes is defined by = 0. Now the second consequently the polar lines are the characteristics. Equation (33) ; reduces to f -f p r = the edge of regression hence the locus of the centers of the osculating spheres 18). is (cf. The envelope is called the polar developable. The of surfaces From ( osculating spheres of a twisted curve constitute a family which is readily studied by the foregoing methods. 18) it follows that the equation of these spheres is The second of equations (32) for this case is which, since spherical curves are not considered, reduces to = 0. And equation (33) is ?? = 0, so that the coordinates of the edge of regression are f = = f = 0. 77 Hence : The osculating lating spheres ; circles of a curve are the characteristics of its oscu and the curve itself is the edge of regression of the envelope of the spheres. 66 ENVELOPES 29. Envelope of spheres. Canal surfaces. We consider now any family of spheres of one parameter. Referred to the moving tri hedral of the curve of centers, the equation of the spheres is By means of (32) we find that a characteristic which a sphere is cut by the plane The r= is the circle in radius of this circle is acteristic s imaginary equal to when r n is &gt; rVl 1. r n . Hence the char 1, reduces to a point when + const., and is real for r f * &lt; By means (34) of (33) we by find that the coordinates of the edge of regression are given f = -n- , , = [l-(rr ) ]p, r two parts with corre Hence the edge of regression consists of sponding points symmetrically placed with respect to the oscu lating plane of the curve of centers (7, unless When is this condition is satisfied the edge is points lie in the osculating planes of C. the case with the osculating spheres of a curve. We a single curve, and its have seen that this We shall show that their when the above condition &lt;7 is satisfied the spheres osculate We (35) edge of regression r write the above equation in the form p[l~-(rr e is )] ^ = er^l-r \ where so that p may be positive. have seen ( 16) that the absolute and relative rates of change with s of the coordinates f, ?;, f of a point on Ct are in the relations or 1, +1 We M = ^_?? + Ss i, ds ^ = ^Z + l + f, &s ^ Ss == ^_!?. ds T p ds p T the values (34) are substituted in the right-hand of these equations, we obtain, in consequence of (35), When members ENVELOPE OF SPHEEES Hence the linear element Ss^ of 67 C l is given by cs 1 = and (36) Uo, Since these are the direction-cosines of the tangent to C^ we see that this tangent is normal to the osculating plane to the curve of centers C. Moreover, these direction-cosines must satisfy the equations /37\ (cf. I, 83) 8a 8s _da ds b 8b 8s _ db ds a p c 8c 8s do ds b p T r Hence we have from which (38) it follows that the radius of curvature p l of Pl C l is =ee rVT^, where 1 or 1 , so that /o 1 may be positive. Since, now, the is direction-cosines of the principal normal have the values e r + it follows that the principal normals to C and Cl are parallel. Furthermore, since these quantities must satisfy equations (37), we have g 3 g 3 , ^^ -. where of (I, denotes the derivative of p l with respect to s r By means 51) we find that the radius of torsion r l of Cl is given by p[ .From (38) we find p[= - so that the radius R^ of the oscu 2 = r and consequently p? p[ TI lating sphere of C^ is given by R* the osculating spheres of Cl are of the same radius as the given + 2 , spheres. 68 ENVELOPES The direction-cosines of the tangent, principal normal, and binor to Cl are found from (36) and (39) to be mal Hence the coordinates of Cl of the center of the osculating (I, 94) sphere are reducible, in consequence of (34), to + l iPi - P( T I\ = we have *? + m iPi ~ friPi = : &gt; ? + W1p1 - XT^ = 0. Therefore the theorem When the edge of regression of a family of spheres of one param eter has only one branch, the spheres osculate the edge. r is Since r does not appear in equation (35), it follows that when given as a function of s, the intrinsic equations of the curve of are where the function/(s) is arbitrary. Moreover, any curve will serve for the curve of centers of such an envelope of spheres. The deter mination of r requires the solution of equation (35) and consequently involves two arbitrary constants. When all the spheres of a family have the same radius, the envelope is called a canal surface. From (34) it is seen that in this case a characteristic is a great circle. Moreover, equation (35) a necessary and sufficient condition that the edge of regression of a canal surface consist of a single curve is that the curve of centers be of constant curvature and the radius reduces to p = r. Hence of the sphere equal to the radius of first curvature of the curve. GENERAL EXAMPLES 1. Let MN be a generator of the right conoid x = u cos u, it y = u sin i&gt;, z = 2 k cosec 2 D, M being the point in which 2. that the tangent plane at meets the surface in a hyperbola which passes through M, and that as moves to the hyperbola describes a plane. along the generator the tangent at z-axis. meets the Show N M N A 2 a2 ft c2 motion always passes through the perpendicular from the center on the tangent plane at the point. Show that the path of the point is the curve in which the ellipsoid point moves on an ellipsoid --h H -- = 1, so that the direction of its is cut by the surface x l y m z n = const. , where 1: m : n --- -: : --- GENERAL EXAMPLES 3. 69 If same angle about the tangent each of the generators of a developable surface be revolved through the to an orthogonal trajectory of the generators at the point of intersection, the locus of these lines is a developable surface whose edge of regression is an evolute of the given trajectory. 4. Show that the edge of regression of the family of planes (1 - w 2 )z + i(l + u*)y + 2uz +f(u) = , is a minimal curve. 5. The developable surface which passes through the 62 , x2 -f z2 6. y = meets the plane x = in circles x2 -f y* = a 2 an equilateral hyperbola. z = 0; surface az Find the edge of regression of the developable surface which envelopes the = xy along the curve in which the latter is cut by the cylinder x2 = by. ellipsoid 7. Find the envelope of the planes which pass through the center of an and cut it in sections of equal area. 8. planes The first and second curvatures ax + /3y + yz p, where /3, &lt;r, of the edge of regression of the family of 7, p are functions of a single parameter u and a 2 + 2 /3 -f y 2 = 1, are given by 1 A3 pp a oc. A2 where aa O A= P P a" p a" f P off , D= -r\ p a y " , ft y 9. Y p, y" p,, " Derive the equations of the edge of regression of the rectifying developable of 28. by the method 10. Derive the results of 11 . 29 without the aid of the moving trihedral. circle Find the envelope of the spheres whose diameters are the chords of a latter. through a point of the 12. Find the envelope and edge of regression of the spheres which pass through a fixed point and whose centers lie on a given curve. 13. Find the envelope and edge of regression of the spheres which have for diametral planes one family of circular sections of an ellipsoid. 14. Find the envelope and edge of regression of the family of &lt;j/2\ ellipsoids 1 H -j2 - = 1, where a is the parameter. (3^2 15. Find the envelope of the family of spheres whose diameters are parallel ellipse. chords of an 16. Find the equations of the canal surface whose curve of centers is a circular helix and whose edge of regression has one branch. Determine the latter. 17. Find the envelope of the family of cones (ax + x + y + z - 1) (ay + z) - ax (x + y + z - 1) = 0, where a is the parameter. CHAPTER III LINEAR ELEMENT OF A SURFACE. DIFFERENTIAL PARAMETERS. CONFORMAL REPRESENTATION 30. Linear element. Upon y a surface , defined by equations in the parametric form (1) x =fi (i*, v), =/ a (w, v), 2 =/, (M, v), we select any curve and write its equations$ (u, v) 0. From we have that the linear element of the curve is given by (2) 3 d? j foj = fa du -- av, t where ax -\ ^M the differentials &lt;^w, ^v j ay -^ = dij au + dy , t , ^w vf , 2 = dz dz , aw H--, 9 du dv dv satisfying the condition 2$^w , c?w + a^ dv = , 0. dv ifweput du dv cu cv du dv or, in abbreviated form, equation (4) (2) becomes oV 2 = Edu? + 2 Fdudv + G dv G thus denned were 2 . The functions E, F, the surface first used by Gauss.* When * is real, and likewise the curvilinear coordinates Disquisitiones generates circa superficies curvas (English translation by Morehead Hiltebeitel), p. 18. Princeton, 1902. Unless otherwise stated, all references to Gauss are to this translation. and 70 LINEAK ELEMENT w, v, the functions 71 understand also that There is, however, an important excep tional case, namely when both E and G are zero (cf. 35). For any other curve equation (4) will have the same form, but the relation between du and dv will depend upon the curve. are real. Vj, # We shall the latter are positive. Consequently the value of c7s, given by (4), is the element of arc of any curve upon the surface. It is called the linear element of the surface (cf. 20). However, in order to avoid circumlocution, we shall frequently call the expression for ds 2 the linear element, is, that the right-hand member of equation (4), which is also called the first ter, fundamental quadratic form. The coefficients of the lat namely E, F, G, are called the fundamental quantities of the for the sake of brevity, first order. If, we put du du (5) d(u, v) dv it dz d(u, v) d(u, v) dv follows from (3) and (5) that (6) EG - F = A + B + C 2 2 2 2 . and likewise the parameters, is different from zero unless J, B, and C are zero. But if A, B, and C are zero, it follows from (5) that u and v are not independent, and consequently equations (1) define a curve and not a surface. However, it may happen that for certain values of u and v all the quantities J, B, C vanish. the quantity Hence when the EGF surface 2 is real The corresponding face. points are called singular points of the sur These points may be isolated or constitute one or more curves upon the surface such curves are called singular lines. In the following discussion only ordinary points will be con ; sidered. From the preceding remarks it follows that for real surfaces, referred to real coordinate lines, the function (?) is real, H defined by 1 and it is positive by hypothesis. 72 LINEAR ELEMENT OF A SURFACE 31. Isotropic developable. and H is zero, is afforded The exceptional case, where the surface is imaginary by the tangent surface of a minimal curve. The equa . tions of such a surface are (cf 22) w2 /I (u) du + 1 w2 (M) u, = ju(f&gt;(u) du + u&lt;j&gt; (u) v, where 0(w) a function of u different from zero. It is readily found that J=v2 2 (w), F 2 = 0. This equation is likewise the sufficient and consequently EG that the surface be of the kind sought. For, when itjs satisfied, the equa condition 2 2 If X denote an tion of the linear element can be_written ds = (Vjdw + V(?dw) factor of \^Edu -f- V(?du, and a function MI be defined by the equation integrating is F= G 0, . $$\fEdu + VGdv) dwi, the above equation becomes ds2 = A"** duf. Hence, if we we const, and any other system for vi take for parametric curves u\ have FI = 0, GI = 0. In other form these equations are = const. , 3v In accordance with the last equation we put 01?! 2 8i?i where By undetermined. integration we have A; is r X, M, v being functions of HI alone. When we these values are substituted in the first of the above equations of condition, get to be satisfied by X, /A, and v. The equation of the tangent plane (1 to the surface (i) is reducible to - 2 W]l ) (X - x) + i (1 + w- 2 )( Y- y) + 2 Ml (Z - z) = 0. Hence the surface Since its edge of regression is a minimal curve is developable. surface is called an isotropic developable. (Ex. 4, p. 69), the theorem is proved. The 32. Transformation of coordinates. It is readily found that the functions E, F, G are unaltered in value by any change of the show that these functions rectangular axes. But now we shall there is a change of the curvilinear change their values when coordinates. TRANSFORMATION OF COORDINATES = u(u^ = v(u ,v l 73 Let the transformation of coordinates be defined by the equations (8) u Vj), v l ); then we have dx du, i _ dx du dx dv dv du. i dx dv. i _ dx du du dv, du du, i 11 dv dv dx dv we find the relations fa (9) E du dv dv dv. 1 du dv dv, du, 11 d^ dv l Hence the fundamental forms when there is quantities of the first order assume a change of curvilinear coordinates. differentiation, new From (8) we have, by du = du du. 11 du l du -\ dv. aVj, dv dv du, 11 du. + - dv dv.. dv, Solving these equations for du v dv^ we get l/dv where (10) , du , \ 1/ dv , . du d(u, v) Hence we have \ "du du 8 dv l dv (du so that (12) 1 (9) ^ From we find the relation LINEAR ELEMENT OF A SURFACE By means equations of this equation (9) into and the relations : (11), we can transform the following ^^\^2F^ EG E (13) l + G( 1 du I* 0V, fa to 1 &lt; 1 __ F (faito1 cu^ ct\ i- du cu cu EG- r^ ,T /" 1 cv JG/Cr Tfi S~1 CM Jj&lt;2 1^1 33. 1 .T WA metric line v in Angles between curves. The element of area. Upon a para = const, we take for positive sense the direction increases, which the parameter u u = const, we the direction in which v increases. the elements of arc of curves v tively, find, and likewise upon a curve If efe,, and e?s M denote and u = const, respec const, from ds v (4), (14) = ^Edu, M, ds u = ^Gdv. Hence, if a w #, y v and , /9M , tangents to these curves respectively, y u denote the direction-cosines of the we have fa du 1 cu cy "*&lt;/ E du dx &lt; 1 dz have seen that through an ordinary point of a surface there passes one curve of parameter u and one of parameter v. and 180, If, as in fig. 11, &) denotes the angle, between We formed by the positive directions of the tangents to these curves at the point, we have (15) cos ft) = aa + , + 77 = -7== W and (16) sin &) = VJSQ--F* H ANGLES BETWEEN CUKVES When two families of curves 75 upon a surface are such that through any point a curve of each family, and but one, passes, and when, moreover, the tangents at a point to the two curves through it are perpendicular, the curves are said to form an orthogonal system. From (15) we have the theorem: A necessary and sufficient condition that the is that a surface form an orthogonal system F = 0. parametric lines upon Consider the small quadrilateral (fig. 11) whose vertices are the points with the curvilinear coordinates (u, v), (u -f du, v), du, v (u dv). To within terms of higher order the sides of the figure are equal. Consequently it is approxi opposite mately a parallelogram whose sides (u, v + dv), + + are of length v E du and is o&gt;. \ G dv and The area the included angle of this parallelogram is called the element of area of the surface. Its expression (17) If its is /(u+du.v) FIG. 11 d^ = sin CD V EG dudv = H dudv. C is any curve on a surface, the direction-cosines tangent at a point have the form dx __ _ _ /dx du I _ I _ a, 7 of _ dx dv _ _ _ /dz /-/ __ _. __ dy ^ ds I ,-^-- /dy ^4^ du , n _ __1 ds cu ds dv ds \cu ds dy dv dv ds g&lt; _ J If ~ dz _ du cz dv ~ds ds~\^uds dv we put dv/du = X and the right-hand member of replace ds by the positive square root of (4), the above expressions can be written dv (18) du dv 7 = du dv 76 LINEAR ELEMENT OF A SURFACE these results it From is not upon the ratio X. absolute values of of seen that the direction-cosines depend du and dv, but upon their obtained by differentiation from the The value (7, X is equation of (19) namely Let Cl be a second curve meeting C the direction-cosines of the tangent to at a point C l at M M, and /3 V let be a v yr They are given by " l du 8s /3 t dvW 8 indicates variation and similar expressions in the direction of If 6 for and 7^ where Cr C and Cl at denotes the angle between the positive directions to M, we have, from (18) and (20), cos i (21) = #tf + ppj + 77j = Eduu + F(du 8v -f dv 8u] +Gdv8v j-^ x and sin 9 = Vl - cos is = H dv 8s (8u ds 8v du 8s ds This ambiguity of sign of due to the fact that 6 as denned is one two angles which together are equal to 360. upper sign, thus determining 6. This gives /nft . We take the (22) Q Tr sm6 = H . /8udv -_ \09 d8 8v du\ -__. 08 ds/ The When have 8v /00 (23) . significance of the above choice will be pointed out shortly. const, through M, we in particular Cl is the curve v = . = and ~ 8s = V E 8u, 1 j so that a cos&lt;9 = T,dv\ = { -^du + .F U ds If dv ds/ 81X100=-= From (24) these equations we obtain tan* = Edu+Fdv Hence there is, The angle metric co between the positive half tangents to the para uniquely denned. in curves has been ANGLES BETWEEN CURVES 77 = const, general, only one sense in which the tangent to the curve v can be brought into coincidence with the tangent to the curve u = const, by a rotation of amount co. We say that rotations in this direction are positive, in the opposite sense negative. From is the angle described in the positive sense (23) it is seen that when the positive half tangent to the curve v const, is rotated = so in the general the angle described in the positive rota case 6, defined by (22), tion from the second curve to the first. into coincidence with the half tangent to C. is And From (26) equations (15), (16), and (23) we find These equations follow also directly from (20) and ering the curve (21) by consid the u = const, as the second line. As an immediate consequence theorem : of equation (21) we have necessary and sufficient condition that the tangents to two curves upon a surface at a point of meeting be perpendicular is (26) A E du Su + F(du Sv + dv 8u) + G dv v = 0. EXAMPLES 1. Show that when ds 2 the equation of a surface is of the form z 2 ) =/(, y), its linear element can be written = (1 + p2 ) dx 2 + 2pqdxdy + (l + q dy 2 , where p = dz/dx, and q = cz/dy. Under what conditions do the y = const, form an orthogonal system ? 2. lines x = const. , Show that the parametric curves on the sphere x = a sin u cos v, y = a sin u sin u, z = a cos u form an orthogonal system. Determine the two families of curves which meet the curves v = const, under the angles ir/4 and 3 7r/4. Find the linear element of the surface 3. when these new curves are parametric. y Find the equation of a curve on the paraboloid of revolution x = wcosu, = w 2 /2, which meets the curves v = const, under constant angle a Determine a as a function of and passes through two points (M O i). (MI, = itsinu, z , i&gt;o), 4. Find the differential equation of the curves upon the tangent surface of a curve which cut the generators under constant angle a. 78 LINEAR ELEMENT OF A SURFACE " . all 5. Show that the equations of a curve which lies upon a right cone and cuts cesinw, the generators under the same angle are of the form x = ce cosu, y the curve upon 2 = 6e", where a, 6, and c are constants. What is the projection of a plane perpendicular to the axis of the cone ? Find the radius of curvature of the curve. 6. Find the equations of the curves which bisect the angles between the para metric curves of the paraboloid in Ex. 3. 34. Families of curves. An equation of the form (27) &lt;(w, v)=c, infinity of curves, or a Through any point of the sur where c is an arbitrary constant, defines an family of curves, upon the surface. face there passes a curve of the family. For, given the curvilinear coordinates of a point, we obtain a value of when c, say of the point. inquire whether this family passes through curves can be defined by another equation. Suppose it is possible, and let the equation be ; CQ these values are substituted in (27) then evidently the curve = c We (28) ^(U,V) c = K. necessarily a function of the if ifr is any same family of define the Since and K are constant along any curve and vary in passing is from one curve to another, each other. Hence i|r is a function of fa Moreover, function of fa equations (27) and (28) curves. From equation (24) it is seen that the direction, at any point, of the the curve of the family through the point is determined by obtain the latter from the equation value of dv/du. We 36 (29 ) , _ d&lt;f&gt; g*H.2*.i.&lt;* which is derived from (27) by differentiation. Let (u, v) = cbe an integral of an ordinary of the first order and first degree, such as (30) differential equation M(u, v) du 4- N(u, v) dv = 0. defined by the former equation are called integral curves of equation (30). From the integral equation we get equation (29) then to obtain equation (30) by differentiation. It must be possible The curves FAMILIES OF CURVES from the integral equation and (29). 79 does not appear in But c (29), consequently the latter equation differs from (30) by a factor at most. Hence M dv N du = 0. Suppose, now, that we have another integral of (30), as ^Hw, v) = e. Then M -^- N -^- = dv cu = d(u,v) o . o , 0. The elimination of it M and N from these equations gives ^ ^ is ; from which follows that ty &lt;. a function of &lt;/&gt;. Moreover, if ^r can But by any function of of the families of curves &lt;, we have seen that -fy = const, &lt;/&gt; and = const, ^ is a function are the same. Hence admit all integrals of equation (30) of the form &lt;f&gt;=c or ^=e may obtain define the of same family of curves. c) However, equation this be solved for (30) an integral in which the constant of integration enters implicitly, as F(u, = t&gt;, 0. But if &lt;?, we one or more integrals of the form (27). Hence an equation of the form (30) defines one family of curves on a surface. Although the determination of the curves when thus integration of the equation, the direction of is given directly by means of (24). If at defined requires the any curve at a point each point of intersection of a curve C with l the curves of a family the tangents to the two curves are perpendicular to one another, Cl is called an orthogonal trajectory of the curves. Sup pose that the family of curves is defined by equation (30). The 7 r\ relation between the ratios tions of the tangents to is du ou the two curves If and &gt; which determine the r\ direc- at the point of intersection, ~~\/T given by equation (26). we replace cu by A 0. , we obtain (31) (EN- FM) du + (FN GM) dv = But any integral curve of this equation is an orthogonal trajectory of the given curves. Hence a family of curves admits of a family of orthogonal trajectories. They are defined by equation (31), when the differential equation of the curves is in the form But when the family is defined by a finite equation, such as the equation of the orthogonal trajectories is (30). (27), (32) 80 LINEAR ELEMENT OF A STJKFACE circles in the plane As an example, we consider the family of on the x-axis whose equation is (i) with centers x2 + y2 - 2 ux - a2 , where u is the parameter of the family and a is a constant. In order to find the orthogonal trajectories of these curves, we take the lines x = const. y = const. , for parametric curves, in which case E = G = 1, and write the equation (i) F = 0, thus 3? in the form (27), x + 2 i (y 2 x - a2 ) = 2 u. Now equation (32) is 2 xy dx (x y2 + a2 ) dy = 0, of which the integral is where v is the constant of integration. Hence the orthogonal whose centers are on the y-axis. trajectories are circles An (33) is ordinary differential equation of the second degree, such as H (u, v) du 2 +2 S(u, v) du dv + T(u, v) dv 2 = 0, equivalent to two equations of the first degree, which are found by solving this equation as a quadratic in dv. Hence equation (33) seek the con defines two families of curves upon the surface. We dition that the curves of one family be the orthogonal trajectories of the other, or, in other words, the condition that (33) be the equa tion of an orthogonal system, as previously defined. If & x and Jc 2 denote the two values of - obtained from du (33), we have From (26) it follows that the condition that the two directions is at a point corresponding to K I and K Z be perpendicular E + FK + If tc + GK = 0. we have the above values are substituted in this equation, it is the condition sought; (34) MINIMAL CURVES 35. is 81 Minimal curves on a surface. equating to zero An equation of the form (33) obtained by the first fundamental form of a surface. This gives Edv?+ ZFdudv + Gdv2 = and it 0, zero which to JSG defines the double family of imaginary curves of length In this case equation (34) reduces lie on the surface. F =0; 2 hence the minimal lines on a surface form an orthogonal system only able ( when the surface is an isotropic develop 31). important example of these lines is furnished by the system on the sphere. If we take a sphere of unit radius and center at the origin, the forms its An equation, x 2 + y*+ z = l, 2 can be written in either of 1 z x -f iy where u and v denote the respective ratios, and evidently are conju ?/, gate imaginaries. If these four equations are solved for z, z, we find u s** -, -h v Ai i -.--, i(v \ u) f_ A uv , 1 9 ~uv+l From these expressions uv +\ uv+1 we find that the linear element, in terms of the parameters u and v^ is given by ,o (36) 4:dudv (1 + v Hence the curves u zero. = const, and const, are the lines of length Eliminating u from the (35), first two and the v*)y last two of equations we get (37) Hence all the = 0, = 0. i(i? + l)z + 2vy+i(l-v points of a curve v = const, lie on the x 4- (1 2 iv z 1 ) line v z )Y 2iv = Q, 82 LINEAR ELEMENT OF A SURFACE F, where X, Z denote current coordinates. In consequence of (35), these equations can be written X-x, , Y-y. Z-z, point. where # y^ z are the coordinates of a particular manner the curves u = const, are the minimal lines In like X-x. Y-y. = Z-z, , EXAMPLES 1 . Show that the most general orthogonal system of circles in the plane 34. is that of the 2. example in Show (w 2 that on the right conoid x dw 2 3. -f a 2 ) dv 2 = = ucosv, y form an orthogonal system. ds* = usinv, z = au, the curves When the coefficients of the linear ds 2 = Erfu* + 2 Fidudv + Gidv 2 , elements of two surfaces, = E 2 du? + 2 F2 dudv + G 2 di; 2 , are not proportional, and points with the same curvilinear coordinates on each of the surfaces are said to correspond, there is a unique orthogonal system on one surface corresponding to an orthogonal system on the other; its equation 0. is (Fi^a 4. 2 FzEi)du* + (E a Gi - EiG z ) dudv 2 +(GiFz - G2 Fi)du = If 61 and are solutions of the equation a^ At/ dag/3 2 /J, da _ u, 8? where X is any function of a and the equations +*5 define a surface referred to its 3 2* . minimal lines. 36. Variation of a function. ?/, system of coordinates v, Let S be a surface referred to any and let (w, v) be a function of u and v. &lt;j&gt; When the values of the coordinates of a point &lt;, Tlif o f the surface are substituted in (38) we obtain a number c ; and consequently the curve VABIATION OF A FUNCTION 83 along this curve the passes through M. In a displacement from remains the same, but in any other direction it changes value of (f&gt; M and the rate of change is given by d d k dv dc) du where k dv/du determines the direction. As thus written it is is understood that the denominator of the right-hand member positive. For the present we consider the absolute value of ~-t and write ds du (39) dv ds e is where negative. direction along the curve (38). In order to find the maximum value we equate to zero the derivative of A with respect to Jc. This gives 1 according as the sign of the numerator is positive or The minimum value of A is zero and corresponds to the From (32) it follows that this value of k determines the direction at right angles to the tangent to this value of k in (39) we get the = c at the point. By substituting maximum &lt;f&gt; value of A. (w, v) at Hence: The differential quotient -^- of a function ds surface varies in value with the direction zero in the direction tangent to the curve absolute value in the direction a point on a &lt;f&gt;= from the point. It equals c, and attains its greatest value being normal to this curve, this 4 &lt; &gt; ; S m^\-ZF-z-z-+G dv du dv A means of ential quotient representing graphically the magnitude of the differ A for any direction is given by the following theorem to : If in the tangent plane tangents at a surface at a point to all M the positive half M, corresponding values of k, positive and negative, 84 be LINEAR ELEMENT OF A SURFACE drawn, and on them the corresponding lengths const. A be laid off from M, the locus of the extremities of these lengths is a circle tangent to the curve &lt;= The proof of this theorem is simplified if we effect a transfor mation of curvilinear coordinates. Thus we take for the new coor const, and their orthogonal trajectories. dinate lines the curves (f&gt; We vl let the former be denoted by u v const, and the latter by = const., and indicate by subscript 1 functions in terms of these parameters. Now F = 0, l -t J. so that 7 where \ direction, denotes the value of dvjdu^ which determines a given and the maximum length is (J&$$~*. From (23) we have cos = . sin = where 6 Q is the angle which the given direction makes with the const. Hence if we regard the tangents tangent to the curve v l const, as axes of coordinates const, and u l at to the curves v l = M = in the tangent plane, the coordinates of the end of a segment of length A are distance from this point to the mid-point of the = segment, measured along the tangent to v t const., The maximum is found to be - readily =&lt;&gt; which proves the theorem. first order. 37. Differential parameters of the If we put (41) A^ = equation (40) can be written (3) where now the normal to the curve &lt;&gt; -** differential quotient corresponds to the direction The left-hand member of this const. = DIFFERENTIAL PARAMETERS equation is 85 evidently independent of the nature of the parameters u and is which the surface is referred. Consequently the same true of the right-hand member. Hence A^ is unchanged in v to value when there is any change , of parameters whatever. The set of full significance of this result is as follows. Given a new , parameters defined by M=/I (M I i^), v=/2 (w 1 v^\ let ^(u^ vj denote the result of substituting these expressions for u and v in (w, v), and write the linear element thus &lt; : ds 2 =E l du* + 2F l du l dv l + G l dv*. The invariance of A^ under , this transformation is expressed by the identical equation EG-F We leave it to the reader to verify this directly with the aid of equations (9). The invariant A^ is ter of the first order ; this name and the called the differential parame notation are due to Lame.* Consider for the (42) moment the partial differential equation A^ = &lt;/&gt; and a solution = const. From the latter we get, by differentiation, d&lt;l&gt; , - du -f , , 3$ dv A = 0. du OJ O J dv in (42) If we replace and by dv and du, which are evi dently proportional to them, we obtain Edu*+ Hence the is 2 Fdudv + Gdv*= 0. integral curves of equation (42) are lines of length zero, = const, is a line of length zero, the function and conversely if (/&gt; &lt;/&gt; a solution of equation (42). Another particular case is that in which A^ is a function of &lt;, say (43) * A,* les = *&lt;*). Lemons sur coordonnees curvilignes et leurs diverses applications, p. 5. Paris, 1859. 86 LINEAR ELEMENT OF A SURFACE (41) it is From seen that when we put equation (43) becomes (44) A denned, 6 is 1 l9=l. As 6 const, is a function of $; hence the family of curves = const. Suppose we have the same as the family &lt;/&gt; const, for the curves such a family, and we take the curves 9 u = const, and their orthogonal trajectories for v = const., thus it follows from effecting a change of parameters. Since Aj%=l, (41) that (45) ^ = 1, and consequently the linear element ds 2 is =du? + Gdv 2 . Since now the linear element of a curve v const, is du, the length of the curve between its u =u and u = u^ const, between these two curves. the segment For this reason the latter curves are said to be parallel. Con = const, of an orthogonal sys versely, in order that the curves u u^ of every curve v is UQ points of intersection with two curves Moreover, this length is the same for . tem be curves v that the linear parallel, it is necessary element of the must be a func const, be independent of v. Hence of coordinates, can be tion of u alone, which, by a transformation made equal to unity. Hence we have the theorem : = E A = (/&gt; the curves of a family necessary and sufficient condition that be a function of const, be parallel is that $$f&gt; &lt;f&gt;. be the equations of two curves upon a surface, through a point M, and let 6 denote the angle between the tangents at M. If we put Let (f) = const, and ^ = const, (46) \(*,*)= -E _F dv dv \dv du + G du dv / du du EG -I" the expression (21) for cos 6 can be written cos , (47) DIFFERENTIAL PARAMETERS an invariant for transformations of coordinates, follows from this equation that A x (0, ^r) also is an invariant. It Since cos 6 is 87 it is called the mixed differential parameter of the first order. diate consequence of (47) is that An imme A,(*. is i/r *= ) the condition of orthogonality of the curves = const, (f&gt; and = const. Now equation (22) can be written r \du dv dv c which by means of the function &lt;e) (w, v), defined thus by Darboux,* can be written in the abbreviated form (49) sin 6 = ^r) the functions in this identity except to be invariants, we have a proof that it also all is Since are (&lt;^, known It is an invariant. a mixed (49) it differential parameter of the 2 ((/&gt;, first order. From (47) and (50) follows that 2 A, (&lt;, f+ ) t) - A^ A^ ; consequently the three invariants denned thus far are not inde pendent of one another. From (41) and (46) ri it follows that ^U = and from these we 2 W find A ^ ^=l^ _ Tf&lt; AlV = ^ jfi (51) (u, v) = A,H V-A 2 2 i (. ) = ^ first order. Consequently @ Hence i, 2 (w, 1 t;) (w, v) ^, and G are differential invariants of the * Lemons, Vol. Ill, p. 197. 88 LINEAR ELEMENT OF A SURFACE Another result of these equations is the following. If the param changed in accordance with the equations Ui eters of the surface are = Ui(u, v), v^v^u, is v), and the resulting linear element ds* written, =E l du* + 2 Fl du^dv^ + 6^ dv*, the value of E l is given by and 7^ and 6^ are found in like manner. In consequence of (51) these equations are equivalent to (13), which were found by direct calculation. Thus far we 38. Differential parameters of the second order. have considered differential invariants of the first order only. We introduce now one of the second order, discovered by Beltrami.* To this end we study the integral n= for an ordinary portion of the surface (cf. bounded by a closed curve C 33). For convenience we put (53) M= -G * z_ F du - d dv , N= we have -dv E z_ F 3 du , so that, in consequence of (46), This may be written If we apply Green reduces to s theorem to the first integral, this equation (54) n= ( C(j&gt;(Mdv-Ndu)- ff(^+ j^} dudv &gt; *Ricerche di analisi applicata (1864), p. 365. alia geometria, Giornale di matematiche, Vol. II DIFFERENTIAL PARAMETERS where the first 89 integral is curvilinear and is taken about C in the Evidently du and dv refer to a displacement C. If we indicate by 8 variations in directions normal to C along and directed toward the interior of the contour, then from (23) customary manner. and (25) it follows that Edu + Fdv _-H8v F du + G dv __ ds 8s ~dT ~ST Hence M dv du Ss dv 8*/ 8s All of the terms in this equation, with the exception of - H\du ( h -r dv J ) &gt; are independent of the choice of parameters. Hence the latter is an invariant. It is called the differential parameter of the second order and is denoted by A 2 i/r. In consequence of (53) we have ..... (56) In the foregoing discussion quantities appear. But all has been assumed that only real these results can be obtained directly it from algebraic considerations of quadratic differential forms * without any hypothesis regarding the character of the variables hence the differential parameters can be used for any kind of ; curvilinear coordinates. In addition to A 2 there are other differential invariants of the c/&gt; second order, such as And are find a AA Q, i/r), A, (A^, A^), (A mixed invariants of the second order. In like manner ; we can group of invariants of the third order AAM&gt;, AA(4&gt;.M&gt;)&gt; for instance, A.A,*, A A*. I, * Cf. Bianchi, Lezioni di geometria differ enziale, Vol. chap. ii. Pisa, 1902. 90 LINEAR ELEMENT OF A SURFACE others, These invariants and their derivatives. extension of this method, involve functions which can be obtained by an evident E, F, G, and A/T, . c/&gt;, , Conversely, T ftw /== /(A we shall show -, * that every invariant of the form v ^ . . r dE tr, dG -, di&gt; *W dw dw 0, -^-, I -, i/r, -^L, ^ du ), where of the . &lt;, -^, are independent functions, is symbols A and . Already we have seen expressible by means that E, F, and G can be expressed in terms of (48) it A x w, A x v, and A 1 (w, v). Moreover, from follows that when X any function whatever. Hence expressed in terms of the symbols A and is all , the terms in / can be applied to Since u and v do not appear explicitly in of parameters, replacing /, we can , effect a u and v by and &lt;/&gt; ty respectively, change and con to these sequently we express / in terms of (/&gt;, ^, and the differential &lt;*) invariants obtained by applying the operators A and functions. In case is the only function appearing in /, c/&gt;, we can such as take for i/r, 2 &lt;, in the change of parameters, any invariant of it is A^ or A so long as not a function of (/&gt;, E, F&gt; or G. EXAMPLES 4 1 . When is the linear element of a surface is in the dv^), form ds 2 = \(du^ + where X both u and v are solutions of the equation A 2 = the differential parameter being formed with respect to the right-hand member. a function of u and D, 0, 2. Show that on the surface x the curves 3. it = u cos u, y = u sin v, form z = av -f- (u), = const, are parallel. is When is the linear element in the ds2 = u, cos^adu 2 + sin 2 a: eh? 2 , where a a function of u and both u and v are solutions of the equation * Cf. Beltrami, I.e., p. 357. SYMMETRIC COORDINATES 4. If the 91 curves the projection = const., \p = const, form an orthogonal system on a surface, on the x-axis of any displacement on the surface is given by dx dx =- d\b = + dx * dd&gt; 2= A0 , where ds and respectively. 5. da- are the elements of length of the curves = const., ^ = const. If /and are any functions of u and u, then . , 0) = a/ --^ a0 ^- du AIU du 2 i if a0 /3A A + (^ - + - - A! (w, . , u) + \cu cv 2 cv du/ a/a0. ^^ cv cv v) Aii&gt;, A2 / = 39. ^A w + ^A u + ^A lW + 2^- Ai(u, SU CM CU 2 0ttCtJ + ^AiU. SV 2 Symmetric co drdinates. We have seen that through every point of a surface there pass two minimal curves which lie entirely on the surface, and that these curves are defined by the differential equation Edv?+2 Fdudv + G dv = 2 0. If the finite equations of these curves be written fi (w, v) a (w, it = const., v) A, ()=&lt;), = const., follows from (42) that (5T) A 1 (/3) = 0. Since for any parameters /^\ w= when the curves a ~ const., ft = const., are taken as parametric, the corresponding coefficients and G are zero, and consequently the linear element of the surface has the form E (59) ds 2 = \ dad/3, where, in general, X is a function of a and {3. Conversely, as fol lows from (58), when the linear element has the form (59) equa tions (57) are satisfied Hence the only transformations form of the linear parametric, that (60) is and the parametric curves are minimal. of coordinates which preserve this element are those which leave the minimal lines a or = - 92 LINEAR ELEMENT OF A SURFACE where F and F l are arbitrary functions. Whenever : the linear ele ment has the form (59), we say that the parameters are symmetric. The above results are given by the theorem are symmetric coordinates of a surface, any two coordi arbitrary functions of a and ft respectively are symmetric nates, and they are the only ones. When a and ft The as the general linear element of a surface can be written product of two factors, namely (61) d**: If denote integrating factors of the respective terms of the of symmetric coordinates a right-hand member of this equation, pair t and t 1 is given by the quadratures (62) When these values are substituted in (61), and the result (59), it is is com pared with seen that X = tt l The first of equations (62) can be replaced by , = da du ^FiH &gt; da&gt; t Eliminating t from these equations, we have E^-F^ du dv 63 &lt; &gt; -IT equation be =l : . ccc, Tu ~ If Z this multiplied by the result can be reduced to r*-o cu dv . dec ISOTHERMIC PARAMETERS From these equations it 93 follows that or, by (56), (65) It is A readily found that /3 2 tf = 0. also satisfies this condition. 40. Isothermic real, and isometric parameters. also, When the surface is and the coordinates imaginary. Hence for t r In this case a and the factors in (61) are conjugate the conjugate imaginary of t can be taken are conjugate imaginary also. In that this choice has been made, and write fi what follows we assume (66) a = &lt;+ty, = &lt;/&gt; iyfr. If these values be substituted in (59), we get (67) ds* = \(d&lt;t&gt; 2 At once we see that the curves +d^). = const, c/&gt; and ^r = const, form an orthogonal system. lines are V\d-*fr Moreover, the elements of arc of these increments (&lt;f) d&lt;f&gt; and respectively. Consequently when the and d^ are taken equal, the four points i/r), ^\d(f&gt; (&lt;, -f c?&lt;, i/r), (, i/r -f efo/r), -f (&lt; tity, = const, and small square. Hence the curves the surface into a network of small squares. (f&gt; ^ -f eityr) are the vertices of a ^|r = const, this divide On and account these curves are called isometric curves, and &lt;f&gt; ty isometric parameters. These lines are of importance in the theory of heat, and are termed isothermal or isothermic, which names are used in this connection as synonymous with isometric. the linear element can be put in the symmetric form, equations similar to (66) give at once a set of isometric parameters. And conversely, the knowledge of a set of isometric parameters leads at once to a set of Whenever symmetric parameters. But we have seen that when (60). one system of symmetric parameters given by equations of the form known, all the others are Hence we have the theorem is : &lt;, Given any pair of real isometric parameters every other pair &lt; -v/r for a surface ; x, ty 1 is given by equations of the form to where F and F Q are any functions conjugate imaginary one another. 94 LINEAR ELEMENT OF A SURFACE Consider, for instance, the case (68) * 1 +^ 1 = ^(0+i. From (69) the Cauchy-Riemann differential equations ?*i d&lt;l&gt; = *i, c^ ?& = _?*i, 3^ 8&lt;f&gt; &lt; it follows that &lt;/&gt; (f) l and the curves &lt;f&gt; 1 const., i T/T = const., ^ = const, are = const. Similar results 1 ^ l are functions of both different and T/T. Hence replaced from the system by in the is argument of the when right-hand member hold +i is of (68). Hence There face; when one system If the value (66) for a double infinity of isometric systems of lines upon a sur is known all the others can be found directly. a be substituted in the is first of equations (57), the resulting equation reducible to Since &lt;f&gt; and ^r are real, this equation is equivalent to (70) A^A.VT, (58) it is A 1 (^,f) = o. From E G, F= 0, when seen that these equations are the condition that and i/r are the parameters. Hence equations &lt;f&gt; (70) are the necessary and sufficient conditions that isometric parameters. and all i/r be Again, when a in (65) tions are real, we have (71) 2 is replaced by &lt;/&gt;+ i^r, and f the func A *=0, when we have a function (f&gt; Conversely, satisfying the first of these equations, the expression cu cv , on ov dv is , an exact differential. Call it d^r ; then jr^ du _ E G du ~~du F c)v H dv^_c^r H ISOTHERMIC ORTHOGONAL SYSTEMS If these equations 95 be solved for du dv we get /r _ ox dv du (&lt;o) H = dd&gt; dv "&gt; du du H ( ) = d&lt;f&gt; dv When we du\dvj and ^ satisfy (70), A 2 -/r=0. Moreover, these two functions in consequence of (72) and (73), and therefore they are isometric dv\duj (f&gt; express the condition = &gt; ( ) we find that parameters. Hence : A the necessary and sufficient condition that be the isometric &lt;f&gt; param c/&gt; eter of one family of an isometric system on a surface is that A 2 = isometric parameter of the other family is given by a quadrature. ; Incidentally we remark that if u and v are a pair of isometric (69). parameters, equations (72) and (73) reduce to 41. Isothermic orthogonal systems. If the linear element of a surface is given in the form (67) and the parameters are changed in accordance with the equations the linear element becomes where the accents indicate differentiation. However, this trans ; formation of parameters has not changed the coordinate lines the coefficients are now no longer equal, but in the relation &lt;&gt; i-f U and V denote functions of u and v respectively. this relation is satisfied the linear element where may Conversely, when be written and by the transformation of coordinates, (75) = 4&gt; C^/lfdu, ^= C 96 it is LINEAR ELEMENT OF A SURFACE brought to the form (67), whatever be : U and V\ and the coor dinate lines are unaltered. Hence necessary and sufficient condition that an orthogonal system of parametric lines on a surface form an isothermic system is that the coefficients A of the corresponding linear element satisfy a relation of the form (74). We and Either curves seek o&gt; now function (w, v) must the necessary and sufficient condition which a = const. satisfy in order that the curves o&gt; their o&gt;, orthogonal trajectories it, or a function of is form an isothermic system. the isothermic parameter of the then &lt;/&gt;; o&gt; = const. We A o&gt; denote this parameter by . &lt;/&gt;=/() Since &lt;/&gt; must be a solution 2 of equations (71), we have, on substitution, (G&gt;) (76) ./ + \a&gt; ./"(a&gt;) = 0, to &&gt;. where the primes indicate differentiation with respect equation is written in the form If this we see that the ratio of the two differential parameters o&gt;, is a func tion of /(a&gt;), co. Conversely, if this ratio is a function of obtained by two quadratures from the function (77) / necessary and const, is (*&gt;) = *-/&gt;, condition that will satisfy equations (71). Hence: a family of curves isothermic sys &&gt;. A a) sufficient = and their orthogonal trajectories form an tem that the ratio of A &&gt; &&gt; 2 and AjO) be a function of ; a function w then the orthogonal tra = const, can be found by quadrature ; for, the curves jectories of the differential equation of these trajectories is Suppose we have such (78 ^ If - "- - \~ dv du/ \~ dv Su equation (76) be written in the form I *f&gt; dv TT- du H * v r\ i , . wu -TT- t/i/ O &lt;7V I *^ f (&)) \ / I I 4/ r (ft)) \ / n = u, ISOTHEEMIC ORTHOGONAL SYSTEMS it is 97 seen that an integrating factor of equation (78) is f (a))/H, where f (co) is given by (77). Hence /() and the function obtained by the quadrature __#&&gt; &lt;f, ^cto s^fo -rJo&lt;* are a pair of isometric parameters. it follows that From these equations and (77) and consequently, by means given the form (80) ds* of (52), the linear element can be d + /sS ctyA 8 " = A^ (da* x The 2 linear element of the plane referred to rectangular axes is ds = dx 2 : 4- dy 2 . Consequently x and y are isothermic parameters, and we have the theorem The plane curves whose equations are obtained by equating to constants the real and imaginary parts of any function of x + iy or x - iy form an isothermal orthog onal system ; and every such system can be obtained in this way. For example, consider where c is 4- ty c = -2 x iy any constant. From this it follows that x2 4- yz x2 4- y 2 = const., = const, form an isothermal orthogonal system, Hence the circles and and ^ are isothermic parameters. The above system of circles is a particular case of the system considered in 34. We inquire whether the latter also form an isothermal system. If we put u 1 = x 4(w 2 i (2/ 2 - 2 ), we find that AIO&gt; = 4- x2 4 a 2 ), A^u = 2d) x2 Hence the ratio of AIW and A2W circles is isothermal. first From tan- 1 a function of w, and consequently the system of follows that the isothermic parameter of the (77) is it family is = 2a , 2a 1 \b and the parameter of the orthogonal family is tanh -1 w &gt; w 2a 2a =y x2 4 4- a2 y 98 LINEAR ELEMENT OF A SURFACE EXAMPLES that the meridians and parallels on a sphere form an isothermal orthog onal system, and determine the isothermic parameters. 1. 2. Show Show Show that a system of confocal ellipses and hyperbolas form an isothermal orthogonal system in the plane. 3. that the surface 2 x a is _ ~ I (a - u) 2 ) \ (a - & 2 (a^v) 2 - c2 (a y b _ ~ I (b* - u) (b 2 2 ) \ (62 - a2 ) (6 - v) c2 ) z c _ ~~ ! ( C2 2 _ U) ( C2 _ 2 ) \f(c -a2 )(c2 -6 an 4. ellipsoid, and that the parametric curves form an isothermal orthogonal system. the surface Find the curves which bisect the angles between the parametric curves on %_u+v y_u_v _ uv a~ = 2 b~"~2~ 2 and show that they form an isothermal orthogonal system. 5. Determine u cos v, y u sin v, z = (v) (v) so that on the right conoid x the parametric curves form an isothermal orthogonal system, and show that the curves which bisect the angles between the parametric curves form a system of &lt;f&gt; = the same kind. 6. Express the results of Ex. 4, page 82, in terms of the parameters and ^ defined by (66). 42. Conformal representation. ence of any kind is a one-to-one correspond established between the points of two sur When faces, either surface may be said to be represented on the other. Thus, if we roll out a cylindrical surface upon a plane and say that the points of the surface correspond to the respective points of the plane into which they are developed, we have a representa upon the plane. Furthermore, as there is no or folding of the surface in this development of it upon stretching the plane, lengths of lines and the magnitude of angles are unal tion of the surface a representation of every surface upon a plane, and, in general, two surfaces of this kind do not admit of such a representation upon one another. tered. It is evidently impossible to make such However, it is possible, as we shall see, to represent one surface upon another in such a way that the angles between correspond ing lines on the surfaces are equal. In this case we say that one surface has conformal representation on the other. In order to obtain the condition to be satisfied for a conformal representation of two surfaces S and S we imagine that they are referred to a corresponding system of real lines in terms of the r , CONFOKMAL KEPKESENTATION same parameters respective forms w, v, 99 and that corresponding points have the same curvilinear coordinates. We 2 write their linear elements in the ds 2 = Edu*+ 2 Fdudv + G dv to , ds 2 = du*+ 2 F dudv + G dv*. co and must be sponding points Since the angles between the coordinate it is lines at corre equal, necessary that F (81) F y/EG and Q r denote the angles which a curve on S and the corre sponding curve on S respectively make with the curves v = const. at points of the former curves, we have, from (23) and (25), If . sin n = , - H dv , . . sin ds a sm 6 = . (to - Q = H du = ) H=, dv s sin , , 1 (to H ai\ - 6[) = -= du V &lt;&? By r hypothesis a&gt; =a) ~ and 6[ Q, according as the angles have the same or opposite sense. Hence we have H H ds ~ H du according to the sense of the angles. From these equations we find which, in combination with (81), may be written where 2 t v in general. denotes the factor of proportionality, a function of u and From (83) it follows at once that ds (84) *=t * And and so when (82) follow. the proportion (83) is satisfied, the equations (81) Hence we have the theorem : necessary and sufficient condition that the representation of two surfaces referred to a corresponding system of lines be conformal is A 100 that the first LINEAR ELEMENT OF A SUKPACE fundamental coefficients of the two surfaces be propor tional, the factor of proportionality being a function of the param eters ; the representation is direct or inverse according as the relative positions of the positive half tangents to the parametric curves on the two surfaces are the same or symmetric. Later we shall find it From (84) means of obtaining conformal representations. follows that small arcs measured from correspond f ing points on S and S along corresponding curves are in the same ratio, the factor of proportionality being in general a function of the position of the point. Conversely, when the ratio is the same for all curves at a point, there is a relation such as (84), with t a function of u and v at most. And since it holds for all directions, we must have the proportion (83). On this account we may say that two surfaces are represented conformally upon one another when in the neighborhood of each pair of homologous points corre sponding small lengths are proportional. par equal to unity, corresponding small lengths are equal as well as angles. In this case the representation is said to be isometric, and the two surfaces are said to be applicable. The ticular the factor t is 43. Isometric representation. Applicable surfaces. When in significance of the latter term is that the portion of one surface in the neighborhood of every point can be so bent as to be made to coincide with the corresponding portion of the other surface with out stretching or duplication. It is evident that such an applica tion of one surface upon another necessitates a continuous array of surfaces applicable to both S and r This process of transformation is called deformation, and Sl is called a deform of S and vice versa. An example of this is afforded by the rolling of a cylinder on a plane. Although a conformal representation can be established between any two surfaces, it is not true, as we shall see later, that any two surfaces admit of an isometric representation upon one another. From time to time we shall meet with examples of applicable sur faces, and in a later chapter we shall discuss at length problems which arise concerning the applicability of surfaces. However, we consider here an example afforded by the tangent surface of a twisted curve. APPLICABLE SURFACES 101 on the We of the recall that if #, y, z are the coordinates of a point curve, expressed in terms of the arc, the equations of the surface are form f =x+ ^ v =y+y t, =z + z% and the linear element of the surface is d &lt;r* = /I + -\ ds + z 2 dsdt + dt\ where p denotes the radius of curvature of the curve. Since this expression does not involve the radius of torsion, it follows that the tangent surfaces to all curves which have the same intrinsic equation p =f(s) are applicable in such a way that points on the curves determined by the same value of s correspond. As there is a plane curve with this equation, the surface is appli cable to the plane in such a way that points of the surface corre spond to points of the plane on the convex side of the plane curve. to a curve are the characteristics of the osculating planes as the point of osculation moves along the curve, and con The tangents it sequently they are the axes of rotation of the osculating plane as moves enveloping the surface. Instead of rolling the plane over the tangent surface, we may roll the surface over the plane and bring all of its points into coincidence with the plane. It is in this sense that the surface it is is called a developable surface developable upon a plane, and for this reason (cf. 27). Later it will be shown is that every surface applicable to the plane a curve ( 64). the tangent surface of 44. Conformal representation of a surface upon itself. We return and remark that another consequence of equations (83) is that the minimal curves correspond upon S and S Conversely, when two surfaces are referred to a corresponding system of lines, if the minimal lines on the two surfaces correspond, equations (83) must hold. Hence to the consideration of conformal representation, r . : necessary and sufficient condition that the representation of two surfaces upon one another be conformal is that the minimal lines correspond. If the A minimal lines upon the two surfaces ds 2 are known and taken as parametric, the linear elements are of the (85) form ds 2 = X dadfr = \ da^dftv 102 LINEAR ELEMENT OF A SURFACE a conformal representation the equations l Hence is defined in the most general way by W = F(a), ft = *;(), or (87) ^ = F(ft), F and ft = *;(), which must be conjugate where F 1 are arbitrary functions real. imaginary when the surfaces are faces referred to their minimal lines, Instead of interpreting (85) as the linear elements of two sur we can look upon them as lines. the linear element of the same surface in terms of two sets of parameters referring to the minimal From this point of (86) (87) define the most general conformal of a surface upon itself. If we limit our considera representation tion to real surfaces and put, as before, view equations and a = + i^, i/r = &lt;-ty, a1 =&lt;^ 1 +i&gt; 1 , ft=0 -*^ 1 1, the functions fa and fa, ^ are &gt; pairs of isothermic parameters. Now (88) equations (86), (87) &lt;#&gt; may 1 be written 1+i = 7^^). : Consequently we have the theorem When and a pair of isothermic parameters fa ty of a surface are known the the surface is referred to the lines = &lt;j&gt; const., = const., ^r most general conformal representation of the surface upon obtained by making a point (fa \fr) correspond to the point into itself is (fa, i^), which it can be transformed in accordance with equation (88). As a corollary of this theorem, we have : When a pair of isothermic parameters is known for each of two surfaces, all the conformal representations of one surface upon the directly. ty other can be found Consider two pairs of isothermic parameters fa a surface S, (89) If and fa, ^ for and suppose their relation is &+*+! = F(t + i+). l two curves C and C are in correspondence in this representa tion, their parametric equations must be the same functional rela tion between the parameters, namely, *,) =0. CONFORMAL REPRESENTATION Denote by 9 and curves l 103 the the the angles which C and C^ ^ = const, and ^ = const, 1 respectively. If make with we write linear element of S in the two forms it follows from (23) that cos a = =- deb y = . , sin a = cos -we i c?i/r =, sin = From these expressions derive the following ^ = d(f&gt; so that in consequence of (89) we have (90) ..,-= :*^ 7^ is the where function conjugate to 7^, and the accents indicate with respect to the argument. If T and F x are differentiation another pair of corresponding curves, and their angles are denoted by 6 and V it follows from (90) that ,, OI&gt; For, the right-hand member of (90) is merely a function of the position of the point and is independent of directions. Hence in any conformal representation defined by an equation of the form (89) the angles between corresponding curves have the same sense. When, now, the correspondence satisfies the equation the equation analogous to (90) is Hence l -0 =0-0 l i consequently the corresponding angles are equal in the inverse sense. 104 LINEAB ELEMENT OF A SURFACE For the plane the be stated thus 45. Conformal representation of the plane. preceding theorem may : The most general real conformal representation of the plane upon to the point itself is obtained by making a point (x, y) correspond where x^iy^ is any function of x + iy or x iy. (x^ y^), We 0) recall the example of 41, namely Xl + iyi = ;rrfc is where c is a real constant. This equation equivalent to and also to C2X Hence the in the on const, and y const., in the xy-plane, are represented parallels x circles which pass through the origin and have their centers the respective axes. Conversely, these circles in the xy-plane correspond to = z^-plane by the parallels in the Xi^/i-plane. If we put o;2 + y* = r2 , x* + y* = rf, equations (ii) and (iii) may be written &lt;*&gt; ?-? f-S. Hence corresponding points are on the same line through the origin, and their On this account equations (iv) are distances from it are such that rr\ = c 2 2 2 2 2 said to define an inversion with respect to the circle x + y = c or, since TI = c /r, , o transformation by reciprocal radii vector-es. From 44 it follows that corresponding angles are equal in the inverse sense. For the case (v) xi + iy\ = x c2 + iy the equations analogous to (iv) are r = ?, n V r = -Vl. ri* line which is the (x, y) lies on the y$$ corresponding to c 2 /r. Evidently reflection in the x-axis of the line OP, and at the distance OPi Hence the point PI (xi, P = this transformation is the combination of an inversion and the transformation *i = *, y\ =- y(i) One finds that the transformations line is and (v) have the following properties : Every straight and conversely. Every circle transformed into a circle which passes through the origin ; is which does not pass through the origin transformed into a circle. CONFOKMAL REPRESENTATION 105 propose now the problem of finding the most general conformal transformation of the plane into itself, which changes circles not We passing through the origin into circles. #, fi, In solving it we refer the plane to symmetric parameters where a =x -f- iy, f$x = iy. The equation (91) &lt;z, of origin is of the any form circle which does not pass through the ca(S+ aa + 5/3 + ; d 0, when the circle is real a and where 5, c, d are constants must be conjugate imaginaries and c real. Equation (91) defines as a function of a. If b @ we differentiate the equation three times with respect to equations, (92) #, and eliminate the constants from the resulting 2 3/3" we find -2/3 /3 "=0, differentiation where it is the accent indicates with respect to a. Moreover, as equation (91) contains three independent constants, We (93) the general integral of (92). know that the most general conformal representation of the plane upon itself is given by ft a 1 = A(a), = (), or (94) ! = (), 13, = A (a). Our problem reduces, therefore, to the determination of functions A and B, such that the equation 3 ft 2 (95) - 2 ft ft" =0, to where the accent indicates differentiation with respect be transformed by (93) or (94) into (92). We consider first equations (93), which we write a v can Now ff^*!L*pta_ * 30 da da In like manner we find ft and stituted in (95) ft". we get, since A( 2 and ") When their values are sub B are different from zero, 4 3 2 ft" -2 ffff" +B (3 B" -2BB + A (3 l A -2 2 A[Al") ft = 0. 106 LINEAR ELEMENT OF A SURFACE it Since equation (95) must be directly transformable into (92), follows that (96) 3 " 2 -2 " /&gt;" = 0, 3 A^~ 2A[A = 0. As these equations are of the form (92), their general integrals are similar to (91). Hence the most general forms of (93) for our problem are &gt; Moreover, when, these values are substituted in an equation in a^ (S l of the form (91), the resulting equation in a and ft is of this form. Equation (91) may likewise be looked upon as defining a in terms of ft, so that a, as a function of ft, satisfies an equation of the form (92) similarly for a l as a function of ft r Hence if we had used (94), we should have been brought to results analogous to (97) and therefore the most general forms of (94) for our problem are ; ; (98) i=!4 T b /3+b^ s : ft- 54 **, + , Hence When a plane general conformal circles is defined in symmetric parameters a, ft, the most representation of the to circles plane upon itself, for which correspond or straight lines, is given by (97) or (98).* EXAMPLES 1. Deduce the equations which define the most general conformal representation cZs of a surface with the linear element 2. 2 = dv? + (a2 z z u^dv 2 upon itself. Show that the surfaces x x u cos v, u cos v, in y = u sin u, y u sin v, = = au, a cosh -* - , which a plane through the z-axis cuts the latter are applicable. Find the curve and deduce the equations of the conformal representation of these surfaces surface, on the plane. 3. When the representation is defined by (97), what are the coordinates of the center and radius of the circle in the &lt;n-plane which corresponds to the circle of center (c, d) and radius r in the or-plane ? * The transformations (97) and (98) play an important role in the theory of functions. For a more detailed study of them the reader is referred to the treatises of Picard, Darboux, and Forsyth. SURFACES OF REVOLUTION 4. 107 Show a ai distinct points, that in the conformal representation (97) there are, in general, two each of which corresponds to itself also that if 7 and 5 are the ; values of at these points, then d a K= 5 ai + ai + V(ai a4 ) 2 + 4 a2 a3 5 Find the condition that the origin be the only point which corresponds to itself, and show that if the quantities 01, ag, ^3, a are real, a circle in the a-plane through and touching the other the origin corresponds to a circle in the a^plane through . circle ; also that a circle touching the x-axis at 6. The equation 2 ai = (a b) a -f - - corresponds to itself. ? where a and 6 are constants, defines a conformal representation of the plane upon itself, such that circles about the origin and straight lines through the latter and hyperbolas in the ai-plane. in the a-plane correspond to confocal ellipses = logo: to lines parallel to the x- and 7. In the conformal representation i y-axes in the ai-plane there correspond lines through the origin and circles about it in the a-plane, and to any orthogonal system of straight lines in the ai-plane an orthogonal system of logarithmic spirals in the a-plane. is By definition a surface of revolution the surface generated by a plane curve when the plane of the curve is made to rotate about a line in the plane. The various 46. Surfaces of revolution. positions of the curve are called the meridians of the surface, and the circles described by each point of the curve in the revolution are called the parallels. and for o&gt;axis and ?/-axis We take the axis of rotation for the 2-axis, and to the z-axis, any two lines^perpendicular to one another, and meeting it in the same point. For any posi tion of the plane the equation of the curve may be written z Avhere r denotes the distance of a point of the curve from the 2-axis. = &lt;/&gt;(r), denote the angle which the plane, in any of its positions, makes with the #2-plane. Hence the equations of the surface are let v We (99) x = r cos,v, is 2 y = rsinv, 2 z=(f&gt;(r). The linear element (100) If ds = [1 + &lt; (r)] dr 2 + we put ) a 01 the linear element (102) is transformed into 108 where X LINEAR ELEMENT OF A SURFACE is a function of u, which shows that the meridians and form an isothermal system. As this parallels change of parameters does not change the parametric lines, the equations x = u, y = v, correspond define a conformal representation of the surface of revolution upon the plane in which the meridians and to the parallels straight lines x = const, and y = const, respectively. By definition a loxodromic curve on a surface of revolution is a curve which cuts the meridians under constant angle. Evidently it is represented on the plane by a straight line. Hence loxodromic curves on a surface of revolution (99) are given by C- Vl + $* where a, , + bv +c= 0, c are constants. Incidentally we have the theorem : When the linear element of a surface is reducible to the form where \ is a function of u or v alone, the surface is applicable to a surface of revolution. For, suppose that X is a function of u alone. Put r = Vx and solve this equation for u as a function of r. If the resulting expression be substituted in (101), we find, bya quadrature, the function for which equations define the surface of y &lt;f&gt;(r) (99) revolution with the given linear element. When, in particular, the surface of revolution r, is the unit sphere, with center at the origin, we have r = sin w, z= Vl r 2 = cos w, where u is the angle which the radius vector of the point makes with the positive z-axis. Now = log tan Hence the equations of correspondence are . | x u = log tan-, , y= v. MERCATOR REPRESENTATION 109 This representation is called a Mercator chart of the sphere upon the plane. It is used in making maps of the earth for mariners. path represented by a straight line on the chart cuts the meridians A at constant angle. Conformal representations of the sphere. We have found ( 35) that when the unit sphere, with center at the origin, is referred to minimal lines, its equations are 47. a (103) "- + /3 () is a/3-l where a and equation of j3 any are conjugate imaginary. real circle on the sphere Hence the parametric of the form ca{3+aa where a and b + b/3+d=Q, From this it follows that the are conjugate imaginary and c and d are real. problem of finding any conformal representation of the sphere upon the plane with circles of the former in correspondence with circles or straight lines of the latter, is the same problem analytically as the determination of this kind of representation of the plane upon 45, it itself. Hence, from the results of follows that All conformal representations of the sphere (103) upon a plane, with circles of the former corresponding to circles or straight lines of the latter, are defined by a.a &lt; + a,, 104) ***-;{?+ ^ = ^A . y bfi+b. * We is wish to consider in particular the case in which the sphere represented on the ^-plane in such a way that the great cir cle determined by this plane corresponds with (103) itself point for point. From we have that the equations of this circle are * The representation with the lower signs is the combination of the one with the upper which from (103) is seen to transform a figure sign and the transformation &i /3, /Si= bn the sphere into the figure symmetrical with respect to the zz-plane. = , 110 LINEAR ELEMENT OF A SURFACE these values are substituted in (104) a it i When is found that we must have i=4 , bl =t&gt; V az =&lt;* 3 =0 =0 r Z 3 A =i), so that the particular form of (104)* is equivalent to *1 =|(+/9), y, = we (-) find From these equations and (103) of the straight lines joining corresponding points are reducible to and that the equations on the sphere plane X For all Y 1-Z Hence values of a and ft these lines pass through the point (0, 0, 1). a point of the plane corresponding to a given point upon the point of intersection with the plane of the line the sphere is P with the pole (0, 0, 1). This form of representation is joining called the stereographic projection of the sphere upon the plane. It is evident that a line in the plane corresponds to a circle on this circle the sphere the given line. ; P is determined by the plane of the pole and will close this chapter with a few remarks about the conformal representation of the sphere upon itself. From the fore We such representation of the going results we know that every of similar form in a^ ft v where sphere (103) is given by equations the latter are given by (86) or (87), and that for conformal repre have the values sentations with circles in correspondence a l and ^ (97) or (98). We consider in particular the case a.a of the sphere are found expressions of the linear elements to be reducible to The 4 dad/3 2 ~ 4 da^ft, 4 dad ft * Here we have used the upper signs in (104). STEREOGRAPHIC PROJECTION 111 define an isometric representation of the Hence, equations (105) are preserved in the same sense sphere upon itself. Since angles this representation may be looked upon as determining by (105), a motion of configuration it. positions if there stationary points in the general motion, upon are any, correspond to values of a and /3, which are roots of the respective equations upon the sphere into new The If t l and 1/ 2 . and 2 are the roots of the former, those of the latter are l/^ Hence there are four points stationary in the motion; their curvilinear coordinates are 1 -L\ /. -*- X / j " \ I J. Ln i " ~ From (103) it is seen that the first two are at infinity, and the last two determine points on the sphere, so that the motion is a rotation about these points. If the z-axis is taken for the axis of must be oo and rotation, we have from (103) that the roots of (106) so that (105) becomes #2 hence 0, ; 3 = = If the rotation is real, these equations must be of the form = where o&gt; e is the angle of rotation. EXAMPLES 1. ds 2 = 2. Find the equations of the surface of revolution with the linear element dw2 + (a2 - w 2 )du 2 . Find the loxodromic curves on the surface X = MCOSU, y = usmv, when z = a cosh- 1 -, i i u and find the equations of the surface referred to an orthogonal system of these curves. Find the general equations of the conformal representation of the oblate spheroid upon the plane. 3. the evolute of 4. Show that for the surface generated by the revolution of to the catenary about the base of the latter the linear element is reducible 2 2 2= ds + u dv du" . 112 5. LINEAR ELEMENT OF A SURFACE A great circle on the unit sphere cuts Find the equation of its the meridian v = in latitude &lt;x under angle a. 6. stereographic projection. y = Determine the stereographic projection of the curve x acos 2 w, z asinw from the pole (0, a, 0). = asinwcosw, GENERAL EXAMPLES 1. When there is cross-ratio of four tangents to one surface at a point the corresponding tangents to the other. 2. a one-to-one point correspondence between two surfaces, the is equal to the cross-ratio of Given the paraboloid x = 2awcosu, y=2&Msinv, z = 2 w 2 (a cos2 u + 6sin 2 u), where a and b are constants. Determine the equation of the curves on the surface, such that the tangent planes along a curve make a constant angle with the xy-plane. Show that the generators of the developable 2, enveloped by these planes, make a constant angle with the z-axis, and express the coordinates of the edge of regression in terms of v. Find the orthogonal trajectories of the generators of the surface S in Ex. 2. that they are plane curves and that their projections on the xy-plane are involutes of the projection of the edge of regression. 3. Show 4. Let C be a curve on a cone of revolution which cuts the generators under constant angle, and Ci the locus of the centers of curvature of C. Show that C\ lies upon a cone whose elements it cuts under constant angle. 5. When the polar developable of a curve degenerates into a point. is developed upon a plane, the curve is 6. When the rectifying developable of a curve line. developed upon a plane, the curve becomes a straight 7. Determine &lt;f&gt;(o) so that the right conoid, x = ucosv, y=usinv, z = (f&gt;(v), shall be applicable to a surface of revolution. Determine the equations of a conformal representation of the plane upon which the parallels to the axes in the ai-plane correspond to lines through a point (a, b) and circles concentric about it in the a-plane. 8. itself for 9. The equation a\ = c sin a, where sentation of the plane upon itself c is a constant, defines a conformal repre such that the lines parallel to the axes in the a-plane correspond to confocal ellipses and hyperbolas in the ai-plane. 10. In the conformal representation of the plane upon itself, given by ai = a2 , to lines parallel to the axes in the ori-plane there correspond equilateral hyperbolas in the a-plane, and to the pencil of rays through a point in the ori-plane and the cir about it there corresponds a system of equilateral hyperbolas through the corresponding point in the or-plane and a family of confocal Cassini ovals. cles concentric 11. When curves, the 12. sum the sides of a triangle upon a surface of revolution are loxodromic of the three angles is equal to two right angles. of a sphere The only conformal perspective representation upon a plane is given by (104). GENERAL EXAMPLES 13. 113 Show interchange of that equations (105) and the equations obtained from (105) by the cc. and /3 define the most general isometric representation of the sphere upon 14. itself. Let each of two surfaces S, S\ be defined in terms of parameters w, u, and points on each with the same values of the parameters correspond. If H\, for S, corresponding elements where the latter is the function for Si analogous to let H H of area are equal and the representation is said to be equivalent.* If the parameters of S are changed in accordance with the equations the condition that the equations u resentation of S and Si is v a), H ^ HI and &lt;f&gt; u (w, v), =$ (u, = M, v =v define an equivalent rep H \[&lt;) du dv 15. cv du HI (0, Under what conditions do x aix the equations + azy + a3 , y = b& + upon b2 y + 63 define an equivalent representation of the plane 16. itself ? Show that the equations determine an equivalent representation of the surface of revolution (99) upon the plane. 17. Given a sphere and circumscribed circular cylinder. If the points at which a perpendicular to the axis of the latter meets the two surfaces correspond, the representation is equivalent. 18. Find an equivalent representation of the sphere upon the plane such that the parallel circles correspond to lines parallel to the y-axis and the meridians to a, 0). ellipses for which the extremities of one of the principal axes are (a, 0), ( * German writers call " it flachentreu." CHAPTER IV GEOMETRY OF A SURFACE 48. IN THE NEIGHBORHOOD OF A POINT we study of it, Fundamental coefficients of the second order. In this chapter the form of a surface in the neighborhood of a point and the character of the curves which lie upon the surface M and pass through the point. all We recall that the tangents at the tangent M to these curves lie in a plane, plane to the surface at the point. The equation may be written (1) of the tangent plane at M(x, y, 2), namely (II, 11), (f- where we have put dz _ _ dx du dx dv dx dti du du dz du dy du dy dv H We for dy^ ~dv H dz H dx dv dv do which the functions X, it define the positive direction of the normal ( 25) to be that 7 I Z are the direction-cosines. From this , const, and follows that the tangents to the curves v = const, at a point and the normal at the point have the same u mutual orientation as the #-, ?/-, and 2-axes. definition From (3) (2) follow the identities F ^ 7T dv fact that the - fl u which express the normal is perpendicular to the tan gents to the coordinate curves. In consequence of these identities the expression for the distance p from a point du, v + dv) (u is of the second order in du and dv. to the tangent plane at M + M It (4) may be written p= ^X dx = 1 (D du +2D dudv + 2 &" dv 2 ) + e, lit COEFFICIENTS OF THE SECOND OKDEK where e 115 denotes the aggregate of terms of the third and higher are defined by orders in du arid dv, and the functions D\ D" Z&gt;, (5) dudv equations (3) If be differentiated with respect to u and v respec tively, we get dv du (6) 1 = dX 0, dx _ ~ dudv dv dv dv And so equations (5) may be written (7) _V Y ~^^ ,,_y The quadratic dudv~ Ztdu d dv du ** -9* ^__y^^ ^ dv dv differential &lt;1&gt; form (8) is = D du* + 2 D dudv 2 -f D"dv called the second fundamental form of the surface, and the func tions D, D , D" the fundamental coefficients of the second order. We leave first it to the reader to show that these coefficients, like those of the order, are invariant for will we shall now be derived. From the equations Later any displacement of the surface in space. have occasion to use two sets of formulas which of definition, (9) ^ \du] get, ^ du dv ~ toy I cv) : we by differentiation and simple reduction, the following a^^_ia^ dx (10) y^^ = ?^_l?^, v dx &x _l dE dG du dv dudv ~2 2 dv 116 Again, GEOMETRY OF A SURFACE ABOUT A POINT if the expressions (9) be substituted in the left-hand mem bers of the following equations, the reduced results may be written by means of (2) in the form indicated : " dv du \ du du, dv du \ dv dv, x, y, z ; Similar identities can be found by permuting the letters X, F, Z. From the fundamental relation we obtain, by differentiation with respect to u and v respectively, the identities These equations and linear in * (7) constitute a system of three equations du ij. du O -y du and a system linear in find, dv , , dv dv Solving -y for and for du dv we by means of (11), dX ~du~ " FD -GDdx H* du dx FD-ED H* FZ&gt; dx ^ dv ^dX _FD"-GD dv 7} -RD"dx H* du H* dv V The and expressions for .. , du ,, 7 -/} are obtained dv by replacing x by y z respectively. first By means of these equations we shall prove that a real surface whose second fundamental coefficients are in proportion, thus (14) and D=V= V" =\ We assume D" where X denotes the factor of proportionality, is a sphere or a plane. that the minimal lines are parametric. In consequence we have E=G=D= so that equations (13) = 0, become dX -\ du du dv \ dv RADIUS OF NORMAL CURVATURE The function X must satisfy the condition 117 dv \ du/ 0. du \ dv which reduces to ---du du dv = dv Moreover, we have two other equations of z respectively. condition, obtained from the above the proportion by replacing x by y and . Since to.ay to = to dv .0y dv to . du du du is dv : == that is, X is a condu dv stant. When X is zero the functions X, F, Z given by (15) are constant, and consequently the surface is a plane. When X is any other constant, we get, not possible for a real surface, we must have by integration from (15), X where \x + a, Y= \y 4- 6, Z = Xz + we c, obtain (\x -f a) 2 4- (\y 4- 6) 2 2 1. Since this is the general equation of a sphere, it follows that the 4- (Xz 4- c) above condition is necessary as well as sufficient. a, 6, c are constants. From these equations v/ 49. Radius of normal curvature. Consider on a surface S any a point M. The direction of its tangent, MT, Let denote the angle which is determined by a value of dv/du. the positive direction of the normal to the surface makes with the curve C through o&gt; positive direction of the principal normal to C at Jf, angles being measured toward the positive binormal. If we use the notation of the first chapter, and take the arc of C for its parameter, we have In terms of the forms ,, . and as as the derivatives in the parenthesis have fo = Zfa/duV ~~ aV du \ds) is 2 2 du dv ds ds equivalent to dv \ds 2 so that the above equation cos (16) w D du + 2 D dudv + 2 D"dv Edu*+ ZFdudv 4- Gdv* As the right-hand member of this equation depends only upon the curvilinear coordinates of the point and the direction of MT, it is the same for all curves with this tangent at M. Since p is cannot be greater than a right angle for one positive, the angle o&gt; curve tangent to MT, if it is less than a right angle for any other 118 GEOMETRY OF A SURFACE ABOUT A POINT MT curve tangent to MT; and vice versa. We consider in particular the curve in which the surface is cut by the plane determined by and the normal tangent to to the surface at M. We call it the normal the the MT, and let p n denote its radius. Since member of equation (16) is the same for C and right-hand normal section tangent to it, we have section (17) P Pn is less or +1 1, according as greater than a right angle; for p and p n are positive. Equation (17) gives the follow where e is or w ing theorem of Meusnier: The center of curvature of any curve upon a surface jection is the pro osculating plane of the center of curvature of the normal section tangent to the curve at the point. upon its In order to avoid the ambiguous sign in (17), we introduce a new function when R which is equal to pn when &lt; o&gt; &lt; TT/%, and to pn 7r/Z&lt;a&gt;&lt;7r, and call it the radius of normal curvature of the surface for the given direction MT. As thus defined, E is given by R Now we may state Edu + 2 Fdudv + Gdv* 2 Meusnier s theorem as follows : If a segment, equal to twice the radius of normal curvature for a given direction at a point on a surface, be laid off from the point on the normal to the surface, and a sphere be described with the segment for diameter, the circle in ivliich the sphere is met by the osculating a curve with the given direction at the point is the circle of plane of curvature of the curve. 50. Principal radii of normal curvature. If we put t = &gt; equa tion (18) becomes I D+2D t+D"t When of t, the proportion (14) is satisfied, R is the same for all values 1/X for the sphere. being oc for the plane, and the constant For any other surface R varies continuously with t. And so we PRINCIPAL RADII OF CURVATURE seek the values of 119 To to t this t for which 11 is a maximum or minimum. end we differentiate the above expression with respect and pnt the result equal to zero. This gives 2 (20) (J} +D"t)(E+2Ft+Gt l )-(F+Gt)(D + 2D t + 2 D"t ) = Q, or 2 (21) (FD"-GD )t +(FD"-GD)t+(ED -FD) = (). Without any loss of generality curves are such that (22) (ED"- E = 0, so that r we can assume that the parametric we have the identity H J 2 GDf- 4 2 7T" (FD" D G) (ED -FD) FD)\. \_ =4 When is E T 2F (FD -FDf+\ ED"GD--(ED E is real, the surface and the parameters is member of this equation positive. also, the right-hand Since the left-hand member the discriminant of equation (21), the latter has two real and distinct roots.* When the test (III, 34) is applied to equation (21), it is found that the two directions at a point determined by the roots of (21) are perpendicular. Hence: At which the radius of it is every ordinary point of a surface there is a direction for which normal curvature is a maximum and a direction for a minimum, and they are at right angles to one another. These limiting values of R are called the principal radii of normal curvature at the point. They are equal to each other for the plane and the sphere, and these are the only real surfaces with this property. From (20) and (19) we have D +D"t_D F+Gt : +D _ 1 E + Ft~~R t Hence the following relations hold between the principal radii and the corresponding values of t E + Ft-R(D + D = Q, = 0. \F+Gt-R(D + f t) D"t) * In order that the two roots he equal, the discriminant must vanish. This is impos sible for real surfaces other than spheres and planes, as seen from (22). For an imaginary surface of this kind referred to its lines of length zero, we have from (21) that or D" D is zero, since F ^ 0. The vanishing condition that the numerator and of the discriminant is also the necessary and sufficient denominator in (19) have a common factor. 120 GEOMETRY OF A SURFACE ABOUT A POINT t is When (24) eliminated from these equations, we get the equation 2 ) (DD" -D A 1 2 )2 ) (ED" + GD-2 FD R + (EG - F = 0, ) whose roots are the principal and /3 2 we have , radii. If these roots be denoted by p l !_ ^^ (25) DD"-D 2 PiP* H and a it Although equations plane, and for no other (14) hold at all points of a sphere surface, may happen that for certain par ticular points of a surface they are satisfied. At such points R, as given by (19), is the same for all directions, and the equa tion (21) vanishes identically. When points of this kind exist they are called umbilical points of the surface. EXAMPLES 1. When the equation of the surface is z =f(x, ?/), show that x,Y,z = D, ^J^, 2z D dz &gt; , D" = where 2. p Show = dz &gt; a r =8 d 2z s dx dy dx* dxdy that the normals to the right conoid along a generator form a hyperbolic paraboloid. 3. Show that the principal radii of normal curvature of a right conoid at a point differ in sign. 4. tion at a point in the direction of the loxodromic curve through Find the expression for the radius of normal curvature of a surface of revolu makes the it, which the meridians. angle 5. a with Show sin i&gt;, y =u that the meridians and parallels on a surf ace of revolution, x = u cosu, in which the radius of normal curvature is z= (w), are the directions ; maximum and minimum Pl that the principal radii are given by (1 + /2 ) P2 M and that the segment of the normal between the point of the surface and the intersection of the normal with the z-axis. /&gt; 2 is 6. Show that AIX =1- X 2 eters are formed with respect and AI (x, y) XY, where the differential to the linear element of the surface. =- param LINES OF CURVATUBE 51. Lines of curvature. 121 Equations of Rodrigues. We have f seen that the curves defined (26) by equation (21), written (ED" (ED - FD) du + 2 - GD) dudv + (FD" - GD ) dv z =0, form an orthogonal system. As defined, the two curves of the sys tem through a point on the surface determine the directions at the point for which the radii of normal curvature have their and minimum values. These curves are called maximum the lines of curvature, and their tangents at a point the principal directions for the point. They possess another geometric property which we shall now find. The normals to face form a ruled surface. a surface along a curve In order that the sur be developable, the normals must be tangent to a curve ( 27), as in fig. 12. If the coordinates of a point on the normal l at a point be denoted by x r y^ z^ we have M M FIG. 12 where r denotes the length MMr If M^ be a point of the edge of regression, we must have dx+rdX+Xdr _dy-{-r dY+ Ydr _dz + r dZ + Zdr X Z Y tive Multiplying the numerators and the denominators of the respec members by X, F, Z, and combining, we find that the common ratio is dr. Hence the above equations reduce to or, when the parametric coordinates are used, ( 8x du du , dx H dv , dv +r dX I du , , dX dv H dv , \du (27) dv fa du dv dv du dv 122 GEOMETRY OF A SURFACE ABOUT A POINT be multiplied by dz dv &gt; If these equations added, and by dx dv ^u ^u ^u respectively and o- ^ dv respectively and added, we get Fdu + Gdv But The normals to r(D du + D" dv) = 0. these equations are the same as (23). Hence: a surface along a curve of it form a ruled surface which is a developable only when the curve is a line of curvature ; in this case the points of the edge of regression are the centers of normal curvature of the surface in the direction of the curve. The (28) coordinates of the principal centers of curvature are the parametric curves are the lines of curvature, equa tion (26) is necessarily of the form (29) When X dudv = 0, and consequently we must have Since ED"GD (30) = ED FD = 0, =Q. FD" GD = 0. 0, these equations are equivalent to ^=0, D Conversely, when these conditions are satisfied equation reduces to the form (29). Hence: (26) A curvature be necessary and sufficient condition that the lines of is parametric that F and D be zero. Let the the principal tions of the lines of curvature v lines of curvature be parametric, and let p^ and p 2 denote radii of normal curvature of the surface in the direc = const, and u = const, respectively. From (31) (19) we find ^ (13) = f By du ~= ^T dz and equations du (32) become dY ri _ dZ du du du These equations are called the equations of Rodrigues. TOTAL AND MEAN CURVATURE 52. Total 123 in and mean curvature. Of fundamental importance the discussion of the nature of a surface in the neighborhood of a point are the product and the sum of the principal curvatures at the point. They are called the total curvature * of the surface at the point and the mean curvature respectively. If they be denoted by K and K m1 we have, from (25), K (33) 1 Mb" ^ JL-i+iPi P* ^ two principal radii When K is positive at a point J/, the have the same sign, and consequently the two centers of principal curva ture lie on the same side of the tangent plane. As all the centers of curvature of other normal sections lie between these two, the lies entirely on portion of the surface in the neighborhood of one side of the tangent plane. This can be seen also in another M way. Since H 2 is positive, we must have DD f D 2 &gt; 0. Hence the distance from a near-by point to the tangent plane at Jf, since it is proportional to the fundamental form ( 48), does not &lt;l&gt; change sign as dv/du is varied. negative at M, the principal radii differ in sign, and consequently part of the surface lies on one side of the tangent plane and part on the other. In particular there are two directions, is When K given by for j&gt;du*+2 D dudv + is D" dv 2 = 0, which the normal curvature zero. In these directions the dis tances of the near-by points of the surface from the tangent plane, as given by (4), are quantities of the third order at least. Hence these lines are the tangents at meets the surface. plane at M M to the curve in which the tangent zero, At infinite. the points for which At these points K &lt; is one of the principal radii -f is has the form (^/J) du vanishes in the direction dv/du passes does not change sign. through the value given by this equation, Hence the surface lies on one side of the tangent plane and is tan gent to it along the above direction. &lt;1&gt; Vl)du + ^/D"dv = ^W dvf and 0. But as * The total curvature is sometimes called the Gaussian curvature, after the celebrated geometer who suggested it as a suitable measure of the curvature at a point. Cf Gauss, p. 15. . 124 GEOMETEY OF A SUEFACE ABOUT A POINT may anchor ring, or tore, is a surface with points of all three kinds. Such a sur be generated by the rotation of a circle of radius a about an axis in the plane of the circle and at a distance b a) from the center of the circle. The points at the distance b from the axis lie in two circles, and the tangent plane to the tore at a point of either of the circles is tangent all along the circle. Hence the surface has zero curvature at all points of these circles. At every point whose dis tance from the axis is greater than b the surface lies on one side of the tangent face (&gt; An plane, whereas, when the distance is less than 6, the tangent plane cuts the surface. There are surfaces for which is positive at every point, as, for example, the ellipsoid and the elliptic paraboloid. Moreover, for the hyperboloid of one sheet and the hyperbolic paraboloid the curvature is negative at every point. Surfaces of the former type are called surfaces of positive curvature, of the latter type surfaces K of negative curvature. is zero at 64) we shall prove that when a surface the latter is developable, and conversely. Later ( K all points of Equation of Euler. Dupin indicatrix. When the lines of curvature are parametric, equation (18) can be written, in con 53. sequence of (34) (III, 23) and (31), in the form 2 cos # I sin 2 6&gt; Pi Pi where the angle between the directions whose radii of normal curvature are and p r Equation (34) is called the equation of Euler. is R When is the total curvature K at a point and p 2 for the point have the same sign, and R has this sign for all positive, p l directions. If the of curvature at the point FlG 13 - tangents to the lines be taken for M coordinate axes, with respect to which % and T? are coordinates, and segments of is length VTIFi be laid off from M in the two directions correspond the ellipse 1. ing to R, the locus of the (fig. end points of these segments 13) whose equation is N + ra = Dupin & This ellipse is called the indicatrix for the point. particular, p l Dupin and p 2 are equal, the indicatrix is a circle. indicatrix at an umbilical point is a circle ( 50). is When, in Hence the For this reason such a point sometimes called a circular point. DUPIN INDICATRIX When 125 negative p l and p 2 differ in sign, and consequently certain values of R are positive and the others are negative. In the directions for which R is positive we lay off the segments is K V.Z2, and in the other directions V R. The locus of the end points of these segments con sists of the conjugate hyperbolas (fig. 14) whose equations are T]_ Pl Pt We (35) or, remark that R is infinite for the directions given by tah 2 0=-^, directions of the is FIG. 14 in other words, in the asymptotes to the hyperbolas. Finally, The above when K = locus the Dupin the equation of the indicatrix e-2_ i indicatrix for the point. is of one of the forms that is, a pair of parallel straight lines. In view of the foregoing results, a point of a surface is called elliptic, hyperbolic, or parabolic, according as the total curvature at the point is positive, negative, or zero. 1 the expression for the distance p upon the of from a near-by point tangent plane to a surface at a point the surface is given by In consequence of (4) M P Edu Pl 2 + Gdv2 = n 2 , to within terms of higher order. But ^/Edu and ^/Gdv are the distances, to within terms of higher order, of in the directions of the lines of curva planes to the surface at P from the normal M ture. Hence the plane parallel to the tangent plane and at a dis tance p from it cuts the surface in the curve Evidently this is a conic similar to the point, Dupin indicatrix at an elliptic or parabolic and to a part of the indicatrix at a hyperbolic point. 126 GEOMETRY OF A SURFACE ABOUT A POINT EXAMPLES 1. Show that the meridians and parallels of a surface of revolution are its lines of curvature, and determine the character of the developable surfaces formed by the normals to the surface along these lines. 2. Show that the parametric lines on the surface X are straight lines. 3. a = -(tt-M), , y = b -( U -v), , z = uv -, Find the is lines of curvature. When a surface denned by z = /(x, ?/), the expressions for the curvatures are and the equation of the [(1 4. lines of curvature [(i is + p2) s - pqr] (to* + 2 i + p-2) t-(l + g2) r -j dxdy + [ pqt _ (1 + ?2) s] dy z = . The principal radii of the surface y cos o 2 _i_ x sin - = at a point (x, y, z) are n2 equal to 5. Find the lines of curvature. Derive the equations of the tore, defined in 52, and prove therefrom the is results stated. 6. 7. The sum of the normal curvatures in two orthogonal directions constant. The Euler equation can be written E= Pi 2plp * + P2 - (PI - pa) cos 2 6 54. Conjugate directions at a point. Conjugate systems. Two are said to have conjugate curves on a surface through a point coincide with conjugate diam directions when their tangents at M M indicatrix for the point. These tangents are also parallel to conjugate diameters of the conicr in which the sur and very face is cut by a plane parallel to the tangent plane to eters of the Dupin M the point in denote a point of this conic and near it. Let which its plane a cuts the normal at M. The tangent plane to to the meets the plane a in the tangent line at the surface at P N P P Moreover, this tangent line is parallel to the diameter conju approaches 3/this tangent line approaches gate to NP. Hence as the diameter of the indicatrix, which is conjugate to the conic. P Dupin diameter in the direction MP. Hence we have (cf. 27) : The of characteristic of the tangent plane to a surface, as the point contact moves along a curve, is the tangent to the surface in the direction conjugate to the curve. CONJUGATE DIRECTIONS By means of this 127 theorem we derive the analytical condition for is conjugate directions. If the equation of the tangent plane f, 77, f being current coordinates, the characteristic is denned by this equation, and where moves. s is If &c, 8y, Bz the arc of the curve along which the point of contact denote increments of #, ?/, z in the direction conjugate to the curve, we have, from the above equations, If Bu and (36) increments of u and v in the conjugate direction be denoted by 8v, this equation may be written D duBu + D (du8v + dvSu) + D"dvv = 0. f The (37) directions conjugate to any curve of the family &lt;(%, v) = const. are given ( by cv 38) du dv first du order and first As it this is a differential equation of the degree, defines a one-parameter family of curves. These curves and the curves const, are said to form a conjugate system. Moreover, = &lt;/&gt; any two families of curves are said to form a conjugate system when the tangents to a curve of each family at their point of inter section have conjugate directions. From (36) it follows that the curves conjugate to the curves v = const, are defined by Sv = 0. Consequently, in order Su + D D that they be the curves u = const., we As the converse also is true, we have must have : D 1 equal to zero. A necessary and sufficient condition form a conjugate system is that D be that the parametric curves zero. 128 GEO vlETRY OF A SURFACE ABOUT A POINT have seen We (51) that the lines of curvature are characterized by the property that, when they are parametric, the coefficients are zero. Hence and F D : The lines of curvature form a conjugate system and the only orthogonal conjugate system. If the lines of curvature are parametric, and the angles which to the curve a pair of conjugate directions v = const, are denoted by make with the tangent and 6 we have , , tan a = [G dv ^J MjE du &lt; ar tan 6 = x N^ Su [GSv - , so that equation (36) (39) may be put in the form tan0tan0 = -?H, is which 55. the well-known equation of conjugate directions of a conic. is equal to 0, Asymptotic lines. Characteristic lines. When reduces to (35). Hence the asymptotic directions are equation (39) = dv/du, we obtain self-conjugate. If in equation (36) we put Sv/Su (40) D du + 2 D dudv + 2 D" dv 2 = 0, at each which determines, consequently, the asymptotic directions point of the surface. This equation defines a double family of curves upon the surface, two of which pass through each point and admit as tangents the asymptotic directions at the point. They are called the asymptotic lines of the surface. The asymptotic lines are imaginary on surfaces of positive curva ture, real on surfaces of negative curvature, and consist of a single real family on a surface of zero curvature. Recalling the results of 52, we say that the tangent plane to a surface at a point cuts the surface in asymptotic lines in the neighborhood of the point. As an immediate consequence, we have that the generators of a ruled surface form one family of asymptotic lines. Since an asymptotic line is self-conjugate, the characteristics of the tangent plane as the point of contact totic line are the tangents to the latter. moves along an asymp Hence the osculating plane of an asymptotic line at a point is the tangent plane to the ASYMPTOTIC LINES 129 surface at the point, and consequently the asymptotic line is the edge of regression of the developable circumscribing the surface along the asymptotic line. This follows also from equation From (40) we have the theorem : (16). necessary and sufficient condition that the asymptotic lines upon a surface be parametric is that D=D"=Q. If these equations hold, and, furthermore, the A parametric curves is are orthogonal, it is seen from (33) that the : mean curvature zero, and conversely. Hence A necessary and sufficient condition an orthogonal system is that the asymptotic lines form that the mean curvature of is the surface be zero. A surface whose mean curvature a minimal surface. sists of At zero at every point is called each of its points the Dupin indicatrix con two conjugate equilateral hyperbolas. means of (39) we find that the angle between conjugate By directions is given by P-/&gt;1 consider only real lines, this angle can be zero only for sur faces of negative curvature, in which case the directions are asymp If totic. we It is natural, therefore, to seek the conjugate directions upon a surface of positive curvature for which the included angle is a minimum. To this end we differentiate the right-hand member of the above equation with respect to 6 and equate the result to zero. The result is reducible to (41) tan 6 (39) = : Then from we have From these equations it follows that # = 0, and Conversely, when = 6 equation (39) becomes (41). Hence: surface of positive curvature there is a unique conjugate system for which the angle between the directions at any point is the Upon a 130 GEOMETRY OF A SURFACE ABOUT A POINT minimum angle between conjugate directions at the point ; it is the only conjugate system whose directions are symmetric with respect to the directions of the lines of curvature. These lines are called the characteristic lines. It is of interest to note that equations (35) and (41) are similar, and that the real upon a surface of negative curvature are symmetric with respect to the directions of the lines of curvature. As just seen, if 6 is the angle which one characteristic line makes asymptotic directions with the line of curvature teristic line v = const, 6. at a point, the other charac makes the angle ture for these directions are Hence the radii of normal curva equal, and consequently a necessary and sufficient condition that the characteristic curves of a surface is be parametric (42) f 7T = iy= By reasoning similar 56. Corresponding to that of systems on two surfaces. 34 we establish the theorem : and sufficient condition that the curves defined by 2 fidu +2S dudv + T dv = form a conjugate system upon a surnecessary 2 A face *s RD" + TD : 2 SD = 0. From this we have at once D If the second quadratic forms of two surfaces S and S are 2 2 dv and D l du + 2 D[ dudv + D[ dv\ and if a du 2 + 2 D dudv + l f D" to the point on the other point on one surface is said to correspond u and tf, the equation with the same values of (43) du2 dudv dv 2 D? D[ DI is D" D D conjugate for both surfaces. real which defines a system of curves By the methods curvature of of 50 we prove that these curves are when If the either or both of the surfaces S, S l S is negative and it is is of positive curvature. referred to its asymptotic lines, the above equation reduces to GEODESIC CURVATUKE Hence the system is real when is, when the curvature of S is l 131 D l and D[ have the same sign, that positive. Another consequence of the above theorem is : necessary and sufficient condition that asymptotic lines on one of two surfaces $, Sl correspond to a conjugate system on the other is (44) A DDJ + D"Di - 2 D D[ = 0. EXAMPLES 1. Find the curves on the general surface of revolution which are conjugate to the loxodromic curves which cut the meridians under the angle a. 2. Find the curves on the general right conoid, Ex. gate to the orthogonal trajectories of the generators. 3. 1, p. 56, which are conju When the equations of a surface are of the form x=U where U\ and l, y=Vi, z=U +V 2 2 , U 2 are functions of u alone, and V\ and F 2 of v alone, the para metric curves are plane and form a conjugate system. 4. Prove that the sum of normal radii at a point in conjugate directions is constant. 5. When a surface of revolution is referred to its meridians and parallels, the asymptotic lines can be found by quadratures. 6. Find the asymptotic x lines on the surface y lines = a(l + cos it) cot v, = a(l + cosw), z z= acosw -and determine 7. Determine the asymptotic upon the surface 3 ?/ y sin a: and their orthog onal trajectories. 8. Show that the x-axis belongs to one of the latter families. their projections 9. Find the asymptotic lines on the surface 2 on the xy-plane. - 2 xyz + z 2 = 0, Prove that the product of the normal radii in conjugate directions is a maxi mum for characteristic lines and a minimum for lines of curvature. 10. When the parametric lines are any whatever, the equation of character istic lines is [D(GD - ED") - 2D (FD - ED )] tin* + 2 [D (GD + ED") - D"(GD - ED")] dv* = 0. + [2D (GD FD") 2 FDD"] dudv 57. Geodesic curvature. Geodesies. Consider a curve C upon a of C. surface and the tangent plane to the surface at a point this tangent plane the portion of the Project orthogonally upon curve in the neighborhood of M, and let C denote this projection. 1 M The curve C is 1 is a curve upon a normal section of the projecting cylinder, and C at M. Hence the theorem the latter, tangent to C" 132 of GEOMETRY OF A SURFACE ABOUT A POINT Meusnier can be applied to these two curves. If l/p g denotes the curvature of C and -^ the angle between the principal normal to C and the positive direction of the normal to the cylinder at Jf, we have (45) i = c ^. P, P it In order to connect this result with others, is necessary to define the positive direction of the normal to the cylinder. This normal lies in the tangent plane to the surface. make the convention that the positive directions of the tangent to the curve, We the normal to the cylinder, and the normal to the surface shall have the same mutual orientations as the positive or-, y-, and 2-axes. From this choice of direction it follows that if, as usual, the direc tion-cosines of the tangent to the curve be dx/ds, dy/ds, dz/ds, then those of the normal to the cylinder are (46 ) Y*-Z*/, ds ds of Z~-*~&gt; ds f ds *f-4ds ds is called the geodesic curvature of (7, and p g the radius of geodesic curvature. And the center of curvature of C is called the center of geodesic curvature of C. The curvature C definition the geodesic curvature is positive or nega tive according as the osculating plane of C lies on one side or the its From other of the normal plane to the surface through the tangent to C. From (45) it follows that the center of first curvature of C is the projection curvature. upon osculating plane of the center of geodesic Moreover, the former is also the projection of the its center of curvature of the normal section tangent to C (49). of (7, normal to the line Hence the plane through a point M curvature at M, is the joining the centers of normal and geodesic intersection with the osculating plane of C for this point, and its is the center of first curvature. join the angle which the positive direction of the normal to the surface makes with the positive By definition ( 49) w denotes direction of the principal normal to (7, angles being measured toward the binomial. Hence equation (45) can be written (47) 1 - = sin w GEODESIC CUEVATUKE 133 These various quantities are represented in fig. 15, for which the tangent to the curve is normal to the plane of the paper, and The directed lines MP, MB, MK, is directed toward the reader. MN represent respectively the positive directions of the principal normal and binomial of the curve and the normals to the projecting cylinder and to the surface. curve whose principal normal at every point coincides with the normal to the surface A upon which it lies, is called a geodesic. From that a geodesic a curve whose may (45) it follows also be defined as geodesic curvature is zero at every point. meridians of a surface of revolution are in geodesies, as follows from the results For example, the 46. A twisted curve is a geodesic on lies 011 its when a straight rectifying developable, and it line a surface, shall to make a geodesic for the surface. Later we an extensive study of geodesies, but now we desire is an expression for the geodesic curvature in terms of the fundamental quantities of the surface and the equation of find the curve. 58. Fundamental formulas. The direction-cosines ( of the prin cipal normal are 8) d 2x H f\ _ ~ ds 2 Q ^ d 2y ds 2 Q r d 2z ds 2 . Consequently, by means of the form 1 ( (46), equation (45) may be put in 48 ) 7 *\ g = ^\ / dz dy\ d x 2 ds~ v, ~ds)~ds Expressed as functions of u and the form the quantities -j- -ji are ^ dx ds _ fa du dx dv dv ds d*x 2 du ds ~ds 2 _ ~ du \ds) 2 g d 2 x dudv dudv ds ds dv \ds) ^Y+ du ds 2 + dv ds 2 134 GEOMETRY OF A SURFACE ABOUT A POINT these expressions are substituted in (48), and in the reduction of (10) and (11), we obtain When we make use ds ds where ~ L and _ M have the significance + . -~ ~ du ds ds^\dv ,_ 29/Vb/ +~ 2 dv \ds) , F &lt;*r (Fv ds^ \\ ^^ du ds ds (\\ F D , 2 ~dv) \ds) ds* * Gdepends From this it is seen that the geodesic curvature of a curve D". upon E, F, G, and is entirely independent of D, Suppose that the parametric lines form an orthogonal system, and that the radius of geodesic curvature of a curve v = const, be denoted by p gu . In this case F= 0, - ds = Vfldu. Hence the above equation reduces to (50) r we is In like manner find that the geodesic curvature of a curve u = const, given by _! As an immediate consequence theorem : c^ these equations f we have the When or the parametric lines upon a surface form an orthogonal system, a necessary and that the curves v sufficient condition is u = const, be geodesies that E be is a function of u alone or = const, G of v alone respectively. of expressible as a function ele differential parameters of v formed with respect to the linear It will now be shown that p gu ment (III, 4). From the definition that when ^=0 of these parameters ( 37, 38) it follows l d IE ~V\G GEODESIC CURVATURE Hence, by substitution in (52) (50), 135 we obtain JUPAL P gn ./ 1 [y&^v In like manner, we find (53) -i= Thus we have shown that the geodesic curvature line is a differential line. of a parametric parameter of the curvilinear coordinate of the Since this curvature is a geometrical property of a line, it is necessarily independent of the choice of parameters, and thus is an invariant. This was evident a priori, but we have just shown that it is an invariant of the differential parameter type. the definition of the positive direction of the normal to a surface ( 48), and the normal to the cylinder of projection, it fol lows that the latter for a curve v = const, is the direction in which v increases, whereas, for a curve u const., it is the direction in which u decreases. Hence, if the latter curves be defined by u = const., equations (52) and (53) have the same sign. From now, we imagine the surface referred to another parametric system, for which the linear element is If, (54) ds* = Edu + 2 Fdudv + G dv\ 2 is given by (50) will be defined = const. And if the sign of$ be an equation such as by (u, v) such that is increasing in the direction of the normal of its pro c/&gt; &lt; the curve whose geodesic curvature jecting cylinder, its geodesic curvature will be given by (55) p. where the differential parameters are formed with respect to (54). If two surfaces are applicable, and points on each with the same curvilinear coordinates correspond, the geodesic curvature of the curve const, on each at corresponding points will be the same &lt;= in consequence of (55). Hence : Upon two applicable surfaces the geodesic curvature of corresponding curves, at corresponding points, is the same. 136 GEOMETRY OF A SURFACE ABOUT A POINT the second When member of equation (55) is developed by (III, 46, 56), we have 1 1 d \ du dv\ d \ dv du ff(V*d cu H dv R du d L cu I I 1 dv 1 H F W\ d^\ d I du dv\ dv du, Hence we have the formula of Bonnet*: (56) i-^ du dv when In particular, the geodesic curvature of the parametric curves, the latter do not form an orthogonal system, is given by dv ft. (57) du The geodesic curvature the differential equation of a curve of the family, defined by has the value 1_ 1 f Mdu + Ndv*sQ, d I 2 " pg \ FN GM H \ du \^EN*- 2 FMN+ GM / d/ FM-EN ZFMN+ GM *Memoire Cahier 32 sur la theorie generale des surfaces, Journal de VEcole Poll/technique, (1848), p. 1. GEODESIC TOKSION In illustration of the preceding results, 137 theorem : we establish the When system is the curves of an orthogonal system have constant geodesic curvature, the isothermal. dsz When the surface is referred to these lines, and = Edu2 + Gdv 2 the condition that the geodesic , the linear element is written curvature of these curves be constant is, by (50) and (51), 1 l/i, dVE = fa VEG KI, where Ui and V\ are functions of u and v respectively. If these equations are differentiated with respect to D and u respectively, we get dudv Subtracting, dv cu we is obtain ""S cucv 75^ dv du CUCV log G = 0. Hence system E/G is isothermic. equal to the ratio of a function of u and a function of u, and the In terms of isothermic parameters, equations (i) are of the form 10X = \2 5U r r, J.3X \2 dV ~ and the (II) linear element is tf It is evident that the such a system. on a sphere. The same meridians and parallels on a surface of revolution form is true likewise of an orthogonal system of small circles have just seen that when a curve is or a differential equation, its geodesic equation curvature can be found directly. The same is true of the normal 59. Geodesic torsion. finite We denned by a curvature of the surface in the direction of the curve by (18). Then from (16) and (47) follow the expressions for p and tw. In order to define the curve for the torsion. it remains for us to obtain an expression it From (59) the definition of sin o&gt; follows that o) = X\ -f Y/JL + Zv, If this where X, /Lt, v are the direction-cosines of the binormal. equation is differentiated with respect to the arc of the curve, and the Frenet formulas (I, 50) are used in the reduction, we get ( 60 ) v ^ _i *- ds 138 GEOMETRY OF A SURFACE ABOUT A POINT (I, From 37, 41) Ave _ ~~~ /dy d"z have dz d*y\ _ ^~ /dz ~ d*x _ dx ~~~~ d"*z d"*z\ ~d~s~dfj j ds 2 ds ds , 2 and Moreover, from r r- d 2x 7 . -V-K / . 6?S^ (13), we obtain the identity ds ds ds ds H\_ \ds/ ds ds Consequently equation (61) (60) is equivalent to eos where l/T has the value 1 " - ED } du2 +(GD ~ (FD 2 ED") dudv + GD ( 2 FD") dv* ~T H(Edu + 2 Fdudv + G dv from zero, that is, ) When cos w is different when the curve is not an asymptotic line, equation (61) becomes hf-F As cients the expression for and dv/du, we T involves only the fundamental coeffi have the following theorem of Bonnet: is the T The function tangent at a ds point. same for all curves which have the same common Among will be these curves there later ( shown one geodesic, and only one, for it one passes 85) that one geodesic and only is direction at the point. through a given point and has a given or 180, and conse of this geodesic w is equal to At every point of T for a given point and direc T. Hence the value T quently is tion The that of the radius of torsion of the geodesic with this direction. function T is therefore called the radius of geodesic torsion of GEODESIC TORSION the curve. 139 From (63) it is seen that T is the radius of torsion of makes a constant angle with the any curve whose osculating plane tangent plane.* When is the numerator of the right-hand member of equation (62) of lines of equated to zero, we have the differential equation curvature. Hence : that the geodesic torsion of necessary and sufficient condition zero at a point is that the curve be tangent to a line of a curve be curvature at the point. A The geodesic torsion of the parametric lines 1 is given by _ FD-ED ~T~ EH _ GD l_ ~ ~TV -FD" GH these lines form an orthogonal system Tu and Tv differ only in sign. Consequently the geodesic torsion at the point of meeting of two curves cutting orthogonally is the same to within the sign. When Thus far in the consideration of equation (61) the case of asymptotic lines. In considering that they are parametric. The direction-cosines of the tangent and binomial to a curve v = const, in this case are we have excluded them now, we assume JL^, ~ =-L^, 7= ; +1 or 1. Consequently the direction-cosines of the normal have the values principal where e is and similar expressions for m and n. When in the Frenet formulas d\ ~~ I dfju _m T dv ds _n r ds T ds we substitute the above values, and in the reduction make use of (11) (65) * and (13), we get Thus far exception must be made of asymptotic lines, but later this restriction will be removed. 140 GEOMETRY OF A SURFACE ABOUT A POINT In like manner, the torsion of the asymptotic lines u = const, is found to be V K. But from (64) we find that the geodesic torsiqn in the direction of the asymptotic lines (63) is is qp V JT. Hence equation true for the asymptotic lines as well as for all other curves on the surface. Incidentally we have established the following theorem of Enneper real asymptotic line at : equal to the abso lute value of the total curvature of the surface at the point; the radii of torsion of the asymptotic lines through a point differ only in sign. is The square of the torsion of a a point The following theorem of Joachimsthal is an immediate consequence of (63) : When two surfaces meet under a constant angle, the line of intersection is a line of curvature of both or neither; and conversely, when the curve of intersection of two surfaces is. a line of curvature of both they meet under constant angle. For, of T, if we denote the values of w for the two surfaces, and Z\, T2 the values by subtracting the two equations of the form (63), that TI = T2 have, o&gt; &&gt;!, 2 , which proves the have ds (wi uz) first part of the theorem. Conversely, if \/T\ = l/T2 = 0, we = 0, and consequently the surfaces meet under constant angle. EXAMPLES 1. Show is revolution that the radius of geodesic curvature of a parallel on a surface of the same at all points of the parallel, and determine its geometrical significance. 2. Find the geodesic curvature of the parametric b a X = -(u + v), y = -(u-v), , / lines on the surface uv &gt; z = 3. Given a family of loxodromic curves upon a surface of revolution which cut all the meridians under the same angle a show that the geodesic curvature of curves is the same at their points of intersection with a parallel. ; these 4. 5. Straight lines on a surface are the only asymptotic lines which are geodesies. Show that the geodesic torsion of a curve 1 is given by 0, T where = -I 1/1 1\ sin . *\fi pj ) 2 6 denotes the angle which the direction of the curve at a point the line of curvature v = const, through the point. 6. 7. makes with Every geodesic line of curvature is a plane curve. line is Every plane geodesic a line of curvature. it is 8. When If the a surface of curvature 9. is cut by a plane or a sphere under constant angle, on the surface, and conversely. a line curves of one family of an isothermal orthogonal system have constant the same property. geodesic curvature, the curves of the other family have SPHERICAL REPRESENTATION 60. Spherical representation. erties of a surface 141 In the discussion of certain prop of advantage to make a representation of * S upon the unit sphere by drawing radii of the sphere parallel to the positive directions of the normals to S, and taking the extrem it is S ities of the radii as spherical images of the corresponding points on a point moves along a curve on $, its image m describes a curve on the sphere. If we limit our consideration to a portion of the surface in which no two normals are parallel, the portions S. As M in a one-to-one correspondence. the sphere is called the spherical map upon representation of the surface, or the Gf-aussian representation. It was first employed by Gauss in his treatment of the curvature of of the surface and sphere will be This of the surface surfaces. f The coordinates of m are the direction-cosines of the if normal to the surface, namely X, F, Z, so that we put the square of the linear element of the spherical representation is In 48 we established the following equations : (68) _f FD GD dx FD ED % H du~ H du dX FD"GD ex FD 1 . dx dv 2 2 ED" dx dv ] v H z du H 2 By means tions (o, of these relations and similar ones in F and Z, the func c^, & may be given the forms = (69) ~ [GIf H _ 2 F(DD" + D + ED 2 ) D"], JL H -2 ( or, in terms of the total and mean curvatures 52), (70) * The sphere of unit radius and center at the origin of coordinates. t L.c., p. 9. 142 GEOMETRY OF A SUEFACE ABOUT A POINT may be In consequence of these relations the linear element (67) given the form (71) doz Itm (Ddu-+ 2D dudv+D"dv 2 ) K(Edi? + 2 Fdudv and, by (18), (72) ^=( (70) \ J* From (73) we have also ff. e is is positive or negative. 1, according as Equations (69) are linear in E, F, G. Solving for the we have where K latter, E= (74) from the definition that the normals In seeking the differential equation of the lines of curvature to the surface along such a curve form a developable surface, we found ( 51) that for a dis placement in the direction of a line of curvature we have fa du 7 du + & dv dv 7 + r(l^ X du + 2X dv \ = 3 , , . - 0, \du 2, dv / and similar equations o in y and QT/" where r denotes the radius of If these equations be multi7} "jf principal curvature for the direction. -T7" Q *7 &gt; plied respectively by - ?., dv _ and dv du du -- and added, and du likewise by - &gt; dv the an(l added, th resulting equations may be written D du +D dv - r(fdu+di&gt;) = 0, D du +D n dv -r(3du + gdv} = 0. Eliminating (75) r, we have 2 as the equation of the lines of curvature - -"dudv D -D"&dv* = 0. SPHERICAL REPRESENTATION principal radii in the form (76) c^ (&lt;o 143 the equation of the Again, the elimination of du and dv gives 2 ) r*(&lt;oD" + 3D 2 &D ) r + (DD" D = 0, ) so that (77) These results enable us to write equations (74) thus : (78) 61. Relations between a surface and its is spherical representation. Since the radius of normal curvature tion except when the surface following theorem : R a function of the direc is a sphere, we obtain from (72) the necessary and sufficient condition that the spherical representation of a surface be conformal is that the surface be minimal or a sphere. A As a consequence of this theorem is onal system on a minimal surface that every orthog represented on the sphere by we have an orthogonal system. From (70) it is seen that if a surface is not minimal, the parametric systems on both the surface and the sphere can be orthogonal only when If is zero, that is, when the lines of curvature are parametric. Hence we have : The lines of curvature of ; by an orthogonal system this is a surface are represented on the sphere a characteristic property of lines of curvature, unless the surface be minimal. This theorem follows also as a direct consequence of the theorem : necessary and sufficient condition that the tangents to a curve upon a surface and to its image at corresponding points be parallel is that the curve be a line of curvature. A In order to prove this theorem we assume that the curve parametric, v is = const. Then du du the condition of parallelism is &.?.du to du ;&.* du du 144 GEOMETRY OF A SURFACE ABOUT A POINT (68) it From But follows that in this case is (FD ED 1 ) must be zero. the latter ( the condition that the curves v = const, be lines of Moreover, from (32) it follows that the positive to a line of curvature and its spherical representa half-tangents curvature 51). tion have the same or contrary sense according as the correspond ing radius of normal curvature is negative or positive. In consequence of (7) the equation (40) of the asymptotic directions may be written dxdX+ dydY+ dzdZ = And so 0. we have the theorem : The tangents property is to an asymptotic line and to its tion at corresponding points are perpendicular spherical representa to one another ; this characteristic of asymptotic lines. It is evident that the direction-cosines of the normal to the sphere are equal to X, Y, Z, to within sign at most. be denoted by then I/, ^; X&gt;&lt;&gt; Let them (79) x&gt; = #\du ( dv ? du dv When in the parentheses, expressions similar to (68) are substituted for the quantities the latter expression is reducible to KHX. (73), Hence, in consequence of (80) x&gt; we have = ex, ^ = er, l = *z, is where e = 1 according as the curvature of the surface positive or negative. From the above is elliptic it follows that according as a point of a surface or hyperbolic the positive sides of the tangent planes at corresponding points of the surface and the sphere are the same or Suppose, for the moment, that the lines of curvature are parametric. From our convention about the positive direction of the normal to a surface, and the above results, it follows that both different. have the the tangents to the parametric curves through a point as the corresponding tangents to the sphere, or both is an elliptic point; but that have the opposite sense, when M same sense M one tangent has the same sense as the corresponding tangent to the sphere, and the other the opposite sense, when the point is GAUSSIAN CURVATURE 145 hyperbolic. Hence, when a point describes a closed curve on a surface its image describes a closed curve on the sphere in the same or opposite sense according as the surface has positive or say that the areas inclosed by these negative curvature. curves have the same or opposite signs in these respective cases. Suppose now that we consider a small parallelogram on the sur face, We whose vertices are the points (u and + du, v gram on (u, v), (u -f du, v), (u, v -f dv), vertices of the corresponding parallelo the sphere have the same curvilinear coordinates, and + dv). The the areas are Ifdudv and e/tdudv, where e 1 according as the sur face has positive or negative curvature in the neighborhood of the point (u, v). The limiting value of the ratio of the spherical and the surface areas as the vertices of the latter approach the point a measure of the curvature of the surface similar to that (u, v) is In consequence of (73) this limiting value is the Gaussian curvature K. Since any closed area may be looked upon as made up of such small parallelograms, we have the following of a plane curve. theorem of Gauss The : limit of the ratio of the area of a closed portion of a surface to the area of the spherical is image of it, as the former converges to a point, in value to the product of the principal radii at the point. equal Since the normals to a developable surface along a generator are parallel, there can be no closed area for which there are not two nor parallel. Hence spherical representation, as defined 60, applies only to nondevelopable surfaces, but so far as the preceding theorem goes, it is not necessary to make this exception mals which are in ; for the total curvature of a developable surface is zero ( 64), and the area of the spherical image of any closed area on such a surface is zero. The fact that the Gaussian curvature is zero at all points of a developable surface, whereas such a surface is surely curved, makes this measure not altogether satis factory, and so others have been suggested. Thus, Sophie Germain* advocated the mean curvature, and Casorati f has put forward the expression 2 [ 1 ) \Pi fill But according a minimal surface is zero, and according minimal surface has the same curvature as a sphere. Hence the Gaussian curvature continues to be the one most frequently used, which may be due largely to an important property of it to be discussed later ( 64). to the first, the curvature of to the second, a * Crelle, Vol. VII (1831), p. 1. f Acta Mathematica, Vol. XIV (1890), p. 95. 146 GEOMETRY OF A SURFACE ABOUT A POINT 62. Helicoids. We apply the preceding results in a study of an important class of surfaces called the helicoids. A helicoid is generated by a curve, plane or twisted, which is rotated about a fixed line as axis, and at the same time translated in the direction of the axis with a velocity which is in constant ratio with the section of the surface by a plane through velocity of rotation. A the axis is called a meridian. All the meridians are equal plane curves, and the surface can be generated by a meridian moving with the same velocities as the given curve. The particular motion described is called helicoidal motion, and so we may say that any helicoid can be generated by a plane curve with helicoidal motion. In order to determine the equations of a helicoid in parametric form, we take the axis of rotation for the 2-axis, and let u denote the distance of a point of the surface from the axis, and v the angle made by the plane through the point and the axis with the #z-plane in the positive direction of rotation. If the equation of the gen erating curve in any position of its plane is z (w), the equations = &lt;f&gt; of the surface are (81) x = u cos v, y = u sin v, z = &lt;/&gt; (u) + av, ; where a denotes the constant ratio of the velocities it is called the parameter of the helicoidal motion. When, in particular, a is zero, these equations define any surface of revolution. Moreover, when &lt; (u) is a constant, the curves v = const, perpendicular to the axis, and so the surface is are straight lines a right conoid. It is called the right helicoid, By (82) calculation we obtain from (81) /2 c/) , F=a&lt;t&gt; ^= l + , G=u +a 2 2 , where the accent indicates differentiation with respect to u. From method of generation it follows that the curves v = const, are meridians, and u = const, are helices on the helicoids, and circles the on surfaces of revolution. From (82) it is seen that these curves form an orthogonal system only on surfaces of revolution and on the right helicoid. Moreover, from (57) it is found that the geo desic curvature of the meridians is zero only when a is zero or In the latter case the meridian is a straight line is a constant. &lt; perpendicular to the axis or oblique, according as is (/&gt; zero or not. HELICOIDS Hence the meridians of surfaces of revolution 147 and of the ruled are helicoids are geodesies. The orthogonal trajectories of the helices (cf. Ill, 2 upon a helicoid determined by the equation afidu 31) ) + (u + ^ / &gt;2 a 2 dv = 0. Hence, if we put v1 = J It -J- Q 2 du + v, are the orthogonal trajectories, and their equations in finite form are found by a quadrature. In terms of the parameters u and v l the linear element is the curves v x = const, 2 (83) t?s u of this result As an immediate consequence we have that the helices and their orthogonal trajectories on any helicoid form an isothermal system. From (83) and ( 46) we have the theorem of Bour: Every helices helicoid on the former correspond We (o4) derive also the following expressions ,r JL, y, z= and (85) -is applicable to to some surface of revolution, and the parallels on the latter. : a sin v u&lt;f&gt; cos v, (a cos v 2 + u$ 2 ! sin v), u V^ 2 (l + f )+a /&gt;,/&gt; .!&gt;"= From (84) it follows that a meridian is a its face of revolution at all curvature (Ex. 7, p. 140). of the lines of curvature of a helicoid, normal section of a sur and consequently is a line of points, This is evident also from the equation namely ] (86) a [1 + 2 &lt;/&gt; + u&lt;t&gt; $] dii 2 - (1 +$ *) u^ + [(u + a +u +a ]dv =Q. -a[u 2 2 ) M&lt;" dudv 2 - 2 2 2 2 (t&gt; Moreover, the meridians are lines of curvature of those helicoids, for which &lt;/&gt; satisfies the condition 148 GEOMETRY OF A SURFACE ABOUT A POINT integration this gives By = &lt;f) Vtf" U2 C log When and D" the surface is vanish. Hence the right helicoid the expressions for the meridians and helices are the asymp D totic lines. Moreover, these lines form an orthogonal system, so that the surface is a minimal surface ( 55). Since the tangent planes to a surface along an asymptotic line are its oscu lating planes, if the surface is a ruled minimal surface, the gener ators are the principal normals of all the curved asymptotic lines. But a circular helix is the only Bertrand curve whose principal normals are the principal normals of an infinity of curves ( 19). Hence we have the theorem Catalan : of FIG. 16 The right helicoid is the only real minimal ruled surface. In fig. 16 are represented the asymptotic lines and lines of is curvature of a right helicoid. For any other helicoid the equation of the asymptotic lines (87) ufi du 2 - Zadudv + uty di? = 0. As the coefficients in (86) and (87) are functions of u alone, : we have the theorem helicoid is referred to its meridians and helices, the asymp totic lines and the lines of curvature can be found by quadratures. When a EXAMPLES 1. Show of revolution 2. surface that the spherical representation of the lines of curvature of a is isothermal. of a line of curvature The osculating planes and of its spherical representa tion at corresponding points are parallel. the asymptotic directions at a point on a surface and as their spherical representation are equal or supplementary, according between at the point. the surface has positive or negative curvature 3. The angles between GENERAL EXAMPLES 4. 149 Show that the helicoidal surface x is = u cos v, y = u sin v, z = bv minimal. 5. The total curvature of a helicoid is constant along a helix. 6. The orthogonal trajectories of the helices upon a helicoid are geodesies. 7. If the fundamental functions E, F, G of a surface are functions of a single parameter w, the surface is applicable to a surface of revolution. 8. Find the equations of the helicoid generated by a circle of constant radius whose plane passes through the axis and the lines of curvature on the surface also find the equations of the surface in terms of parameters referring to the meridians and their orthogonal trajectories. ; GENERAL EXAMPLES to a surface, and if 1. If a pencil of planes be drawn through a tangent to the sections of the surface by these lengths be laid off on the normals at planes equal to the curvature of the sections, the locus of the end points is a M MT straight line normal to the plane determined by surface at M. MT and the normal to the 2. If P is a point of a developable surface, P the point where the generator through P touches the edge of regression, t the length PoP, p and r the radii of curvature and torsion of the edge of regression, then the principal radii of the surface are given by -i = 0, -=- -i 3. For the surface of revolution of a parabola about its directrix, the principal radii are in constant ratio. helices The equations x = a cos it, y = asinw, z = uv define a family of circular which pass through the point A (a, 0, 0) of the cylinder each helix has an involute whose points are at the distance c from A (cf. I, 106). Find the surface which is the locus of these involutes show that the tangents to the helices are 4. ; ; normal 5. to this surface ; find also the lines of curvature upon the latter. The surfaces defined by the equations q*f(y), (cf. 25) l+p 2 + ?2 = y-axis respectively. 6. x + pz=&lt;t&gt;(p) have a system of lines of curvature in planes parallel to the xz-plane and to the The equations y - ax = 0, a: 2 + y2 + z2 - 2 px - a2 = 0, where a and (0,0, a). characteristics of a family of spheres, except when f(a) is a linear function also that the circles are lines of curvature on the envelope of these spheres. 7. ft are parameters, define all the circles through the points (0, 0, a), Show that the circles determined by a relation ft=f(a) are the ; If the other nonrectilinear lines of curvature one of the lines of curvature of a developable surface lies upon a sphere, lie on concentric spheres. 150 8. GEOMETRY OF A SURFACE ABOUT A POINT If the center of normal curvature of the line is a point on a surface, the angle between the lines of curvature, and PI, P 2 the centers of normal bisecting curvature in two directions equally inclined to the first, then the four points P P P, PI, PO, PZ form a harmonic range. R m denote the radii of normal curvature of m sections R a R-s, which make equal angles 2 tr/m with one another, and m 2, then a surface 9. If EI, , , &gt; of I /-I m \Ri 10. If the V--1 R2 + J_\ = * RJ 2 /i + T Vx P IV J is Dupin indicatrix at a point P of a surface an ellipse, and through either one of the asymptotes of its focal hyperbola two planes be drawn perpen dicular to one another, their intersections with the tangent plane are conjugate directions on the surface. and whose osculating All curves tangent to an asymptotic line at a point 11 for a point of inflection. surface at 3f, have planes are not tangent to the . M , M 12. The normal curvature equal to the 13. mean curvature of an orthogonal trajectory of an asymptotic line of the surface at the point of intersection. is The surface x of revolution whose equations are u sin w, z = u cos w, y = = a log (u 4- vV 2 a2 ) called the catenoid. is Show a catenary about its axis generated by the rotation of that it is the only minimal surface of revolution. ; it is a constant angle 14. When the osculating plane of a line of curvature makes with the tangent plane to the surface, the line of curvature is plane. line of curvature is represented on the unit sphere by a circle. 15. A plane 16. The cylinder whose tion p = a - s 2 /6, it where a and lie the intrinsic equa right section is the curve defined by b are positive constants, has the characteristic prop erty that I/Void. 17. upon curves of curvature a + a26 b , ^l ^ whose geodesic curvature is When a surface is the curves v geodesic curvature of curvature of the curve the referred to an orthogonal system of lines, and the radii of = const, and u - const. are p^, pgv respectively, geodesic which makes an angle cos0 Pgu sin &gt; Q with the lines v = const, is given by 1 _ dd ~~ Pg dS Pgv s referred to an orthogonal system of lines, and p vi for one system of isogonal denote the radius of geodesic curvature and the arc and pj, 8 the similar functions for the trajectories of the parametric lines, then whatever be the direction of the first trajectories of the former, orthogonal 18. When a surface is curves the quantity 19. If p 4. to constant at a point. first and p denote the radii of also p, fi s curvature of a line of curvature and its and spherical representation, curves, then and ff curvature of these p g the radii of geodesic fa dtr 77 where ds and d&lt;r =d P~8~P? are the linear elements of the curves. GENERAL EXAMPLES , 151 20. When a surface is referred to its lines of curvature, and #o denote the angles which a curve on the surface and its spherical representation make with the curves v = const., the radii of geodesic curvature of these curves, denoted by pg and pg respectively, are in the relation ds ddo Py dcr = d&o --- 21. When Pg the curve x=f(u)cosu, is y=f(u)sinu, z =and also geodesies subjected to a helicoidal motion of parameter a about the z-axis, the various positions of this curve are orthogonal trajectories of the helices, on the surface. 22. When a curve is subjected to a continuous rotation about an axis, and at the same time to a homothetic transformation with respect to a point of the axis, such that the tangent to the locus described by a point of the curve makes a con stant angle with the axis, the locus of the resulting curves is called a spiral surface. Show that if the z-axis be taken for the axis of rotation and the origin for the center of the transformation, the equations of the surface are of the form z x = f(u) e hv cos (u + a constant. spiral surface line, v) , y = f(u) e hv sin (u + v) , = v (u) e* , where h 23. is can be generated in the following manner: Let C be a on C describes an a point on the latter if each point isogonal trajectory of the generators on the circular cone with vertex P and axis I in such a way that the perpendicular upon I, from the moving point revolves about I curve, I A any and P ; M M , with constant velocity, the locus of these curves 24. is a spiral surface (cf . Ex. 5, 33). that the orthogonal trajectories of the curves u = const., in Ex. 22, can be found by quadratures, and that the linear element can be put in the form Show where A is a function of a alone. 25. Show that the lines of curvature, minimal lines, and asymptotic lines upon a spiral surface can be found by quadrature. CHAPTER V FUNDAMENTAL EQUATIONS. THE MOVING TRIHEDRAL 63. Christoffel symbols. and sufficient equations of condition to be satisfied In this chapter we derive the necessary by six func tions, E, F, G ; D, D\ D", in order that they may be of the fundamental quantities for a surface. For the sake of brevity we make use two sets of symbols, suggested by Christoffel,* which represent certain functions of the coefficients of a quadratic differential form and their derivatives of the first order. If the differential form is a n du 2 -f- 2 a^du^du^ + the first c set of symbols is defined by R&l [l J !/^ + ^_&lt;HA 2\du k i, du ( duj where each of the subscripts k, I has one of the values 1 and 2.f From this definition it follows that When these symbols are used in connection ,with the first fun 2 2 2 damental quadratic form of a surface ds = E du + 2F dudv + G dv they are found to have the following significance : , [iriia* L1J 2Su 2dv [""L^ia* 2 Su L2J (1) 2J O & cu ,* 2 di&lt; I ~l I L * Crelle, Vol. t . J- I cv Q/* O L^J I I ^ 2~dv LXX, pp. 241-245. This equation defines these symbols for a quadratic form of any number of vari n. u n In this case i, k, I take the values 1, ables wi, 152 , , CHRISTOFFEL SYMBOLS The second set of 153 symbols is defined by the equation where A vl denotes the algebraic complement of a vl in the discrimi a 22 divided by the discriminant itself. With reference nant a n a 22 to the first fundamental quadratic form these symbols mean and 4i--Jv du f) ^ =:^ du A =-!r* dv TT 2 fill I du ," dv Ti-O du n I ri2\ ^g^ du llJ f" 2^ G ^v 3t* I 12 \_ 2 du dv l2/~ f221 dv G dv du dv From these equations we derive the following identities : With the aid of these identities we derive from (III, 15, 16) the expressions &gt; (MVKff}&gt; From the above definition of the symbols : Sm-K?))f^l &lt;! &gt; WQ obtain the following important relation 64. The equations of Gauss and of Codazzi. equations (IV, 10) and the equation The first two of 154 FUNDAMENTAL EQUATIONS set of equations linear in 2 , form a consistent determinant is du 2 , du 2 du 2 and the equal to H. Solving for -, we get similar equations hold for y and z. Proceeding the other equations (IV, 10) and (6) in like manner with we get the following equations of G-auss : a 2 * _ri2\az fiaiat awa li/a* la/a* ^ For convenience of reference we recall from 48 the equations dx dv dX_FD = ~du (8) GDdx H 2 a^ 4 FDED n FD 2 dX __ ~ 3v FD"-GD dx -ED" dx dv (7) are H 2 du H 2 The conditions of integrability of the Gauss equations du\dudv dv \dudv du\dv 2 By means (9) of (7) and &lt;* (8) these equations are reducible to the forms x 1 a v o 2 ^ 4. A du ?E 2 _i_ x ; 1 dv v , D" D, D c 2 are determinate functions of E, F, G where a v a 2 similar to (9) hold for y and and their derivatives. Since equations z, we must have , &gt;, (10) !=0, a 2 =0, &j=0, 62 =0, ^=0, &lt;? 2 =0. EQUATIONS OF GAUSS AND OF CODAZZI When , 155 the expressions for a r a 2 b^ and 5 2 are calculated, it is found that the first four equations are equivalent to the following : 12 12\ d jMlll d |12jfl2j , fill J221 /in ri2i rni/121 ri2i 2 , rnif22 fi2i fill 221 HiAaJ-ta-J d M21 , /221 /121 , /221 Ml f!2\ a /22\ , fl2\fl2\ f22\/ll When the expressions for the Christoffel symbols are substituted in these equations the latter reduce to the single equation Z?D"-jP a H = 1 f gr F dF du 2 ~2N \du \_EH j^r dv dE__ 1 3G1 dv H du\ _ 2 i IH H dv EH du_ This equation was discovered by Gauss, and is called the G-auss equation of condition upon the fundamental functions. The left- hand member of the equation is the expression for the total curva ture of the surface. Hence we have the celebrated theorem of Gauss * : The expression for the total curvature of a surface is a function of the fundamental coefficients of the first order and of their deriva tives of the first and second orders. When the last the expressions for c^ and c2 are calculated, we find that two of equations (10) are v (13) " du dv *L.c., p. 20. 156 FUNDAMENTAL EQUATIONS These are the Codazzi equations, so called because they are equiva lent to the equations found by Codazzi * ; however, it should be mentioned that Mainardi was brought to similar results some what earlier. f It is sometimes convenient to have these equations written in the form D~ (13 ) d D , f22\D " f!2\D fin ~~ which reduce readily to (13) by means of (3). With to (13). the aid of equations (7) we find that the conditions of integrability of equations (8) and similar ones in Y and Z reduce From surfaces the preceding theorem and the definition of applicable ( 43) follows the theorem : Two the same total curvature at corre applicable surfaces have sponding points. As a consequence we have : Every surface applicable twisted curve. to a plane is the tangent surface of a For, when a surface 2 is applicable to a plane its linear element is reducible to ds is = du 2 + dv\ and consequently its total curvature zero at every point by (12). 2 From (IV, 73) it follows that Hence X, Y, Z the surface is are functions of a single parameter, and therefore the tangent surface of a twisted curve (cf. 27). Incidentally we have proved the theorem : When and K is zero at all the latter points of a surface is developable, conversely. una Vol. II superficie e dello spazio, Annali, Ser. 3, p. 395. * Sulle coordinate curvilinee d (18(W) t , p. 269. Giornale dell Istituto Lombardo, Vol. IX, FUNDAMENTAL THEOREM 65. 157 Fundamental theorem. When the lines of curvature are and Codazzi equations (12), (13) reduce to parametric, the Gauss DP" _ (14) G 2_ ( dv D "\ _^ E ~ du The v direction-cosines of the tangents to the parametric curves, = const, and u = const., have the respective values (15) 2 By means of equations (7) and (8) we find D" (16) . du V ax ( du and similar equations obtained by replacing X^ X^ Z. From (15) we have respectively, and by Z^ Z^, X by Yv T Y 2 , = C^EX (17) = CVflY = CVEZ We proceed to the proof of the converse theorem D", : Given four functions, E, G, D, exists a surface for which E, 0, quantities of the first G ; satisfying equations (14); there 0, D" are the fundamental Z&gt;, and second order respectively. 158 In the FUNDAMENTAL EQUATIONS first place we remark that all the conditions of integraof the equations (16) are satisfied in consequence of bility (14). Hence these equations admit sets of particular solutions whose values for the initial values of u and v are arbitrary. From the form of equations (16) it follows (cf. 13) that, if two such sets of particular solutions be denoted by X^ 2 X X and Y^ F z, 2, F, then XI + XI + X* = 2 const., I 0, 0, 1. Y?+ F + F = const., X F + X Y + XY= const. 1 x 2 2 From the theory of differential equations we know that there exist three particular sets of solutions X^ X\ Fp F2 F; Z^ Z^ Z, z which for the initial values of u and v have the values 1, 0, 0; 0, 1, 0; X , , In this case equations (18) become X + X + ^ = 1, 2 1 2 2 2 2 (19) +F + F = X Y +X Y + XY=Q, F 2 2 2 l, i l 2 2 which are true for all values of u and v. In like manner we have (19 ) follows that the expressions in the right-hand mem bers of (17) are exact differentials, and that the surface denned by these equations has, for its linear element and its second quadratic From (16) it form, the expressions (20) Edu?+G dv\ we had D dy? + 2 D"dv respectively. Suppose, now, that a second system of three sets of solutions of equations (16) satisfying the conditions (19), (19 ). s, F s, and Z s equal By a motion in space we could make these to the corresponding ones of the first system for the initial values X of u and v. But then, because of the relations similar would be equal for all values of u and v, as shown in motion in space, a surface (20). is to (18), they to within a ratic 13. Hence, determined by two quad forms As in 13, it can be shown that the solution of equations (16) reduces to the integration of an equation of Riccati. FUNDAMENTAL THEOEEM 159 Later * we shall find that the direction-cosines of any two per and of the normal a pendicular lines in the tangent plane to surface, to the surface, satisfy a system of equations similar in form to (16). Moreover, these equations possess the property that sets of solu tions satisfy the conditions (18) when the parametric lines are any whatever. Hence the choice of lines of curvature as parametric lines simplifies the preceding equations, but the result is a general one. Consequently we have the following fundamental theorem: the coefficients of two quadratic forms, When Edu* + to 2 Fdudv + G dv\ Ddu +2 D dudv + 2 D"dv\ satisfy the equations of Gauss and Codazzi, there exists a surface, unique within its respectively the first forms are and second fundamental quadratic forms ; and a Riccati which these position in space, for the determination of the surface requires the integration of equation and quadratures. are the funda From (III, 3), (5) and (6), it follows that if E, F, G; D, D mental functions for a surface of coordinates (x, y, z), the surface symmetric with the coordinates ( x, y, z), has respect to the origin, that is, the surface with , D" - D". Moreover, in consequence the fundamental functions E, F, G; - D, two surfaces whose fundamental quantities bear such a rela of the above theorem, D , tion can be Two moved in space so that they will be symmetric with respect to a point. surfaces of this kind will be treated as the same surface. EXAMPLES 1. When is surface 2. said to be isothermic. the lines of curvature of a surface form an isothermal system, the Show that surfaces of revolution are isothermic. Show that the hyperbolic paraboloid x is =a + -(t* is t&gt;), y b = -(*-), * = uv in terms of isothermic. 3. When a surface isothermic, and the linear element, expressed is parameters referring to the lines of curvature, of Codazzi and Gauss are reducible to Pl i ds 2 = \2 (du 2 + dv 2 ), the equations dp 2 a _ PZ i api Find the form of equations 39). symmetric coordinates (cf 4. . (11), (13) when the surface is defined in terms of * Cf . 69. Consult also Scheffers, Vol. II, pp. 310 et seq. ; Bianchi, Vol. I, pp. 122-124. 160 FUNDAMENTAL EQUATIONS K is equal to zero for the tangent surface of a twisted 5. Show that curve, taking the linear element of the latter in the form (105), 20. 6. Show its that the total curvature of the surface of revolution of the tractrix about 7. axis is negative and constant. Establish the following formulas, in which the differential parameters are formed with respect to the form Edu? + 2Fdudv + Gdv 2 : )=~where the quantities have the 8. same significance as in JT, 65. Deduce the identity A2x = ( 1 ) and show therefrom that the curves in which a minimal surface is cut by a family of parallel planes and the orthogonal trajectories of these curves form an isothermal system. 66. Fundamental equations if in another form. We have seen in 61 that X, F, Z denote the direction-cosines of the normal to a surface, the direction-cosines of the normal to the spherical rep 1 according resentation of the surface are eX, eF, eZ, where e is as the curvature of the surface is positive or negative. If, then, the second fundamental quantities for the sphere be denoted by A^ (21) , 3", we have =-&lt; ^ = -e^, ,&" = -e^ so that for the sphere equations (7) become rnvax_ 2X (22) J + 12J dv 2 f 12V l2J F -^x; where the Christoffel symbols T \ ?\ are formed with respect to the linear element of the spherical representation, namely conditions of integrability of equations (22) are reducible by means of the latter to The = 0, EQUATIONS OF CODAZZI where a v a2 , 161 A., A 2J B^ B b lt b 2 , are the functions obtained from the quantities 2 DD"D 64 by replacing of E, F, G respectively 2 &gt; by &, ^respectively. Since the above equations must be sat isfied by Y and Z, the quantities A^ A 2 B v Bz must be zero. This 1, , gives the single equation of condition (24} J J_rA/^^__L^V-^- fa -- 2v --^- du)\ = 1 ^1 ft 2 ft to &ft dv ft du [du \ft ) W Moreover, the Codazzi equations (13 of (21), ) become, in consequence (26). 3w o \ W //"/ o / v \/// V // / 1 \.\jft I //" ~ * ~* * f ~* * "" which vanish If identically. dx equations (IV, 13) be solved for f and dx cv we du get dx _ " . (26) i_ dv ft* du By means of equations (22) the condition of integrability of these g equations, namely /^\ ^ and similar conditions in y and 2, reduce to (27) - OU ^Hu v^ dv Hence two quadratic forms (odu 2 -f 2 & dudv + dv 2 , D du 2 +2Z&gt; dudv + 2 D"dv , whose coefficients satisfy the conditions (24), (27), may be taken as the linear element of the spherical representation of a surface and as the Si3cond quadratic form of the latter. When X, F, Z are 162 FUNDAMENTAL EQUATIONS of the surface can be found by however, the determination of the former requires known, the cartesian coordinates ; quadratures (26) the solution of a Riccati equation. If the equations t D = _^fc*x by (7) be differentiated with respect to u and to the form * and means of (22) : v, the resulting equations may be reduced 55 cu (ii ( 11 1 2 12 l = a I2 [ 2) -f c i Jl 2 cD ^-= cu cv * JX + D+ 12 2 22 ) D cu -f 2 D cv D" D". (V surface may be 67. Tangential coordinates. Mean evolute. not only as the locus of a point whose position looked upon but also as the envelope of its two A depends upon parameters, tangent planes. the surface is developable or not. We parameters according as case in 27, and now take up the latter. considered the former distance from the origin to the tan denotes the algebraic If This family of planes depend* upon one or two W S at the point M(x, gent plane to a surface (29) y, z), then W=xX+yY+zZ. with respect to u and v, If this equation is differentiated the to in consequence of (IV, resulting equations are reducible, 3), X dW I, p. *Cf. Bianchi, Vol. 157. TANGENTIAL COORDINATES The 163 three equations (29), (30) are linear in #, y, z, and in con Hence sequence of (IV, 79, 80) their determinant is equal to e/ we have and similar expressions for y and identities z. From (IV, 11) we deduce the Y ( dz du a- Z dY du y. e = TH/ ^dX ex 1 , r ~dX\ &lt;o 01 dl ) -r rf \ , du C^\- dv / zr&lt; ) rC rrV^ v I ~Y j^G-A- By means of these equations the above expression for x is reducible to cv Hence we have (32) x = WX+k[(W,X), y= WY+k((W,Y), z = WZ+ &((W,Z], the differential parameters being formed with respect to (23). of u and Conversely, if we have four functions X, F, Z, W i&gt;, such that the (33) first three satisfy the identity x +r +^ = l, 2 2 2 equations (32) define the surface for which X, F, Z are the directionis the distance of the latter from cosines of the tangent plane, and the origin. For, from (33), we have W = 0, dv in consequence of which and formulas &gt;TA (22) we find from (32) that dx Moreover, equation (29) also follows from (32). Hence a surface is completely defined by the functions X, F, Z, W, which are called the tangential coordinates of the surface.* * Cf Weingarten, Festschrift der Technischen Hochschule zu Berlin (1884) Darboux, Vol. I, pp. 234-248. I, pp. 172-174 . ; ; Bianchi, Vol. 164 FUNDAMENTAL EQUATIONS equations (30) are differentiated, When we obtain ffw _ dv* By means of (22), (29), and (30) these equations are reducible to du (34) 2 D =- tfW \_dudv are substituted in the these expressions for D, for p^+ p 2 the latter becomes expression (IV, 77) When A &" , By means (35) of (25) this equation can be written in the /&gt; form 1 4-^ 2 = -(A;TF4-2^), is where the differential parameter formed with respect to the linear element (23) of the sphere. Moreover, if A^ 2 denotes the following expression, _ r22y^__ 1 1 J an 2J 12 follows from (34) that 12 it (3T) MEAN EVOLUTE In passing 165 a differential parameter we shall prove that it is A 22 is by showing that (38) expressible in the form Without (39) loss of generality we take 2 Edu*+Gdv Then /I dG 1 as the quadratic form, with respect to which these differential parameters are formed. 1 1 dE\ 1 dE u = -F dv \du By substitution we find _ eters, their Since the terms in the right-hand member are differential param values are independent of the choice of parameters v, u and is in terms of which (39) is expressed. Hence equation (38) an identity. The coordinates # face halfway y Z Q of the point on the normal to a sur between the centers of principal curvature have , , the expressions The is surface enveloped by the plane through this point, which is parallel to the tangent plane to the given surface, mean evolute of the latter. If called the W denotes the distance from the origin to this plane, we have (40) W,= ZXf9 =W+^(p +Pt ). 1 By means (41) of (35) this^may be written TFO =-JA;TF. 166 FUNDAMENTAL EQUATIONS EXAMPLES Derive the equations of the lines of curvature and the expressions for the principal radii in terms of W, when the parametric lines on the sphere are 1. (i) meridians and parallels ; (ii) the imaginary generators. lie Show that in the latter case the curves corresponding to the generators metrically with respect to the lines of curvature. sym 2. Let Wi and 2 denote the distances from the origin to the planes through the normal to a surface and the tangents to the lines of curvature v = const. , u = const, respectively, so that we have W Show Wi = xX l + yYi + that zZi, W 2 = xX2 + yYz + zZ 2 Pi . the differential parameters being formed with respect to 3. Edu* + 2 Fdudv + If 2 q = x2 + yz + z2, then we have 4. Show that when the lines of curvature are parametric = Pi cu cu ~ = P2 v dv is 5. The determination of surfaces whose mean evolute is a point problem as finding isothermal systems of lines on the sphere. the same dition trihedral. The fundamental, equations of con be given another form, in which they are frequently may used by French writers. In deriving them we refer the surface to 68. The moving a moving set of rectangular axes called the trihedral T. Its ver tex is a point of the surface, the a^-plane is tangent to the surface at M, and the positive 2-axis coincides with the positive M direction of the normal to the surface at x- M. and ?/-axes is z-axis, U being a function of u and v. given In Chapter I we considered another moving trihedral, consisting of the tangent, principal normal, and binormal of a twisted curve. the curve v = const, determined by the angle U makes with the through M position of the which the tangent to The Let us associate such a trihedral with the curve v const, through THE MOVING TRIHEDRAL 16T M and we call call it the trihedral t u. We have found , tions of the direction-cosines a 6 , c 16) that the varia ( of a line L, fixed in space, with reference to (7M , M, as its vertex moves along the curve which are given b by (42 ) -, &lt;^= ds u pu f W- + dsu \ Pu ; where p u r u denote the radii of first and second curvature of Cu1 and dsu its linear element evidently the latter may be replaced , by V^ du. 1 The direction-cosines of r L with respect to the trihedral &gt; T have the values (43) a _a a &gt; Ib _ ^ s n ^ _ c cos c cos sin U + (b sin w u cos jj j r ^ o&gt; s j n ^r ?7, M ) cos =6 [&lt;7 f cos o&gt; w -f c sin o) tt , where w u makes the angle which the positive direction of the z-axis with the positive direction of the principal normal to Cu at Jf, is the angle being measured toward the positive direction of the binormal of Cu From equations (42) and (43) we obtain the following . : (44) da - = br , db cq, du j9, q, du = cp do ar, = : , cu aq op, where r have the following significance p= (45) cosU rr/ sm7( c?&&gt; 1 . )-f sin U coso) coso). = 1\ 2 I rr cosU ds,. V If, in like u = const, manner, we consider the trihedral tv of the curve through M, denoted by Cv we obtain the equations , da , db do , where ^^ Pui * q v r l can be obtained from (45) by replacing Vjg; Z7, M , S ^ denotes the angle which the Tu ^7 ^^^ ^ 8 Pv TV pv - v-&gt; i; A tangent to the curve (46) Cv at M makes with the V-U=G. a&gt;axis, we have 168 If the line, FUNDAMENTAL EQUATIONS vertex is, that along a curve other than a parametric along a curve determined by a value of dv/du, the , M moves c are variations of a, evidently given by ^ da du da dv dv ds ^_ dv ds do du do dv dv ds du ds in du ds du ds which the 69. differential quotients have the above values. Fundamental equations with the trihedral ciate T Suppose that we asso a second trihedral TQ whose vertex is of condition. it fixed in space, about which revolves in such a manner that its edges are always parallel to the corresponding edges of T, as the vertex of the latter moves over the surface in a given manner. The position of T is completely determined by the nine directioncosines of its edges with three mutually perpendicular lines L v L 2 , L s Call these direction-cosines a v b { c l through These functions must satisfy the equations 3 0. , &lt;? ; a2 5 2 , , c 3 , . da - = 6r, ^w (47) dv~ If = br TI the we equate make use cucv two tion of these equations, and in the reduction of the resulting equa of (47), we find Since this equation must be true b , &lt;? when b and c have the values ; 5 , ** ; 63 , &lt;? 8, the expressions in parenthesis T ^&gt;O must be equal /^9 to zero. Proceeding in the same manner with obtain the following fundamental equations * : - and d*c we dudv dp dpi dq (48) dq l dr dr l Ser. * These equations were first obtained by Combescure, Annales de VEcole Normale, 1, Vol. IV (1867), p. 108; cf. also Darboux, Vol. I, p. 48. ROTATIONS - 169 , These necessary conditions upon the six functions p, r 1? in cs may determine the order that the nine functions a x position of the trihedral T are also sufficient conditions. The proof of this , , , is similar to that given in 65. * Equations (47) have been obtained by Darboux from a study of the motion of the trihedral TQ He has called jt?, q, r l the . rotations. We t u . Let return to the consideration of the moving trihedrals T and ?/, z ) denote the coordinates of a point P (x, y, z) and (# , with respect to u respectively. the following relations hold : T and t Between these coordinates / =x =# f cos U &lt;W (y sin o&gt; M z cos 2 cos W M ) sin &&gt; 7, sin CT 1 (x z If in a y cos + (# M+ 2 sin Wu o&gt; M) cos /, sin M. the trihedral these displacement of P absolute increments with respect to t be indicated by S, and increments relative to u at M moving axes by c?, we have, from 16, ^L^-S^+i, d ds u u pu dsu (45) = ^_ + - + -, dsu pu TM ^. = C?S M : dz L-y-. ru ds u From (49), (50), and $x we obtain the following! du = dx 4- VrE cos Ury + qz, du -^ ^W 2 = ^M + VfismUpz + ra, ox = dz aw a% + PV. Equations similar to these follow also from the consideration t Hence, when the trihedral T moves over the v surface with its vertex describing a curve determined by a of the trihedral . M value of dv/du, the increments of the coordinates of a point P(x, y, z), in the directions of the axes of the trihedral, in the I, chaps, i and v. In deriving these equations we have made use of the fact that equations (49) define a transformation of coordinates, and consequently hold when the coordinates are replaced by the projections of an absolute displacement of P. * t L.c., Vol. 170 FUNDAMENTAL EQUATIONS may also be absolute displacement of P, which * to these axes, have the values moving relative where we have put The coordinates of M are (0, 0, 0), so that the increments of its displacements are (53) Sx = 1; du + ^dv, y = vidu + , ri l dv, Sz = 0. with respect to the y v 2J denote the coordinates of L 3 previously defined, it the lines L v fixed axes formed by 2 If fa, M follows that and similar expressions ag , for y and z^ where a l t, 6 1? ^; 2, 62 , 2 6 8 , c 3 are to the moving the direction-cosines of the fixed axes with reference axes. Since the latter satisfy equations (47), the conditions that the a 2 two values * dz z of cu ^- obtained from (54) be equal, and similarly for J and ---1 are dudv (55) have ten functions f f p ?/, 77^ p, p^ q^ r, r satis -, c s can be and (48), the functions a 1? fying these conditions of a Riccati equation, and x^ y# z l by quad found by the solution as well as ratures. Hence equations (48) and (55) are sufficient to the Gauss and are equivalent necessary, and consequently Codazzi equations. When we , &lt;?, x, * Cf. Darboux, Vol. II, p. 348. LINES OF CURVATUKE 70. Linear element. 171 (53) Lines of curvature. is From we see that the linear element of the surface (56) ds 2 = (%du + ^ dvf +(ndu + r )i dv}\ Hence a necessary and lines be orthogonal is sufficient condition that the parametric (57) ff!+^i==0. - c), it being For a sphere of radius c the coordinates of the center are (0, 0, that the positive normal is directed outwards. As this is a fixed point, it assumed follows from equations (51) that whatever be the value of dv/du we must have and consequently /KQ\ + hdv - (qdu + qidv)c = ydu + dv + (pdu + pidv)c = du 171 0, 0, zr i) q -P = 1 = ^i =: C. qi -Pi in space, c) is fixed Conversely, when these equations are satisfied, the point (0, 0, therefore the surface is a sphere. Moreover, suppose that we have a propor and tion such as (58), where the factor of proportionality is not necessarily constant. For the moment call it t. When the values from (58) are substituted in (55) and reduction is made in accordance dt r? with dt (48) we get dt l 7?1 ^" ^ = 3t *to~* to 31, is seen to - f^ is zero, which, from (56) and t is constant unless be possible only in case the surface is isotropic developable. Hence ^ By definition (51) a line of curvature is a curve along which the the normals to the surface form a developable surface. When vertex is move in a point (0, 0, p) must displaced along one of these lines, are zero. Hence we must have such a way that Bx and % ^dv + (qdu + q dv r]du + (pdu+p f du -h i] 1 l dv)p l = 0, dv)p = 0. the equation of Eliminating p and dv/du respectively, we obtain the lines of curvature, (59) (f du + ^dv) (p du + p z ) v dv) + (17 du + rj^v) (q du + ) q^v) = 0, 0. and the equation of the principal (60) radii, p - qPl + p (qrj, - q,rj + p^ - p (pq, + (fa - ^) = From (59) it follows that a necessary and sufficient condition that the parametric lines be the lines of curvature is (61) fe&gt; + i# = 0, f^i+ih^O. 172 FUNDAMENTAL EQUATIONS may replace these equations by \rj, We P= ? = -xf, ^^V?!, ?i = -\fn When these thus introducing two auxiliary functions X and \. values are substituted in the third of (55), we have X and \ are equal, the above equations are of the form (58), which were seen to be characteristic of the sphere and the isotropic developable. Hence the second factor is zero, so that equa If tions (61) (62) or (63) may be replaced by ffi+ ^=0, 0, m+??i=&lt;&gt; ^=17 = (52) it P= q1 =Q- From follows that in the latter case the x- and ?/-axes are tangent to the curves v this case later. = const, we and u = const. We shall consider From (60) and (52) find that the expression for the total curvature of the surface is where denotes the angle between the parametric curves. the third of equations (48) may be written co Hence /g4\ V76rsin PiP* co H PiP dr dr. 71. Conjugate directions representation. We and asymptotic directions. Spherical have found ( 54) that the direction in the tangent plane conjugate to a given direction is the characteristic of this plane as it envelopes the surface in the given direction. Hence, from the point of view of the moving trihedral, the direc is tion conjugate to a displacement, determined by a value of dv/du, the line in the #?/-plane which passes through the origin, and which does not experience 2;-axis. an absolute displacement in the it direction of the From the third of equations (51) is is seen that the equation of this line (65) (p du H- p^v) y (q du + q^dv) x = 0. CONJUGATE DIRECTIONS If the 173 increments of u and v, corresponding to a displacement in the direction of this line, be indicated by d^ and d^v, the quan tities x and y are proportional to (f d^u f ^v) and (r; d^ 4- rj^v). When x and ?/ in (65) are replaced by these values, the resulting + equation (66) may be reduced to ]l - gf) dudjU + (pr - qgj dudy + (p& - qg) d^udv (prj In consequence of (55) the coefficients of dud^v and d^udv are equal, so that the equation is symmetrical with respect to the two tion sets of differentials, thus establishing the fact that the rela between a line and its conjugate is reciprocal. In order that the parametric lines be conjugate, equation (66) must be satisfied by du = and d^v = 0. Hence we must have (67) It should be noticed that equations (61) are a consequence of the of (62) first and (67). Hence we have the result that the lines of curvature form the only orthogonal conjugate system. From (66) it follows that the asymptotic directions are given by (68) (prj - gf ) du* + (prj l - q^ +p^ - q) dudv + (p^ - q^) dv = 0. 2 spherical representation of a surface is traced out by the point m, whose coordinates are (0, 0, 1) with respect to the tri hedral T of fixed vertex. From (51) we find that the projections of a displacement of m, corresponding to a displacement along the surface, are (69) The SX=qdu + q dv, l &Y= (pdu+p 2 l dv), &= 0. Hence the (70) linear element of the spherical representation is da 2 = (qdu + by q^v) + (pdu+ p^dv)\ The line defined (65) is evidently perpendicular to the direc tion of the displacement of m, as given by (69). Hence the tangent to the spherical representation of a curve upon a surface is perpen dicular to the direction conjugate to the curve at the corresponding point. Therefore the tangents to a line of curvature and its rep resentation are parallel, whereas an asymptotic direction and its representation are perpendicular ( 61). 174 72. FUNDAMENTAL EQUATIONS Fundamental relations (69) and we have, for the point and formulas. From equations on the surface, M (53) = du (71) "" * ^= =i?i, \7~ = = ?, = 0; ^\ and 5v ~V 7 cu (72) du du .. Consequently the following relations hold between the fundamental coefficients, the rotations, and the translations: F= f + ^, (73) G= f in particular, the parametric system on a surface is orthog the x- and y- axes of the trihedral are tangent to the curves onal, and v = const, and u = const, through the vertex, equations (52) are When, ( 74) f=V5, 17 = =0, and equations (55) reduce to (76) r L - Moreover, equations (45) and the similar ones for p lt q^ r t become P (76) ""^ T.. . The first two of equations (75) lead, 1 " by means 1 of (76), to sin w d^/~E fo shift\ PU VEG PV which follow also from 58. The third of in remarked equations (75) establishes the fact, previously 59, that the geodesic torsion in two orthogonal directions differs only in sign. FUNDAMENTAL RELATIONS The u const, are represented 175 variations of the direction-cosines X\, Y\, Z\ of the tangent to the curve by the motion of the point (1, 0, 0) of the trihedral T with fixed vertex. From (51) we have 5^1 cu dZ\ 5-5Ti du (78) du dv cv see that as a point describes a curve v = const. , namely the tangent to this curve undergoes an infinitesimal rotation consisting of two components, one in amount rdu about the normal to the surface and the other, From these equations we CM , qdu, about the line in the tangent plane perpendicular to the tangent to C u Consequently, by their definition, the geodesic and normal curvature of Cu are r/^/E and q/^/E respectively. Moreover, it is seen from (72) that as a point describes Cu the normal to the surface undergoes a rotation consisting of the com . ponents q du about the line in the tangent plane perpendicular to the tangent, and p du about the tangent. Hence, if Cu were a geodesic, the torsion would be p/VE to within the sign at least. Thus by geometrical considerations we have obtained the fundamental relations (76). We From that suppose now that the parametric system is any whatever. the definition of the differential parameters ( 37) it follows E= if G= denote functions similar to p, . Consequently general curve P,$, ^ q, r, for a v) = const. and whose tangent makes the angle which passes through with the moving z-axis, we have, from (45), M P^VA^T cos (79) ) T, &lt; ( /^ 1\ + sm ^ cos ^1 --, &lt;I&gt; \ds idco 4&gt; r/ 1\ cos &lt; p cos P , = H. V A.cf) T sin , \ds rj sn A 51) ff~ any other family of curves where by 2 (III, = A 2 X ((, T|T) and ty = const, defines Moreover, equations analogous to (44) are da _ bE cQ db __ cP aR dc ds aQ 176 T. FUNDAMENTAL EQUATIONS . If now in - da as da =- du H da dv dv as cu as we for db/ds replace the expressions for and dc/ds, we obtain / l l (f&gt;(p and from (47), and similarly Pd8=H \ A From du +p 1 dv), VA~C/&gt; Qds=H^/~K^&gt;(qdu (r + q^dv), Eds = 7^ du -f r^ dv). these equations and (79) : we derive the following funda mental formulas ( ds j TI = cos = sin d&lt;& \ as &lt;&(p du +p +p l l dv) + sin cos &lt; (q du + q l dv), (80) ds &lt;I&gt; (p du 1 dv) 4&gt; (&lt;? c?t* -f q^ dv), P sin oj du ds dv ds p ds of the last of equations (80) we shall express the geodesic curvature of a curve in terms of the functions E, F, G, of their derivatives, and of the angle 6 which the curve makes By means with the curve v tangent to the = const. If we take the rr-axis of the trihedral curve v = const., we obtain from the last of (80), 1 in consequence of (45), d0 ^/E du Pg /V G \Pffv dco\dv Po~~ ds ds dv/ds From (III, 15, 16) we obtain dv If 2 EG \ dv dv/ for p gu dv When this value and the expressions and p gv (IV, 57) are substituted in the above equation, we have the formula desired: - L __ 2dvds + 2H\du __ E dv ds EXAMPLES 1. A necessary is only point in the tangent, be the sufficient condition that the origin of the trihedral which generates a surface to which this plane is moving zy-plane and T that the surface be nondevelopable. PARALLEL SURFACES 2. Determine p so that the point of coordinates (p, describe a surface to which the x-axis of T is normal 0, 0) ; 177 with respect to T shall const. examine the case when the lines of curvature are 3. parametric and the x-axis is tangent to the curve v = When it is sphere, the parametric curves are minimal lines for both the surface and the necessary that or in this case the 77 = i, ifji = ii, q = ip, q\ = ipi\ parametric curves on the surface form a conjugate system, and the (cf. surface 4. is minimal 55). When the asymptotic lines on a surface form an orthogonal system, we must have in ^+^= is ^ ^+^= cosw Q&gt; which case the surface 5. minimal. When the lines of curvature are parametric, and the x-axis of T is tangent to the curve v 1 = const., equations (80) reduce to dw -j- T = sin p /I ( 1\ ) . sin as * cos 1 = cos 2 * H pi sin 2 * , 4&gt;, \PI PZ/ d&lt; p / q dp\ PZ w _ du ds Pz Pi \pi cv ds is p\ dPz dv\ q du ds) s, 6. When cos P2 resulting equation the second equation in Ex. 5 is reducible to differentiated with respect to the u dp ds sinw/ dw p \ 2\ T/ _ 2 s dp\ /du\ 2 dPi/du\ 2 dv dv \ds/ ds ds du \ds/ dpo du /du\ 2 dp /dv\ 8 7. On a surface a given curve makes the angle * with the x-axis of a trihedral T; the point of coordinates cos sin with reference to the parallel trihedral TO with fixed vertex, describes the spherical indicatrix of the tangent to the curve the direction-cosines of the tangent to this curve are P &lt;t&gt;, &lt;J&gt;, ; sin * sin w, cos &lt; sin w, cos w, is where w has the significance indicated in 49, and the linear element therefrom by means of (51) the second and third of formulas (80). 8. ds/p; derive The point #, whose coordinates with reference sin to T of Ex. 7 are * cos w, cos $cos w, sin w, describes the spherical indicatrix of the binormal to the given curve on the surface, and its linear element is ds/r; derive therefrom the first of formulas (80). 73. Parallel surfaces. We inquire under what conditions the t normals to a surface are normal to a second surface. In order that this be possible, there must exist a function such that the point 7", of coordinates (0, 0, Q, with reference to the trihedral describes a surface to which the moving 2-axis is constantly normal. Hence 178 FUNDAMENTAL EQUATIONS 8z we must have and consequently, by equations (51), t must may have any value whatever. We have, therefore, the theorem 0, = be a constant, which : If segments of constant face, these segments being other end points is length be laid off upon the normals to a sur measured from the surface, the locus of their a surface with the same normals as the given surface. These two surfaces are said to be parallel. Evidently there is an infinity of surfaces parallel to a given surface, and all of them have the same spherical representation. Consider the surface for which t has the value a, and call it$. follows that the projections on the axes of (51) on S have the values placement it From T of a dis r f du (82) = du 77 + ^dv -f (q du + q^dv) a, jj^dv (p du + Pidv) a. -f- Comparing these results with (53), we see that the displacements on the two surfaces corresponding to the same value of dv/du are parallel only in case equation (59) is satisfied, that is, when the point describes a line of curvature on S. But from a characteristic property of lines of curvature ( 51) it follows that the lines of curva ture on the two surfaces correspond. Hence we have the theorem : The tangents to corresponding lines of curvature of two parallel surfaces at corresponding points are parallel. From first (82) and (73) we have the following expressions for the Y fundamental quantities of /S : y or, in consequence of (IV, 78), (84) i / / PI p PARALLEL SURFACES The moving (82) it 179 trihedral for S can be taken same and thus the rotations are the parallel to trihedrals ; for both T for , and from follows that the translations have the values = + 00, li=i+ fl ?i&gt; ^ = i?-op, *?i=&gt;?i-api- analogous to (59), (60), (66), we obtain the fundamental equations for S in terms of the functions for S. Also from (73) we have the following expressions for the equations for On substituting in the second fundamental coefficients for S: (85) D = D-a, D = D -a&, D" = D" - ag. Since the centers of principal curvature of a surface and its are the same, it follows that parallel at corresponding points (86) Pi = Pi+ a &gt; P2 = P2+ a Suppose that we have a surface whose total curvature is constant and equal to 1/c 2 Evidently a sphere of radius c is of this kind, but later (Chapter VIII) it will be shown that there is a large group of surfaces with this property. We call them spherical surfaces. . From so that (86) if we have take a ^_ ^ = c, ( ^_ a = ^ ) we we obtain I+l-i. Pi P* c Hence we have the theorem ciated two surfaces of of Bonnet * : With every surface of constant total curvature 2 1/c there are asso mean curvature from it. 1/ey they are parallel to the former and at the distances :p c And conversely, is constant and different zero there are associated two parallel surfaces, one of which has from constant total curvature and the other constant mean curvature. With every surface whose mean curvature M moves over a surface S the corresponding centers of principal curvature M and M describe 74. Surfaces of center. As a point l z two surfaces S and S2 which are called the surfaces of center of S. Let C l and (72 be the lines of curvature of S through M, and D l and 1 , 7&gt; 2 the developable surfaces formed by the normals to *Nouvelles annales de mathematiques, Ser. 1, S along Cl Vol. XII (1853), p. 433. 180 FUNDAMENTAL EQUATIONS and C2 respectively. The edge of regression of D v denoted by I\, is a curve on Sl (see fig. 17), and consequently Sl is the locus of one set of evolutes of the curves Cl on S. Similarly $2 is the locus of a set of evolutes of the curves Cz on is S.$2 are said to constitute the evolute of S, also Evidently any surface parallel to S For this reason S1 and and S is their involute. an involute of S and S2 . l The line M^M^ as a generator of Dv is 2 tangent to I\ at and, as a generator of D it is tangent to F at Hence it is a 2 z common tangent of the surfaces S and S From this it follows that the developable surface D meets S along T and envelopes Sz along a curve F 2 Its generators are con , . M Mv z . l 1 { l . sequently tangent to the curves conjugate to Fg ( 54). In particular, the generator -flfjJfg is tangent to directions of at Jf2 are conjugate. Similar results follow from the considera Z&gt; F F 2 and T 2 : 2, and therefore the tion of 2 . Hence On the surfaces of center of a surface S the curves corresponding to the lines of cur vature of S form a conjugate system. Since the developable the tangent plane to $at FIG. 17 D M 1 envelopes is *Sf , 2 2 plane at it is 1 Z( tangent to D l all along 25), and consequently determined by M^MZ and the tangent to C[ at M. Hence the M plane to is D l at this point. But 2 MM the tangent the tangent normal to S at M 2 is parallel to the tangent to manner, the normal to S l at M C2 at M. In like l is parallel to the tangent to C l at M. Thus, through each normal to S we have two perpendicular planes, of which one is tangent to one surface of center and the other to the second surface. But each of these planes is at the same time tangent to one of the developables, and is the osculating plane of its edge of regression. Hence, at every point of one of these curves, the osculating plane is perpendicular to the tangent plane to the sheet of the evolute upon which it lies, and so we have the theorem : The edges of regression of normals to the developable surfaces formed by a surface along the lines of curvature of one family are the SURFACES* OF CENTER 181 geodesies on the surface of center which is the locus of these edges ; and the developable surfaces formed ly the normals along the lines of curvature in the other family envelope this surface of center along the curves conjugate to these geodesies. In the following sections we shall obtain, in an analytical manner, the results just deduced geometrically. 75. Fundamental quantities over the surface for surfaces of center. As the trihe dral T moves surface of center Sr Let S the point (0, the lines of curvature on 0, p^ describes the S be parametric, and the z-axis of T be J. tangent, to the curve v const. Now \ / & L J. L ri f^ -I. A rz f\ so that the first two of equations (48) may be put in the form JI, ___,,=,_ (88) -,_ __. The projections on the moving axes of the absolute displace on S are found ment of J/J corresponding to a displacement of M from (51) to be (89) Bx l = 0, S^ = (rj l p^pj dv = V6r ( 1 )dv, Szj = dp r Hence the (90) linear element of S l is ds*= dri.+ / pV Q(I-^]dfi consequently the curves p^= const, on Sl are the orthogonal tra = const., which are the edges of regression, jectories of the curves v of the developables of the normals to S along the lines of I\, curvature v = const. Let us consider the moving trihedral T^ for Sl formed by the = const, and p l const, at M^ and the nor tangents to the curves v mal at this point. From (89) it follows that the first tangent has the same direction and sense as the normal to S, and that the sec ond tangent has the same direction as the tangent to u = const, on S, the sense being the same or different according as (1 p l /p 2 ) is 182 FUNDAMENTAL EQUATIONS And the normal to positive or negative. as the tangent to v = const, on , Sl has the same direction and the contrary or same sense accordingly. If then we indicate with an accent quantities referring to the moving trihedral Tv we have (a =c, where (89) it l (1 =bj is c = ea, e is 1 according as follows that pjp^ positive or negative. From (92) When the values (91) are substituted in equations for 2\ similar to equations (47), we find Since / is zero, it follows from v (76) that the curves = const, are found geometrically. various fundamental equations for St may now be obtained of the by substituting these values in the corresponding equations geodesies, as The preceding sections. Thus, from (73) we have which follow likewise from (90); and also Hence the parametric curves on S form a conjugate system l (cf. 54). The equation of the lines of curvature may be written and the equation of the asymptotic directions is ^^-41^=0. pl$u p? du SURFACES OF CENTER The expression for 183 is K^ the total curvature of S^ (98) ^-L-Jj. ~du From (80) the curve on and (93) it follows that the geodesic curvature at l of Sl which makes the angle l with the curve v = const, &lt;& M through M^ is given by Hence is, the radius of geodesic curvature of a curve p l = const., that is a right angle, has, in consequence of a curve for which t &lt;J&gt; In accordance with 57 the center of geo (87), the value p l p in the desic curvature is found by measuring off the distance p l 2 negative direction, on the 2-axis of the trihedral T. Consequently . , /&gt; M z is this center of curvature. Hence we have the following theo rem of Beltrami: The centers of geodesic curvature of the curves p^ = const, on St and of p 2 = const, on S., are the corresponding points on $2 and Sl respectively. For the sheet$2 of the evolute we (90 d** find the following results : ) = E\- du 2 + is the equation of the lines of curvature (96 ) r**^ is the equation of the asymptotic lines ^ 5* -BS" is the expression for the total curvature 8ft *"5FS-5 dv 184 FUNDAMENTAL EQUATIONS : In consequence of these results we are led to the following theorems of Ribaucour,* the proof of which we leave to the reader A necessary and sufficient condition that the lines of curvature upon and S2 correspond is that p p 2 = c (a constant); then K^ K^ 2 1/c and the asymptotic lines upon S and $2 correspond. A necessary and sufficient condition that the asymptotic lines on Sl and S2 correspond is that there exist a functional relation between p^ and p 2 Sj l = , 1 . complementary to a given surface. We have just seen that the normals to a surface are tangent to a family of geo desies on each surface of centers. Now we prove the converse 76. Surfaces : The tangents to to a family of geodesies on a surface S l are normal an infinity of parallel surfaces. Let the geodesies and their orthogonal trajectories be taken for const, and u = const, respectively, and the param the curves v eters chosen so that the linear element has the form refer the surface to the trihedral formed by the tangents to the parametric curves and the normal, the z-axis being tangent to the curve v = const. Upon the latter we lay off from the point l denote the other extremity. of the surface a length X, and let We M P As M moves over the surface the projections of the corresponding 1 displacements of (99) P have the values d\ + du, V +X X ~l = u dv, - X (y,du + q,dv). In order that the locus of P be normal to the lines J^P, we must have d\ + du = 0, and consequently X where c + , denotes the constant of integration whose value determines a particular one of the family of parallel surfaces. If the directioncosines of M^P with reference to fixed axes be v Yv Z^ the X coordinates of the surface /S, for which c= 0, are given by where x^ y^ z l are * the coordinates of Mr (1872), p. 1399. Comptes Rendus, Vol. LXXIV COMPLEMENTAKY SUKFACES The is 185 S. surface S 1 is , one of the surfaces of center of In order to find the other,$2 we must determine X so that the locus of trihedral. P tangent at P to the zz-plane of the moving The con dition for this is Hence S2 is given by y\ ^L V i *!&gt; ^i ^2 ^i / y&lt;i aV^ dw gVg. dM tfM and the principal radii of S are expressed by ( 10 ) Pl = u, Pz =udu Bianchi* calls S2 the surface complementary to S l for the given geodesic system. Beltrami has suggested the following geometrical proof of the above theorem. Of the involutes of the geodesies v const, we consider the single infinity which meet S^ in one of the orthogonal = UQ shall prove that the locus of these curves trajectories u . We is a surface S, normal to the tangents to the geodesies. Consider the tangents to the geodesies at the points of meeting of the latter with a second orthogonal trajectory u = u r The segments of these and the points P of tangents between the points of contact meeting with S are equal to one another, because they are equal M to the length of the geodesies between the curves moves along an Hence, as u lines JfP, orthogonal trajectory l describes a second orthogonal trajectory of the latter. moves along a geodesic, describes an involute Moreover, as M UQ and u = u r u = u of the P M P necessarily orthogonal to MP. Since two directions on are perpendicular to JfP, the latter is normal to S. which is S EXAMPLES 1. Obtain the results of 73 concerning parallel surfaces without making use of the moving trihedral. 2. Show that the surfaces parallel to a surface of revolution are surfaces of revolution. *Vol. I, p. 293. 186 FUNDAMENTAL EQUATIONS 3. Determine the conjugate systems upon a surface such that the corresponding curves on a parallel surface form a conjugate system. 4. Determine the character of a surface S such that its asymptotic lines corre spond to conjugate lines upon a parallel surface, and find the latter surface. 5. Show that when the parametric curves are the lines of curvature of a surface, the characteristics of the 7/z-plane and zz-plane respectively of the moving trihe dral whose x-axis is tangent to the curve v = const, at the point are given by (r du (r du + ri dv) y + r\ dv) x q (z pi) p%) du dv pi(z = = 0, ; and show that these equations give the directions on the surfaces Si and S2 which are conjugate to the direction determined by dv/du. 6. Show that for a canal surface ( 29) one surface of centers is the curve of centers of the spheres and the other 7. is the polar developable of this curve. The surfaces of center of a helicoid are helicoids of the same axis and parameter as the given surface. GENERAL EXAMPLES 1. If t is an integrating factor of ^Edu-\--- imaginary function, then A 2 log V# is equal to the total curvature of the quadratic form E du 2 + 2 Fdudv + Gdv 2 all the functions in the latter being real. , v^ dv, and t the conjugate the only real surface such that its first and second fundamental quadratic forms can be the second and first forms respectively of 2. Show that the sphere is another surface. 3. Show that there exists a surface referred to its lines of curvature with the is linear element ds 2 = eau (du* + du 2 ), where a is a constant, and that the surface developable. 4. When a minimal surface is referred to its minimal lines hence the lines of curvature and asymptotic lines can be found by quadratures. 5. formed with respect Establish the following identities in which the differential parameters are to the linear element : . 6. Prove that (cf. Ex. 2, p. 1G6) A2 * = - 4k VJE^PI I- + -}f* -VGCV\PI -(- + -}- x( f) l + -V PZ/ \PI GENEEAL EXAMPLES 7. 187 Show that z2 + 2 ?/ + z2 = 2 "FT + Ai TF, (23). the differential 8. parameter being formed with respect to A necessary and sufficient condition that all the curves of is an orthogonal system on a surface be geodesies 9. that the surface be developable. If the geodesic (different from zero) all curvature of the curves of an orthogonal system is constant over the surface, the latter is a surface of constant negative curvature. 10. When the linear element of a surface ds 2 is in the form = du 2 + 2 cos u dudv + dto 2 , the parametric curves are said to form an equidistantial system. case the coordinates of the surface are integrals of the system Show that in this du dv dy dz dz dy du dv cz dx dz dx du dv ex dy _ dx dy_ cu dv du dv cu dv dv cu cu dv dv cu 11. If the curves v = const., u = const, form an equidistantial system, the tan the lines joining the centers of geo gents to the curves v = const, are orthogonal to desic curvature of the curves u = const, and of their orthogonal trajectories. 12. Of all ment occur of its vertex when M the displacements of a trihedral T corresponding to a small displace over the surface there are two which reduce to rotations they describes either of the lines of curvature through the point, and the M ; axes of rotation are situated in the planes perpendicular to the lines of curvature, each axis passing through one of the centers of principal curvature. 13. When a surface 3 a 2 irl d M is referred to its lines of curvature, the curves defined by + 3 g2 dv duz dv + 3p? dudv 2 du + P? dv s dv = du in these directions at a point are possess the property that the normal sections or are superosculated by their circles of curvature (cf Ex. 9, p. 21 straight lines, These curves are called the superosculating lines of the surface. Ex. . ; 6, p. 177). 14. Show Show : that the superosculating lines on a surface and on a parallel surface correspond. 15. that the Gauss equation (64) can be put in the following form due to Liouville du dv p gu du\ p gv ) where p gu and p gv denote the radii of geodesic curvature of the curves u = const, respectively. 16. v = const, and When may Ex. 15 the parametric curves form an orthogonal system, the equation of be written _!\_J:___L VE du\ pgv ) p%u P%V 17. Determine the surfaces which are such that one of them and a parallel divide harmonically the segment between the centers of principal curvature. 188 FUNDAMENTAL EQUATIONS 18. Determine the surfaces which are such that one of them and a parallel admit of an equivalent representation (cf. Ex. 14, p. 113) with lines of curvature . corresponding. 19. Derive the following properties of the surface a2 _ ab ft2 uv Va 2 u b 62 v u (i) + V& 2 M+ w2 U _ Va 2 a &2 u Vu 2 w ; a2 _ + ; v (ii) (iii) the parametric lines are plane lines of curvature the principal radii of curvature are p\ = p% = i&gt;, u algebraic of the fourth order the surfaces of center are focal conies. (iv) the surface is ; 20. Given a curve C upon gents to M. N are perpendicular to C at its points at which the tangent plane to the ruled surface S which a surface S and the ruled surface formed by the tan the point of each generator M ; is perpendicular to the tan gent plane at when the ruled the center of geodesic curvature of C at surface is developable, this center of geodesic curvature is the point of contact of with the edge of regression. is ; M to S M MN 21. If two surfaces have the same spherical representation of their lines of in con curvature, the locus of the point dividing the join of corresponding points is a surface with the same representation. stant ratio 22. The locus of the centers of geodesic curvature of a line of curvature is an evolute of the latter. 23. Show that when E, is F, G D, ; IX, IX of a surface are functions of a single parameter, the surface a helicoid, or a surface of revolution. CHAPTER VI SYSTEMS OF CURVES. GEODESICS 77. Asymptotic lines. We have said that the asymptotic lines on a surface are the double family of curves whose tangents at any point are determined in direction by the differential equation D du + 2 D dudv + 2 2 D"dv = 0. These directions are imaginary and distinct at an elliptic point, real and distinct at a hyperbolic point, and real and coincident at a from our discussion, parabolic point. If we exclude the latter points the asymptotic lines neces may be taken for parametric curves. condition that they be parametric is (55) sary and sufficient (1) A D = D n =Q. (IV, 25) Then from we have _ _D^_ where p as thus !_ denned is called the radius of total curvature. ) The Codazzi equations (V, 13 may be written of which the condition of integrability a is ri2i i d ri2i 2 4 &lt; &gt; ail -hail is r In consequence of (V, 3) this equivalent to In tives. 64 we saw that K is a function of E, F, G and their deriva coefficients Hence equations form (3) are two conditions upon the 2 , of a quadratic (6) E du + 2 Fdudv + G dv 2 189 190 that it SYSTEMS OF CURVES may be the linear element of a surface referred to its asymp totic lines. When these conditions are satisfied the function D is given by (2) to within sign. tween a surface and its follows the theorem : Hence, if we make no distinction be symmetric with respect to a point, from 65 A the linear element of its coefficients necessary and sufficient condition that a quadratic form (6) be a surface referred to its asymptotic lines is that satisfy equations (3); when they are satisfied, the surface is unique. is For example, suppose that the total curvature of the surface every point, thus j a2 the same at where a is a constant. In this case equations (3) are cv cu cv du which, since H 2 ^ 0, are equivalent to dv du u alone, and G a function of v alone. By a suitable choice of the parameters these two functions may be given the value a2 so that the linear element of the surface can be written Hence E is a function of , (7) ds 2 = a 2 (du 2 + 2 cos o&gt; dudv + dv 2), where w denotes the angle between the asymptotic lines. Thus far the Codazzi equa tions are satisfied and only the Gauss equation (V, 12) remains to be considered. When the above values are substituted, this becomes (8) sinw. dudv to every solution of this equation there corresponds a surface of constant Hence curvature whose linear element a2 is given by (7). The equation of the lines of curvature is du 2 dv 2 = 0, so that if we put 2 M!, u u -f v v = 2 !, the quantities u\ and v\ are parameters of the lines of cur vature, and in terms of these the equation of the asymptotic lines is du} dv} = 0. Hence, when either the asymptotic lines or the lines of curvature are known upon a surface of constant curvature, the other system can be found by quadratures. the asymptotic lines are parametric, the Gauss equations (V, 7) may be written When ^ + ^ + 5^1 = du du dv 0, y () /OX i dv 72 ai fru l dv~ ASYMPTOTIC LINES where a, 5, 191 v, ax , b 1 are (5) determinate functions of u and da Jo and in consequence of (10) if du real linearly inde the equations =/&gt;(!*, Conversely, two such equations admit three v), pendent integrals f^u, &lt;* f z (u, v), f 3 (u, v), 1 ^ ) # =/2 (M, V), =/l(w V), define a surface on which the parametric curves are the asymptotic lines. For, by the elimination of a, 6, a^ b from the six equations l obtained by replacing 6 in (9) by x, y, z we get n ju 7 uy 7 v*&gt; 7 = f\ " = 0, which are equivalent As an example, to (1), in consequence of (IV, 2, 5).* consider the equations is auv + bu + cv + d, where a, b, c, rf are constants. of the choosing the fixed axes suitably, the most general form of the equations surface may be put in the form of which the general integral By From these equations is it is seen that that the surface a quadric. all the asymptotic lines are straight lines, so Moreover, by the elimination of u and v from these equations we have an equation of the form z the surface is a paraboloid. = ax- -\- 2hxy + by 2 + ex + dy. Hence 78. Spherical representation of asymptotic lines. From (IV, 77) we have that the totic lines, total curvature of a surface, referred to its asymp may be expressed in the form (ii) A ff-" =-^ + where = (o o^ 2 , the linear element of the spherical represen- tation being da 2 = (odu 2 + 2 &dudv * Darbonx, Vol. I, p. 138. It should be noticed that the above result shows that the condition that equations (9) admit three independent integrals carries with it not only (10) but all other conditions of integrability. 192 SYSTEMS OF CURVES this result From and (2) it follows that * 5! # Hence the fundamental /- relations (IV, 74) reduce to Jf o\ Jf n^/"" _ r? "- and equations (V, 26) may be written " A ^ ^X P / **dX e$X\ 3X__^__ /{ \ _ du cv Moreover, the Codazzi equations (V, 27) are reducible to Consider now the converse problem : To determine the condition to be satisfied by a parametric system serve as the spherical of lines on the sphere in order that they representation of the asymptotic lines may on a surface. First of bility. all, Then x, values of equations (15) must satisfy the condition of integrap is obtainable by a quadrature. The corresponding y, z found from equations (14) and from similar ones are the coordinates of a surface are parametric. upon which the asymptotic lines For, it follows from (14) that du ** dv dv ; Furthermore, p is determined to within a constant factor is conse true of x,y,z\ therefore the surface is unique quently the same to within homothetic transformations. Hence we have the following theorem of Dini : A necessary and sufficient condition that a double family of curves lines upon the sphere be the spherical representation of the asymptotic upon a surface is that &, &,$ satisfy the equation V d ri2 the corresponding surfaces are homothetic transforms of one another, and * as is their Cartesian coordinates are found by quadratures. The choice seen from P = D /ft gives the surface symmetric to the one corresponding to (12), and hence may be neglected. (14), FORMULAS OF LELIEUVKE 193 When equations (1) obtain, the fundamental equations (V, 28) lead to the identities |in = 121 (18) r 11 v 2 ri2V 1 |22 = r 22V_ r = ri2i ri2v rm ri2i l2/ - ri2v i rny \2J r22j llJ- py llJ The (3) third and fourth of these equations are consequences also of and (15). Tangential equations. In conse quence of (V, 31) equations (14) may be put in the form 79. Formulas of Lelieuvre. where e is 1 according as the curvature of the surface is positive or negative. (20) Hence, if we put v2 ^ = V-/)X, = V^epY, vs = ^/ * : we have the following formulas due to Lelieuvre ~ du dv _ du dv _ dudv v* v* Bulletin des Sciences Mathtmatiques, Vol. XII (1888), p. 126. 194 ^ SYSTEMS OF CURVES and (15) By means , of (V, 22) we find from (20) that the common Consequently i^, i/ is ratio of these equations i =^ -- &lt;& 2, v s are solutions of the equation ,. . , , ,. dudv Conversely, \^/p dudv : we have the theorem G-iven three particular integrals v^ 2 i&gt; 2 , v s of an equation of the form (22) d -^- = X0, where \ is any function whatever of u and v ; the surface, whose co ordinates are given by the corresponding quadratures (21), has the and the total curvature of the parametric curves for asymptotic lines, surface /93\ is measured by K~ , to For, from (21), it is readily seen that v^ i/ 2 v z are proportional if these the direction-cosines of the normal to the surface. And direction-cosines be given by (20), we are brought to (19), from which we see that the conditions (16) are satisfied. Take, for example, the simplest case j / = i 0, \ and three solutions //; The coordinates of the surface are and similar expressions for y and 0,- /r \ / / V{ 0| (U) -(- Yi(V). (I = 1 *j &, &) 9 Q\ j z. When, in particular, we take (u) = a,-w + &, $i (v) the expressions for x, y, z are of the form auv surface is a paraboloid. + bu = a to + + cv + /S,-, d, and consequently the the asymptotic equations (V, 22, 34) it follows that when are the tangential coordinates X, Y, Z, lines are parametric, solutions of the equations From W HVd0 18^-llJ du \ I ^2/j ^22") 30 CONJUGATE SYSTEMS EXAMPLES 1. 195 Upon a nondevelopable surface straight lines are the only plane asymptotic lines. 2. The asymptotic lines on a minimal surface form an orthogonal isothermal system, and their spherical images also form such a system. 3. Show that of all the surfaces with the linear element ds2 = du* + (u 2 + a 2 ) du2 , one has the parametric curves for asymptotic lines and another for lines of curva ture. Determine these two surfaces. 4. The normals to a ruled surface along a generator are parallel to a plane. Prove conversely, by means of the formulas of Lelieuvre, that if the normals to a surface along the asymptotic lines in one system are parallel to a plane, which differs with the curve, the surface is ruled. 5. If we take v^ = u, vz = D, j&gt; 3 = 0(u), the formulas of Lelieuvre define the most general right conoid. 6. If the asymptotic lines in one system on a surface be represented on the sphere by great circles, the surface is ruled. 80. Conjugate purpose now systems of parametric lines. Inversions. It is our to consider the case where the parametric lines of a surface form a conjugate system. As thus defined, the character istics of the tangent plane, as it envelops the surface along a curve const, at their points are the tangents to the curves u of intersection with the former curve ; and similarly for a plane const. enveloping along a curve u v = const., = = The analytical condition that the parametric lines form a conju is ( gate system (25) It follows 54) D =0. 7) that x, y, z are solu tions of immediately from equations (V, an equation of the type (26) cudv b are J^ + a^ + 6^0, du dv v, where a and functions of u and or constants. By : a method similar to that of 77 we prove v) be the converse theorem Iffi(u, v),/2 (M, v),/3 (w, tions of three linearly independent real solu an equation of * the type (26), the equations (27) = /&gt;,*), y=f (u,v), z *=f (u,v) t define a surface upon which . the parametric curves form a conjugate I, system.* 9 * Cf Darboux, Vol. p. 122. 196 SYSTEMS OF CURVES have seen that the lines of curvature form the only orthog onal conjugate system. Hence, in order that the parametric lines on the surface (27) be lines of curvature, we must have We F^fa + tyty+tete^^ du du du dv dv dv But this is equivalent to the condition that xz +yz +z 2 also be a solution of equation (26), as is seen by substitution. have the theorem of Darboux * : Hence we If x, y, z, # 2 -{- 2 ?/ -f- z* are particular solutions of an equation of the form (26), the first three serve for the rectangular coordinates of a surface, upon which the parametric lines are the lines of curvature. Darboux theorem : f has applied this result to the proof of the following When of the a surface is face, the lines of curvature of the latter. transformed ly an inversion into a second sur former become lines of curvature By radii, definition is an inversion, or a transformation by reciprocal given by * where (29) c denotes a constant. (if From these equations ) , we find that + f + z*) (x? + y + z, = c z, and by solving for x, y, ( ,f ~ If, " yt+*l now, the substitution Q *?+**-- , __ " ?+*+? , be effected upon equation (26), the resulting equation in or will 4 admit, in consequence of (29) and (30), the solutions x v y^ z v c and therefore (31) is of the form * Vol. I, p. 136. t Vol. I, p. 207. SURFACES OF TRANSLATION and consequently x* + yl + the theorem. z? is a solution of (31), 197 Moreover, equation (26) admits unity for a particular solution, which proves As an example, we consider a cone of revolution. Its lines of curvature are the elements of the cone and the circular sections. When a transformation by recip rocal radii, whose pole is any point, is applied to the cone, the transform S has two families of circles for its lines of curvature, in consequence of the above theorem and the fact that into circles. circles and straight is lines, not through the pole, are transformed the envelope of a family of spheres whose cen ters lie on its axis, and also of the one-parameter family of tangent planes the latter pass through the vertex. Since tangency is preserved in this transformation, the surface S is in two ways the envelope of a family of spheres all the spheres Moreover, the cone ; : of one family pass through a point, and the centers of the spheres of the other family lie in the plane determined by the axis of the cone and the pole. 81. Surfaces of translation. The simplest form of equation (26) is dudv in which case equations x , (27) are of the type (32) =U 1 +r y= u^+V e = V&gt;+r where U^ Z7 Us are any functions whatever of u alone, and V^ F2 F3 any functions of v alone. This surface may be generated by effecting upon the curve , X l= UV Vl= U 21=^3 a translation in which each of its points describes a curve con gruent with the curve *, = F,, y,= r,, Z2 . =F 3 . In like manner the it may curve in which each of first be generated by a translation of the second its points describes a curve congruent with curve. For this reason the surface is called a surface of translation. From this method of generation, as also from equa tions (32), it follows that the tangents to the curves of one family at their points of intersection with a curve of the second family are parallel to one another. Hence we have the theorem of Lie * : The developable enveloping a surface of translation along a gener ating curve is a cylinder. * Math. Annalen, Vol. XIV (1879), pp. 332-367. 198 SYSTEMS OF CURVES is Lie has observed that the surface defined by (32) the mid-points of the joins of points on the curves the locus of be that these two sets of equations define the same curve in terms of different parameters. In this case the surface is the It may locus of the mid-points of all chords of the curve. These results are only a particular case of the following theorem, whose proof is immediate : The locus of the point which divides in constant ratio the joins of points on two curves, .or all the chords of one curve, is a surface of translation ; in the latter case the curve is an asymptotic line of the surface. When the equations of a surface of translation are of the form x=U, y = V, 9=Ui+V the generators are plane curves whose planes are perpendicular. leave it to the reader to show that in this case the asymptotic lines can be found by quadratures. We 82. Isothermal-conjugate systems. When the asymptotic lines upon a surface are parametric, the second quadratic form may be written X dudv. When the surface is real, so also is this quadratic form. Therefore, according as the curvature of the surface is posi tive or negative, the parameters u and v are r conjugate-imaginary or real. We when consider the former case and put and v l are real. In terms of these parameters the second Hence the curves M = const., quadratic form is $$du+dvj). == const, form a vl conjugate system, for which u^ t (33) D= D", D =0. Bianchi * has called a system of this sort isothermal-conjugate. Evi an ana dently such a system bears to the second quadratic form lytical relation similar to that of * Vol. an isothermal-orthogonal system 107. I, p. ISOTHERMAL-CONJUGATE SYSTEMS to the first quadratic form. 199 that DD" EGF* be I) 2 &gt; positive, In the latter case it was only necessary and the analogous requirement, namely by surfaces of positive curvature. Hence the theorems for isothermal-orthogonal systems ( 40, 41) are translated into theorems concerning isothermal-conjugate systems 0, is satisfied all by substituting In particular, Z&gt;, IX, D" for E, F, G respectively in the formulas. we remark that if the curves u = const., all v = const. = const., on a surface form an isothermal-conjugate system, isothermal-conjugate systems are given by the quantities u l and v l being defined by other real v1 u = const., u i+ where &lt; il \ () t( u i w) is is any analytic function. negative and When the curvature of the surface we put in the second quadratic form \dudv, it becomes \(du* dv*). In this case (34) 1 D= -!&gt;", D =0. Hence the curves w const, and i\ = const, form a conjugate sys tem which may be called isothermal-conjugate. With each change of the parameters u and v of the asymptotic lines there is obtained a new isothermal-conjugate system. Hence if u and v are parame ters of an isothermal-conjugate system upon a surface of negative curvature, the parameters of all such systems are given by where &lt; and ^r denote arbitrary functions. parameters for a surface are such that It is evident that if the (35) D" = -, V Z/=0, where u and v respectively, then by a which does not change the parametric curves we can reduce (35) to one of the forms (33) or (34). Hence equa tions (35) are a necessary and sufficient condition that the para metric curves form an isothermal-conjugate system. Referring to are functions of U and V change of parameters 200 77, SYSTEMS OF CUKVES we see that the lines of curvature upon a surface of constant form an isothermal-conjugate system. When equation (35) is of the form (33) or (34), we say that the parameters u and v are isothermal-conjugate. total curvature 83. Spherical representation of conjugate systems. When the parametric curves are conjugate, equations (IV, 69) reduce to - GI? ~-~W^ FDD" ~W ., ED m -JiCD these equations and (III, 15) it follows that the angle between the parametric curves on the sphere is given by COS , o&gt; From = &= - qp F - == if COS O), where the upper sign corresponds to the case of an elliptic point and the lower to a hyperbolic point. Hence we have the theorem: The angles between two conjugate directions at a point on a sur and between the corresponding directions on the sphere, are equal face, or supplementary, according as the point is hyperbolic or elliptic. curves form a conjugate system, the Codazzi equations (V, 27) reduce to the When parametric and equations (V, 26) become (dx D / X du dv dX Hence, when a system of curves upon the sphere is given, the a problem of finding the surfaces with this representation of system reduces to the solution of equations (36) and conjugate been determined quadratures of the form (37), after X, Y, Z have of a Riccati equation. By the elimination of D by the solution or from equations of the second order. D" (36) we obtain a partial differential equation CONJUGATE PARAMETRIC SYSTEMS From the general equations (V, 28) we derive the when the parametric curves form a conjugate system 201 following, * : (fii\_8io S D = li/ ~w D" D" rnv ~\ir i2/ f!2\ /22\ = aiogD" /22V ~fo~ ~i2/ (38) /12\ = fllV f!2 ilJ 22 "T12/ l2/ ll = D /22V ~I7 ll/ !2V The prob 84. Tangential coordinates. Pro jective transformations. lem of finding the surfaces with a given representation of a con jugate system is treated more readily from the point of view of tangential coordinates. For, from (V, 22) and (V, 34) it is seen that -3T, r, Z, and W are particular solutions of the equation &lt;&gt; Hence every A", F, Z solution of this equation linearly independent of determines a surface with the given representation of a conjugate system, and the calculation of the coordinates 2-, y, z does not involve quadratures ( 67). Conversely, it is readily seen that if the tangential coordinates satisfy an equation of the form d*e ha 30 --h b 00 f-c#=0. -- du dv the coordinate lines form a conjugate system on the surface. As an example, we determine the surfaces whose lines of curvature are repre sented on the sphere by a family of curves of ccinstant geodesic curvature and their orthogonal trajectories. If the former family be the curves v = const., and if the linear element on the sphere be written da- 2 Edu 2 -f Gdu 2 we must , have (IV, 60) where (u) is a function of v alone. made equal to unity. By a change of the parameter v this In this case equation (39) is reducible to may be du *Cf. Bianhi, Vol. I, p. 167. 202 The general SYSTEMS OF CUKVES integral of this equation is where and u v denotes a constant value of t&gt;, and U and V are arbitrary functions of u respectively. Hence : the sphere by The determination of all the surfaces whose lines of curvature are represented on a family of curves of constant geodesic curvature and their orthogonal two quadratures. all trajectories, requires In order that among tation of a conjugate system there the surfaces with the same represen may be a surface for which the system is conjugate, isothermal-conjugate, and the parameters be isothermalit is necessary that equations (36) be satisfied by iX ssiD, according as the total curvature is positive or negative. In this case equations (36) are 01og.D_/12\ The fllV is alocrT) 12V 22V u condition of integrability a rri2V ^Llim2)rdl2mi)J When this is satisfied my-i may z 17121 /22V ] D be found by quadratures, and then the coordinates, by (37). Hence we have the theorem: A and the sphere represent necessary and sufficient condition that a family of curves upon an isothermal-conjugate system on a surface, v be isothermal-conjugate parameters, is that satisfy (40); then the surface is unique to within its homothetics, , &lt;^, that u and and its coordinates are given by quadratures. ,-&lt; The following theorem concerning directions the invariance of conjugate : and asymptotic is lines is to due to Darboux When totic lines a surface subjected a protective transformation or a transformation by reciprocal polars, conjugate directions and asymp are preserved. We When prove this theorem geometrically. Consider a curve C on a surface the developable circumscribing the surface along C. a projective transformation is effected upon S we obtain a and D S19 corresponding point with point to S, and C goes into a curve CjUpon S^ and D in to a developable l circumscribing Sl along surface D PROJECTIVE TRANSFORMATIONS ; 203 moreover, the tangents to C and (7X correspond, as do the gener Cj ators of and r Since the generators are in each case tangent to D D the curves conjugate to C and Cl respectively, the theorem is proved. In the case of a polar reciprocal transformation a plane corre sponds to a point and vice versa, in such a way that a plane and a point of into of Sv C D go into a point and a plane through it. Hence S goes D1 D into Cv and the tangents to C and generators into the generators of J\ and tangents to r Hence the it into , &lt;7 theorem is proved. EXAMPLES 1. Show that the parametric curves on the surface 2 - u* + Fs u + V* where the I7 s are functions of u alone and the F s of v alone, form a conjugate system. F where U\, U are functions of 2. On the surface x U\V\, y = U V\, z u alone and FI, F2 of v alone, the parametric curves form a conjugate system and - ^I : +F U+V I _ - C7 _ AZ "1TTF" Z 2 , 2 the asymptotic lines can be found 3. (cf. by quadratures. form an equidistantial system The generators p. 187). of a surface of translation Ex. 10, 4. Show that a paraboloid is a surface of translation in more than one way. is 5 . The locus of the mid-points of the chords of a circular helix is a right helicoid. Q. Discuss the surface of translation which the locus of points dividing in constant ratio the chords of a twisted cubic. 7. From (28) it follows that 2 dx? _ + , dy? + a dz? , = c4 (dx (x* 2 + dy 2 + dz 2 ) : + y* + is 22) 2 consequently the transformation by reciprocal radii conformal. Determine the condition with the linear element 8. to 9/ be satisfied by the function u so that a surface &lt;&gt;,79, = a? (cos 2 w du 2 + that if shall have the total curvature I/a 2. Show the parametric curves are the lines of curvature, they 9. form an isothermal-conjugate system. A necessary and sufficient condition that the linear element of a surface referred to a conjugate system can be written is upon the curves on the unit sphere same that the parametric curves be the characteristic lines. Find the condition imposed in order that they may represent these lines. 10. Conjugate systems and asymptotic lines are transformed into curves of the sort when a surface is transformed by the general protective transformation X = D ABC y = D * = D Xi, y\, z\. where A, 2&gt;, C, D are linear functions of the new coordinates 204 GEODESICS 85. Equations of geodesic lines. We is have defined a geodesic to zero at every point is ; be a curve whose geodesic curvature its conse osculating plane at any point quently tangent plane to the surface. perpendicular to the follows that every geodesic an integral curve of the differential equation From (IV, 49) it upon a surface is (41) , ds . ds/\ ds 2 ds*/ \ Y-4- ds /V \ + ds ds T 7&lt; ds)l\du W -- 2 to/\ds) Y 4i du ds ds fi f 2 dv \ds _/ \ If the \\ds/[2 du \ds) (\ 2 -i- dv ds ds \dv -\( 2 ~ Y= du)\ds fundamental identity \ds/ +2+ ds ds v, s \ds/ which gives the relation between w, entiated with respect to s, we have along the curve, be differ du d*u d*v\ dv d*u d*v L+ dv If this / , 2 4-24du)\ds) ds \ dv du) ds \dsj -4--= dv equation and (41) be solved with respect to 72 2 72 \ and \ T,tfu ~d*v\ F- + G -^ ds ) we 1^ obtain ,, . ds*/ 4- F ds 2 +F ds - 2 Y+ dv ds ds +( \dv -dv 2 du \ds) 2 du)\ds/ V V= 2 cFv (W_lMy&lt;faV dGdu IdG/dv \ = 2 Bv \ds) Q dt If these equations \9* 2 dv )\ds) du ds ds - be solved with respect to 2), and ^ we have, in consequence of (V, d*u riii/&lt;fov 2 i 12 \ (fw ^+{ 22 V Y=o dv . (42) 111 /du\* . rt fl21 f 22- EQUATIONS OF GEODESICS Every pair of solutions v 205 of these equations of the form u =/j(), determines a geodesic on the surface, and s is its arc. =/2 (), But a geodesic may be defined in terms of u and v alone, without s. the introduction of the parameter curve, then If v = &lt;f&gt;(u) defines such a dv du d * v /du\* .,d*u ct Substituting these expressions in (42) and eliminating 2 have, to within the factor (du/ds) , u &gt; we (43) &lt;// From (42) it follows that when du/ds is zero, Hence, when this condition the geodesies on a surface ; is not satisfied, equation (43) defines it is satisfied, and when equations (43) and u = const, define them. From exists a , the theory of differential equations it follows that there unique integral of (43) which takes a given value for for u MO and whose first derivative takes a given value Hence we have the fundamental theorem : u =U Q . , Through every point on a surface there passes a unique geodesic with a given direction. As an example, we found ( consider the geodesies on a surface of revolution. We have and 46) that the linear element of such a surface referred to its meridians parallels is of the form (45) ds2 = (1 + 2 ) du 2 -f wW, If where z (46) = (u) is the equation of the meridian curve. we put and indicate the inverse of (47) this equation 2 &lt;Zs by u ^(wi), , we have = d^ + fdw 2 still and the meridians and tions (42) are (48) ( parallels are the parametric curves. For this case equa ^i - w*y = " &lt;W W &lt;W + -= ds ds - 206 The first GEODESICS integral of the second is 2 &lt;// .do ds = c, where c is a constant. Eliminating ds from we have (49) c this equation and (47), and integrating, f / - = 2 w , _ + C2 where Ci is a constant. The meridians Hence we have the theorem : v = const, correspond to the case c = 0. The geodesies upon a surface of revolution referred can be found by quadratures. It to its meridians and parallels should be remarked that equation (49) defines the geodesies upon any surface applicable to a surface of revolution. 86. Geodesic parallels. Geodesic parameters. From (43) it fol lows that a necessary and sufficient condition that the curves v = const, on a surface be geodesies is that parametric system be orthogonal, this condition makes it 2 be a function of u alone, say E = U By replacing necessary that If the E . I U du by u we do not change the parametric lines, and E becomes equal to unity. (51) And G the linear element has the form d**=du*+Gdi?, in general is where a function of both u and v. From this it follows that the length of the segment of a curve v the curves u = U Q and u = u^ is given by = const, between /! I ds u = X! I du V0 = u^u^ WO Since this length is independent of v, it follows that the segments of all the geodesies v = const, included between any two orthog onal trajectories are of equal length. In consequence of the funda mental theorem, we have that there is a unique family of geodesies which are the orthogonal trajectories of a given curve C. The above results enable us to state the following theorem of Gauss * : If geodesies be drawn orthogonal to a curve C, measured upon them from trajectory of the geodesies. C, the locus of their * L.C., p. 25. and equal lengths be ends is an orthogonal GEODESIC PARALLELS 207 This gives us a means of finding all the orthogonal trajectories of a family of geodesies, when one of them is known. And it sug these trajectories. Referring gests the name geodesic parallels for to 37, we see that these are the curves there called parallels, and so the theorem of 37 may be stated thus : A (52) necessary and sufficient condition that the curves is = const, &lt;f&gt; be geodesic parallels that A,* =/(*), formed with respect to the linear any function. In order that = 0, curves measured from the curve geodesic is where the differential parameter element of the surface, and be the length of the it f denotes &lt;f&gt; &lt;/&gt; is necessary and sufficient that (53) A,* = l. (52), a Moreover, we have seen that when a function satisfies new function satisfying (53) can be found by quadrature. this function is When taken as shall call u, the linear element has the form (51). v geodesic parameters. In this case we u and 87. Geodesic polar coordinates. The following theorem, due to Gauss,* suggests an important system of geodesic parameters: the locus of the If equal lengths be laid offfrom a point P on the geodesies through P, end points is an orthogonal trajectory of the geodesies. In proving the theorem we take the geodesies for the curves and let u denote distances measured along these geo v = const., from P. The points of a curve u = const, are consequently at the same geodesic distance from P, and so we call them geodesic circles. It is our problem to show that this parametric system is desies orthogonal. From the choice of it u we know that E = \, At P, follows that Zv F is dv independent of u. are zero. dv that and hence from (50) is for u = 0, the derivatives for Consequently F and G are zero u = 0, and the former, being independent of w, is always zero. Hence the theorem is proved. We consider such a system and two points Q (u, 0), J/^w, vj on the geodesic circle of radius u. The length of the arc Q 1 M MM *L.c.,p. 24. 208 is GEODESICS / ^Gdv. As u approaches zero the ratio Jo to the geodesies approaches the angle between the tangents at v= and v = v^ If 6 denotes this angle, we have given by .. a 6 = lim v P = r I dv. u =o /o In order that v be 6, it is necessary and sufficient that L : = 1. du Ju= These particular geodesic coordinates are similar to polar coordi nates in the plane, and for this reason are called geodesic polar coordinates. The above results may now be stated thus The necessary and sufficient conditions that a system of geodesic coordinates be polar are (54 ) L J=o =0, L -i. Bu J M=0 It should be noticed, however, that it may be necessary to limit the part of the surface under consideration in order that there be a one-to-one correspondence between a point and a pair of coordinates. For, it may happen that two geodesies starting defined from P meet again, in which case the second point of meeting would be by two sets of coordinates.* For example, the helices are geodesies on a cylinder ( 12), and it is evident that any number of them can be made to pass through two points at a finite distance from one another by varying the angle under which they cut the elements of the cylinder. Hence, in using a system of geodesic polar coordinates with pole at P, we consider the portion of the surface inclosed by a geodesic circle of radius r, where r is such that no two geodesies through P meet within the circle, t When (55) the linear element is in the form (51), the equation of Gauss (V, 12) reduces to denotes the total curvature of the surface at the pole P, which by hypothesis is not a parabolic point, from (54) and If Q K (55) it follows that _ o ~ L &lt;* _ K * * Notice that the pole is tDarboux also (Vol. II, p. 408) a singular point for such a system, because H* = for u = 0. shows that such a function r exists; this is suggested by 94. GEODESIC POLAK COOKDINATES Therefore, for sufficiently small values of w, . 209 we have O Hence the circumference and area have the values * of a geodesic circle of radius u = /* 2 I 2 ITU + Jo where e t and e2 denote terms of orders higher than the third and fourth respectively. EXAMPLES 1. 2. Find the geodesies of an ellipsoid of revolution. The equations x u, linear element ds2 = v (du 2 on the former are represented by parabolas on the 3. Find the total curvature of a surface US T XV* (a - -- latter. 2 + v define a representation of a surface with the y dv 2 ) upon the xy-plane in such a way that geodesies = with the linear element - v 2) du? + 2 w&gt; dudv a2 ( _ W2 _ - + (a 2 - u2 ) dv* 9 W2)2 where 4. R and a are constants and integrate the equation of geodesies for the surface. twisted curve is A a geodesic on its rectifying developable. its 5. 6. The evolutes of a twisted curve are geodesies on polar developable. Along a geodesic on a surface of revolution the product of the radius of the parallel through a point and the sine of the angle of inclination of the geodesic with the meridian is constant. 7. Upon a Upon a surface of revolution a curve cannot be a geodesic and loxodromic cylindrical. at the 8. same time unless the surface be helicoid the orthogonal trajectories of the helices are geodesies and the other geodesies can be found by quadratures. 9. If a family of geodesies and their orthogonal trajectories on a surface form an isothermal system, the surface is applicable to a surface of revolution. 10. The radius varies as the cube of the distance of of curvature of a geodesic on a cone of revolution at a point from the vertex. P P 88. Area of a geodesic triangle. With the aid of geodesic polar coordinates Gauss proved the following important theorem f : 180 of the sum of the angles of a triangle formed on a surface of positive curvature, or the deficit from 180 by geodesies The excess over * Bertrand, Journal de Mathematiques, Ser. 1, Vol. XIII (1848), pp. 80-86. t L.c., p. 30. 210 of the GEODESICS sum of the angles of such a triangle on a surface of negative curvature, is measured by the area of the part of the sphere which represents that triangle. In the proof of this theorem Gauss geodesic lines in the made use of the equation of form where 6 denotes the angle which the tangent to a geodesic at a const, through the point. This point makes with the curve v When equation is an immediate consequence of formula (V, 81). the parametric system is polar geodesic, this becomes (57) M = - -*. Let ABC be a triangle whose sides are geodesies, and let a, /3, 7 denote the included angles. From (IV, 7 3) it follows that the inclosed area on the sphere is given by (58) d= f f// dudv = Ipcff dudv, 1 according as the curvature is positive or negative, where e is and the double integrals are taken over the respective areas. Let A be the pole of a polar geodesic system and AB the curve v = 0. From (55) and (58) we have rr -- dvdu , o Jo Jo o ] In consequence of (54) we have, upon integration with respect to u, which, by (57), is equivalent to &= For, at v e f Jo dv + e dd. Jn-ft f B the geodesic at = 0, and C it BC makes the angle TT fi with the curve makes the angle 7 with the curve v = a. Hence (7i we have = e(a + /3 + 7 - TT), which proves the theorem. AREA OF A GEODESIC TRIANGLE 211 Because of the form of the second part of (58) Ci may be said to measure the total curvature of the geodesic triangle, so that the above theorem may also be stated thus : The over total curvature of a geodesic triangle is equal to the excess 180, or deficit from 180, of the sum of the angles of the tri is positive or negative. angle, according as the curvature The extension is of these theorems to the case of geodesic polygons straightforward. In the preceding discussion it has been tacitly assumed that all the points of the can be uniquely denned by polar coordinates with pole at A. We triangle shall show that this theorem is true, even if this assumption is ABC not made. If the theorem is not true for ABC, it cannot be true for both of the triangles ABD and ACD obtained by joining A and the middle point of BC with a geodesic (fig. 18). For, by adding the results for the two triangles, we should have the AD ABC. Suppose that it is not true for A BD. Divide the latter into two triangles and apply the same reason should obtain a triangle as ing. By continuing this process we theorem holding for we please, inside of which a polar geodesic system would not uniquely determine each point. But a domain can be chosen about a and any other point point so that a unique geodesic passes through the given point of the domain.* Consequently the above theorem is perfectly general. small as By means of the above result we prove the theorem : Tivo geodesies on a surface of negative curvature cannot meet in two points and inclose a simply connected area. a Suppose that two geodesies through a point A pass through second point B, the two geodesies inclosing a simply connected portion of the surface (fig. 19). two segments Take any geodesic cutting these AB in points C and D. Since the four angles ACD, ADC, BCD, BDC are /D together equal to four right angles, the sum of the angles of the two triangles ADC arid four right angles by the sum of the angles at A and B. Therefore, in consequence of the above theorem of Gauss, the total curvature of the surface cannot be negative at all points of the area ADBC. BDC exceed On the contrary, it curvature geodesies through a point meet again in general. * Darboux, Vol. II, p. can be shown that for a surface of positive In 408 ; cf. 94. 212 fact, the GEODESICS exceptional points, if there are any, lie in a finite portion which may consist of one or more simply connected parts.* For example, the geodesies on a sphere are great circles, and all of these through a point pass through the diametrically of the surface, Again, the helices are geodesies on a cylinder evident tnat any number of them can be made to ( 12), pass through two points at a finite distance from one another by varying the angle under A hich they cut the elements of the cyl opposite point. and it is inder. is Hence the domain restricted oi a system of polar geodesic coordinates on a surface of oositive curvature. 89. Lines of shortest length. Geodesic curvature. : We are now in a position to prove the theorem that only one geodesic passes through them, the segment of the geodesic measures the shortest dis If two points on a surface are such tance on the surface between the two points. Take one and the of the points for the pole of a polar geodesic system 0. The coordinates of the geodesic for the curve v = second point are (u^ 0). The parametric equation of any other and the curve through the two points is of the form v = ^&gt;(w), length of its arc is f Jo Since tegral G is &gt; 0, the value of this in necessarily greater than By means of equation (57) we is proved. Wj, derive another definition of geo and the theprem desic curvature. Consider two points and upon a curve C, and the unique geodesies g, g tangent to C at these points (fig. 20). Let P denote the point of intersection of g and g\ and Sty the angle under which they cut. Liouville f has called Sty the angle of M M geodesic contingence, because of its analogy to the ordinary angle of contingence. Now we shall prove the theorem: The limit of the ratio Sty/Ss, as M approaches M, is the geodesic curvature of C at M. by H. v. Vol. * For a proof of this the reader is referred to a memoir XCI (1881), pp. 23-53. t Journal de Mathtmatiques, Vol. XVI (1851), p. 132. Mangoldt, in Crelle, GEODESIC ELLIPSES AKD HYPERBOLAS In the proof of this theorem we take for parametric curves the given curve (7, its geodesic parallels and their geodesic orthogonals, the parameter u being the distance measured along the latter from C. Since the geodesic g meets the curve v = v orthogonally, the angle under which it meets v W approaches dd given by approaches angles of the triangle M PQ approaches 18C. Jf, = v may be denoted by ?r/2 y/&gt;7), 4- SO. As M and the sum S^/r of the Hence approaches dQ, so that we have Ss ds v- which is the expression for the geodesic curvature of the curve C. 90. Geodesic ellipses parametric lines for a surface is parallels. and hyperbolas. An important system of formed by two families of geodesic Such a system may desic parallels of two curves be obtained by constructing the geo C^ and (72 which are not themselves , geodesic parallels of one another, or by taking the two families of geodesic circles with centers at any two points F^ and 2 Let u and F . v measure the geodesic distances from C^ and C2 or from , F l and F 2 . They must be we must have solutions of (53). Consequently, in terms of them, Q. ^ EG-F*~~ EG-F*~ If, as usual, o&gt; denotes the angle between these parametric lines, we have, from (III, 15, 16), -U v= n = U * &gt; sin 2 ft) T? JF = COSft) . ? sin &lt;o so that the linear element has the following form, due to Weingarten /rrix : (59) ,9 du ds*=- + 2 2 cos . ft) dudv &) -f- dv - 2 sin a 2 Conversely, when the linear element is reducible to this form, u and v are solutions of (53), curves are geodesic parallels. and consequently the parametric v =u In terms of the parameters u v and v^ denned by u l i\, the linear element (59) has the form = i^-h v l and ( 60) df^^aL + JuL. sm . o o S 2 214 GEODESICS i^ The geometrical significance of the curves of parameter is seen when the above equations are written The curves w and vl are respectively the loci of the sum and difference of whose geodesic distances from C1 points and Cg, or from t and z are constant. In the latter case these 1 = const, and vl = const, F F , and hyperbolas in the plane, the points Fl and F2 corresponding to the foci. For this reason they are called geodesic ellipses and hyperbolas, which names are given curves are analogous to ellipses likewise to the curves u^ are = const., v l = const., when . the distances at measured from two curves, Cl and C2 From (60) follows once the theorem of Weingarten * A system of geodesic ellipses and hyperbolas is orthogonal. : By means of (61) equation (60) can be transformed into (59), thus proving that when the linear element of a surface is in the form (60), the parametric curves are geodesic ellipses and hyperbolas. If 6 denotes the angle which the tangent to the curve v^= const, through a point makes with the curve v (III, 23) that cos u = cos : ft) i sin ... = = const., , it follows from sin Hence we have the theorem Given any two systems of geodesic parallels upon a surface ; corresponding geodesic ellipses included by the former. 91. Surfaces of Liouville. the and hyperbolas bisect the angles Dini f inquired whether there were any surfaces with an isothermal system of geodesic ellipses and a sur hyperbolas. A necessary and sufficient condition that such face exist is that the coefficients of (60) satisfy a condition of the form (41) ^ 8in2 = V r r/i CQ8 , | , where U^ and denote functions of u l and i\ respectively. In this case the linear element may be written V i *\ *Ueber die Oberfliichen fur welche einer der beiden Hauptkrummungshalbmesser eine Function des anderen ist, Crelle, Vol. LXII (1863), pp. 160-173. t Annali, Ser. 2, Vol. Ill (1869), pp. 269-293. SURFACES OF LIOUVILLE By the change of parameters defined by 215 1 this linear element is transformed into (63) ds* = U+r ( 2 8) (du* + dvt), where U 2 and V z are functions of u 2 and v 2 respectively, such that Conversely, if the linear element is in the form (63), it may be changed into (62) by the transformation of coordinates Surfaces whose linear element first is reducible to the form (63) were studied by Liouville, and on that account are called surfaces of Liouville.* To this class belong the surfaces of revolution and the quadrics ( 96, 97). We may state the above results in the is form : When the linear element of a surface in the Liouville form, the parametric curves are geodesic ellipses and hyperbolas ; these systems are the only isothermal orthogonal families of geodesic conies.^ 92. Integration of the equation of geodesic lines. Having thus discussed the various properties of geodesic lines, and having seen the advantage of knowing their equations in finite form, we return to the consideration of their differential equation and derive certain theorems concerning its integration. in the first place, that we know a particular first inte Suppose, gral of the general equation, that is, a family of geodesies defined by an equation of the form (64) From _2_ (IV, 58) it follows that \ 2 M and N must d_ satisfy the equation \ 2 / du \^/EN 2 - 2 * t ___ FN-GM I 2 FM-EN FMN + GM / ^ \^EN - 2 FMN + GM / p. 345. p. 208, for = Journal de Mathematiques, Vol. XI (1846), The reader is referred to Darboux, Vol. II, under which a surface is of the Liouville type. a discussion of the conditions 216 GEODESICS know that there exists a func In consequence of this equation we tion &lt;f&gt; denned by ._. (DO dc#&gt; ) = ====================== EN-FM d&lt;f&gt; FN-GM ==: ? -=============================== du ^EN*-1FMN+GM* A^=l. to V EN*-2FMN+GM* Moreover, we find that (66) From (III, 31) and (65) it follows that the curves &lt;/&gt; = const, are the orthogonal trajectories of the given geodesies, and from (66) it is seen that measures distance along the geodesies from the curve = 0. Hence we have the theorem of Darboux * &lt; : When a one-parameter family trajectories of geodesies is defined by a differ ential equation of the first order, the finite equation of their orthogonal can be obtained by a quadrature, which gives the geodesic at the parameter same time. Therefore, geodesies is the general first integral of the equation of known, all the geodesic parallels can be found by when when is now the converse problem of finding the geodesies the geodesic parallels are known. Suppose that we have a solution of equation (66) involving an arbitrary constant a, which not additive. If this equation be differentiated quadratures. consider We with respect to a, we get (67) where the element. the curves differential parameter is But &lt;/&gt; this is a necessary formed with respect to the linear and sufficient condition ( 37) that = const, and the curves (68) ^ = const.= a da form an orthogonal system. are geodesies. Hence the curves defined by (68) In general, this equation involves two arbitrary a and a which, as will now be shown, enter in such constants, a way that this equation gives the general integral of the differ , ential equation of geodesic lines. * Lemons, Vol. II, p. 430; cf. also Bianchi, Vol. I, p. 202. EQUATIONS OF GEODESIC LINES Suppose that a appears (69) in in equation (68), 217 : and write the latter thus f (u, (67) l v, =a a) r , which case equation becomes (70) A (*,^)=0. direction of each of the curves (69) is The ratio given by is -^- / If this be independent of a, so also by (70) Write the latter in the form the ratio -^-1 36 /dd&gt; If this equation and a, (66) be solved for and cu dv we obtain values independent of so that a would have been additive. Hence / iJL/l_, an d therefore a direction at involves a, and cu / dv If then aQ a point (MO v ) determines the value of a; call it be such that = so also does , . ^. (, r\ *,) 4, the geodesic -v/r (%, v, passes through the point (w ?; ) and ) has the given direction at the point. Hence all the geodesies are defined by equation (68), and we have the theorem: a =^ , Criven a solution of the equation II A 1 &lt;^ = 1, involving an arbitrary constant a, in such a way that da involves a; the equation da for all values of a arc of the geodesies is the finite equation of the geodesies, and the is measured by (/&gt;.* By means to Jacobi : of this result we establish the following theorem due If a first integral known, the of the differential equation of geodesic lines be finite equation can be found by one quadrature. integral is Such an of the - form dv du Cf. = ^(u, v, a), Darboux, Vol. II, p. 429. 218 GEODESICS is where a (64), the an arbitrary constant. c/&gt;, As this equation is of the form function defined by = is P , (# + a solution of equation (66). As involves a in the manner the finite equation of the specified in the preceding theorem, d(f&gt; geodesies is = a. The surfaces of Liouville 93. Geodesies on surfaces of Liouville. ( We of the theorem of Jacobi. 91) afford an excellent application in the form * take the linear element (71) ds 2 = (U- 2 V) (U?du + 2 V?dv ), which evidently is tion (66) becomes no more general than (63). In this case equa When this equation is written in the form u*\du. one sees that differential equa belongs to the class of partial tions admitting an integral which is the sum of functions of u In order to obtain this integral, we put each side and v alone, it f equal to a constant a and integrate. (72) This gives &lt;/&gt; = C l\ -\/Ua du of geodesies is f F! Va Vdv. Hence the equation (73) If 6 denotes the angle which a geodesic through a point makes with the line v = const, through the point, tan 6 it follows from (III, 24) and (71) that = y- dv * Cf. Darboux, Vol. Ill, p. 9. t Forsyth, Differential Equations (1888), p. 310. SURFACES OF LIOUVILLE If the value of 219 dv/du from equation following 2 (73) be substituted in this first equation, we obtain the integral of the Gauss equation (56): (74) ?7sin is + Fcos 2 = a. This equation due to Liouville. * EXAMPLES 1. On portional to the difference between the a surface of constant curvature the area of a geodesic triangle is pro sum of the angles of the triangle and two right angles. 2. Show that for a developable surface the be found by quadratures. first integral of equation (56) can 3. Given any curve C upon a surface and the developable surface which is the envelope of the tangent planes to the surface along C; show that the geodesic curvature of C is equal to the curvature of the plane curve into which C is trans formed when the developable is developed upon a plane. 4. When the plane is whose foci are at the distance 2 c apart, the linear referred to a system of confocal ellipses and hyperbolas element can be written 5. A d&lt;p ds 2 6. 2 necessary and sufficient condition that be a perfect square. , be a solution of Ai0 = 1 is that If 1, Ai0 =: = did + 62 where 6\ and 62 are functions of u and v, is a solution of the curves 0i = const, are lines of length zero, and the curves B\a -j- 62 = const, " are their orthogonal trajectories. 7. the equation Ai0 satisfies When the linear element of a spiral surface is in the form ds 2 = e 2 (du 2 = 1 admits the solution e Z7i, where U\ is a function of ? 2 -\U" do 2 ), M, which an equation of the first order whose integration gives thus all the geodesies on the surface. 8. For a surface with the linear element where (f&gt; of v alone, the equation Ai0 = 1 admits the solution and ^ 2 requiring the solu the determination of the functions tion of a differential equation of the first order and quadratures. V and V\ are functions v ( ), u\fsi (v) -f ^2 \f&gt;i 9. If denotes a solution of Ai0 = 1 involving a nonadditive constant a, the linear element of the surface can be written ca where (0, } indicates the mixed differential parameter (III, 48). *i.c.,p. 348. 220 GEODESICS 94. Lines of shortest length. Envelope of geodesies. can go a step farther than the first theorem of 89 and show that whether one or more geodesies pass through two points and 2 on a sur l face, the shortest distance on the surface between these points, if it We M M measured along one of these geodesies. Thus, =f(u) and v =f (u) define two curves C and Cl passing through the points M^ M# the parametric values of u at the points being u^ and u 2 The arc of C between these points has the length exists, is let v l . (75) = v 2Fv +Gv 2 du, where denotes the derivative of v with respect to venience we write the above thus : u. For con (76) s= f Jiti we put 1 *4&gt;(ui v, v )du. Furthermore, f (u)=f(u) + , ea&gt;(u), where w(u) is a function of u vanishing when u is equal to u and M 2 and e is a constant whose absolute value may be taken so small l that the curve C l will lie in Hence the length of the arc MM 1 any prescribed neighborhood of 2 C. of C v l is = fc/tt (u, v -f- e tw, -f- e CD ) C?M. Thus j is a function of e, reducing for it is e = to s. Hence, in order that the curve C l be the shortest of pass through M and J/2 , the neai?-by curves which necessary that the derivative of s l all 0. with respect to e be zero for e = This gives On the assumption that admits a continuous first derivative in the interval (u^ tives, u z ), and continuous &lt;f&gt; first and second deriva the left-hand member of this equation may be integrated by parts with the result " 1 wl-2- /& \v d , n ^lauaaO: du v d&lt;l&gt;\ LINES OF SHORTEST LENGTH for o&gt; 221 &&gt; vanishes when u equals u^ and u 2 . As the function is arbi is trary except for the above conditions upon it, this equation * equivalent to the following equation of Euler : (77) du this result is applied to the particular When tkm (75), form of (f&gt; in equa- we have d F+ Gv __ dv I / _ cv 71 I _ dv ?? " _ ~ which readily reducible to equation (43). shortest distance between two points, if existent, measured along a geodesic through the points. This geodesic is Hence the if is is the surface has negative total curvature at all points. unique For other surfaces more than one geo may pass through the points if the latter are sufficiently far apart. shall now investigate the nature of this desic We problem. Let v f(u, a) define the family of geo , desies through a point J/ (w v ), and let v g (u) be the equation of their envel = ope let Cl and C2 (fig. 21), and and M,,(u v z ) denote their points of contact with the MI(UV vj is greater than Jf Jfr The envelope. Suppose that the arc 2 to distance from measured along C and ^ equal to Q (o. We consider two of the geodesies , MM M l &lt;~is D= f JttQ f J^ If is 3/2 is considered fixed and a. M l variable, the position of the latter determined by The variation of D with M 1 is given by j da J M=MI * 21 (Lausanne, 1744) Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, chap, ii, cf. Bolza, Lectures on the Calculus of Variations, p. 22 (Chicago, ; 1994). 222 GEODESICS u., B u t for u = u=u and f = g and f = g first x \ consequently the last term f)f is zero. Integrating the u=u ( member hy parts, and noting 26), we have that -f- is zero for Since C^ hence D the geodesies through the evolute of a curve does to a family of normals to the curve. not a geodesic, for at each point of it there Moreover, the curve ("is a geodesic, the expression in parenthesis is zero, and does not vary with r This shows that the envelope of a point bears to them the relation which is M is is an arc connecting M^ and 2 tangent a geodesic. Hence there In this way, by taking different which is shorter than the arc of &. M points M and on & we obtain any number of arcs connecting Q which are shorter than the arc of C2 each consisting of an arc z of a geodesic such as C and the geodesic distance M^ It is then of to a point distance from Q necessarily true that the shortest l , M M l M M Q V C2 beyond M within the arc that the arc MM M M of Q 2 is lies not measured along 2 However, when a domain can be chosen about (72 so small Q 2 &lt;7 . M M , &lt;7 2 is shorter than the arc M M of any other curve within the domain Another historical and passing through these points.* this problem associated with problem is the following : t Given an arc length joining C A and B, and joining two points A, B on a surface ; inclosing with Co a given area. It is evident that to find the curve of shortest The area is given by CClfdudv. two functions r &lt; M and N can be found in an infinity of ways such that __ 8N ~ du dM dv By the application of Green s theorem //lldudv where the Since =- we have and is taken around the contour of the area. our problem reduces to the determination of a curve C along fixed, the which the integral C *Mdu + Ndv is constant, and whose arc AB, that is, /A last integral is curvilinear CQ is * For a more complete discussion of this problem the reader is referred to Darboux, Vol. Ill, pp. 86-112; Bolza, chap. v. Vol. (18.30), tin fact, it was in the solution of this problem that Minding (Crelle, function to which Bonnet (Journal de I Ecole Poly technique, p. 297) discovered the Vol. XIX (1848), p. 44) gave the name geodesic curvature.. V ENVELOPE OF GEODESICS integral 223 C V2? -f 2 Fv + Gv ^du, is a minimum. From the calculus of variations we know that, so far as the differential equations of the solution is concerned, this as finding the curve C along which the integral is the same problem B JA is f VE + 2 Fv + Gv *du s + c(M + Nv )du is a minimum, c being a constant. Euler equation for this integral d / F 4- GV \ + cto &gt; ^ cu + ^.!? dy_ _ 2 -f 6rv / V E + 2 .FV -f Crt/ 2 with the formula of Bonnet (IV, 56), we see that C has con stant geodesic curvature 1/c, and c evidently depends upon the magnitude of the area between the curves. Hence we have the theorem of Minding :* Comparing this result In order that a curve C joining two points shall be the shortest which, together with a given curve through these points, incloses a portion of the surface with a given area, it is necessary that the geodesic curvature of C be constant. GENERAL EXAMPLES 1. When the parametric curves on the unit sphere satisfy the condition 12 1 ) a I 12 2 ) ( 12 i J dv - n ) j 12 ) 2 r is surface whose total curvature they represent the asymptotic lines on a 2. When the equations of the sphere have the form (III, 35), the parametric is (1 curves are asymptotic and the equation (22) + wu) 2 =CU vV 20, of which the general integral is ^ \f/ 2 ^(K) + ^(.)_ 1 + uv where 3. (u) and (v) denote arbitrary functions. The sections of a surface by all the planes through a fixed line and on L, form a conjugate system. 4. the curves of contact of the tangent cones to the surface whose in space, vertices are L Given a surface of translation x = u, y = v, z =f(u) + 0(0). Determine the lines of so that ( Pl + P2 )Z = const., where Z functions/ and the z-axis, and determine the angle which the normal makes with denotes the cosine of the curva ture on the surface. 5. Determine the relations between the exponents x m&lt; and n - t in the equations = U mi V ni , y = Um *V"* t z = U m3 V Hs , so that on the surface so defined the parametric curves shall form a conjugate sys can be found by quadratures. tem, and show that the asymptotic lines *Z.c., p. 207. 224 6. GEODESICS The envelope (Ui of the family of planes + Fi)z + (Uz + V2 )y + (Us + F8 )z + (U* , + F4 = ) 0, where the U" s are functions of u alone and the F s of is a surface upon which the parametric curves are plane, and form a conjugate system. 7. The condition that the parametric curves form a conjugate system on the envelope of the plane x cos u is + y sin u -f z cot v =/(u, u), that / be the sum of a function of u alone and of v alone ; in this case these curves are plane lines of curvature. 8. Find the geodesies on the surface of Ex. 7, p. 219, and determine the expres sions for the radii of curvature and torsion of a geodesic. 9. representation of two surfaces upon one another is said to be conformalconjugate when it is at the same time conformal, and every conjugate system on one surface corresponds to a conjugate system on the other. Show that the lines of A curvature correspond and that the characteristic lines also correspond. 10. Given a surface of revolution z defined by = ucosu, y = wsinw, zf(u), and the function (i) A and c are constants a conf orjual-conjugate representation of the surface is defined by MI sin I?!, z\ = upon a second surface x\ = MI cosi, y\ where ; &lt;f&gt;(ui) V CUi, C log Ui = - du where F denotes the function of M found by solving (i) for &lt; . 11. If two families of geodesies cut under constant angle, the surface is developable. 12. If a surface with the linear element ds* = (aM 2 - bv 2 - c) (du? + cto 2 ), where a, 6, c are constants, is represented on the xy-plane geodesies correspond to the Lissajous figures defined by by u = x, v = y, the where A, -Z?, C are constants. 13. When there is upon a surface more than one family of geodesies which, together with their orthogonal trajectories, form an isothermal system, the curva ture of the surface is constant. 14. If the principal normals of a curve meet a fixed straight line, the curve is a the case where geodesic on a surface of revolution whose axis is this line. Examine the principal normals meet the line under constant angle. GENERAL EXAMPLES 15. 225 representation representation of two surfaces upon one another is said to be a geodesic when to a geodesic on one surface there corresponds a geodesic on the other. Show that the representation is geodesic when points with the same A parametric values correspond on surfaces with the linear elements where the IPs are functions 16. of u alone, the F s of v alone, and h is a constant. A surface with the linear element ds 2 = (w* - v 4 ) [0 /itself. where 17. is any function whatever, admits of a geodesic representation upon face necessary and sufficient condition that an orthogonal system upon a sur regarded as geodesic ellipses and hyperbolas in two ways, is that when the curves are parametric the linear element be of the Liouville form in this case A may be ; these curves may be so regarded in an infinity of ways. 18. Of all the curves of equal length joining two points, the one which, together with a fixed curve through the points, incloses the area of greatest extent, has con stant geodesic curvature. draw 19. Let T be any curve upon a surface, and at two near-by points P, the geodesies g, g perpendicular to T; let C be the curve through conjugate and Q the intersection of the tangents to g to gr, P" the point where it meets g P P , and g at the limiting position of Q, as of geodesic curvature of T at P. P" ; P and P f approaches P, is the center 20. such a Show that if a surface S admits of geodesic representation upon a plane in way that four families of geodesies are represented by four families of par each geodesic on the surface is allel lines, p. 209). represented by a straight line (cf Ex. . 3, CHAPTER QUADRICS. VII RULED SURFACES. MINIMAL SURFACES Elliptic coordinates. 95. Confocal quadrics. Two quadrics are confocal when tions coincide. the foci, real or imaginary, of their principal sec Hence a family of confocal quadrics is defined by the equation a) where u such that (2) is -A+/-+-A-1, u u u a 2 b 2 c 2 the parameter of the family and a, 6, c are constants, a 2 &gt; b 2 &gt; c 2 . For each value of (1) defines a quadric u, positive or negative, less than a 2 equation , which is 2 &gt; ellipsoid when c u &gt; oo, &gt; &gt; 2 u an hyperboloid of one sheet when b 2 u an Ian hyperboloid of two sheets when a &gt; c*, &gt; b 2 . As u zero. approaches Hence the smallest axis of the ellipsoid approaches 2 the surface u = c is the portion of the zy-plane, c 2 counted twice, bounded by the 2 ellipse (4) a 2 c 2 b 2 2 c 2 Again, the surface u twice, = b is the portion of the ^-plane, counted bounded by the hyperbola which contains the center of the curve. define the focal ellipse and focal hyperbola Equations (4) and (5) of the system. CONFOCAL QUADEICS Through each point (x, y, z) 227 of the family; they are determined roots of the equation 2 in space there pass three quadries by the values of w, which are (6) &lt; (u) = (a - u) (b - u) (c - u) - x 2 2 2 -y Since &lt;/&gt; 2 2 (a -u) (c 2 2 - u) - z 2 &lt; 2 2 (a - u) (c - u) - u) (b - u) = 0. 2 2 (b 2 2 (a ) &lt; 0, c/&gt; (b ) &gt; 0, (c ) &lt; 0, &lt;/&gt; (, oo) &gt; 0, the roots of equation (6), denoted by u l9 u the following intervals : , w 3 are contained in (7) a 2 &gt; u, &gt; b 2 , b 2 &gt;u,&gt; c 2 , c 2 &gt; u^&gt; oo. seen that the surfaces corresponding to u v w 2 u s are respectively hyperboloids of two and one sheets and an ellipsoid. Fig. 22 represents three confocal quadrics; the curves on the (3) it is , From ellipsoid are lines of cur vature, and on the hyperboloid of one sheet they are asymptotic lines. From the definition of u v w2 u s , it follows that &lt; (u) is u). equal to (u^u) (u 2 in (6) is u) (u s When by is (/&gt; replaced this expression and u given successively the 2 2 2 values a b c we obtain * , , , FIG. 22 or = (8) = (*)(&gt;-?) These formulas express the Cartesian coordinates of a point in space in terms of the parameters of the three quadrics which pass through the point. These parameters are called the elliptic coordinates of the point. It is evident that to each set of these * Kirchhoff , Mechanik, p. 203. Leipsic, 1877. 228 QUABEICS coordinates there correspond eight points in space, one in each of the eight compartments bounded by the coordinate planes. in (8) be made constant, and the one of the parameters = others u^ %, where i Ar, be allowed to vary, these equations j If - . t = define in parametric form the surface, also defined in (1), 1^.= by equation which u has this constant value u r The parametric curves const, are the curves of intersection of the given const., u k Uj to the quadric and the double system of quadrics corresponding parameters If ,r\\ and uk 12 , we put o 7* 2 . ng .. the equation of the surface becomes (10) - + ^ + -=1, a b c (8) and the parametric equations " reduce to u)(a v) la(a N fb (a - b) (a - c) u)(b v) (11) y ~~~ \ N - a) (b - c) (b u) (c (c (b Z ~~ \c(c v) b} a}(c Moreover, the quadrics which cut (10) in the parametric curves have the equations: a (12) u b u c u = 1, 1. av bv cv = In consequence of or (11) define (3) and (9) we have that equations (10) (13) - an ellipsoid when a&gt;u&gt;b&gt;v&gt;c&gt;0, an hyperboloid of one sheet when a&gt;u&gt;b&gt;Q&gt;c&gt;v, an hyperboloid of two sheets when a&gt;0&gt;b&gt;u&gt;c&gt;v. FUNDAMENTAL QUANTITIES 96. 229 culation (14) Fundamental quantities we find from (11) *. for central quadrics. By direct cal U ( U -V) _A ^_V(V-U) /() f(0) where for the sake of brevity we have put (15) = 4 (a - 6) (b - 0) (c - 6). : We derive also the following (a-b)(a(16) = and lobe \ ^ a)(c b) u v abc u v (17) JL&gt; \ uv f(u) N uv f(v) Since ture. F and D And are zero, the parametric curves are lines of curva since the change of parameters (9) did not change the curves, we have the theorem parametric : The quadrics of a confocal system cut one another along curvature, and the three surfaces lines of a point cut one another through orthogonally at the point. This result is illustrated (17) lobe by fig. 22. From ., (14) and we have 1_ ~ \abc 1 Pl ~ 1 /&gt;^ NtfV all points, p2 N^ 3 2 ~wV is _ abc Hence the ellipsoid and hyperboloid of two sheets have positive whereas the curvature negative at all curvature at points of the hyperboloid of one sheet. If formulas (16) be written \abc &gt; x abc z uv a \ uv b uv is c the distance W from the center to the tangent plane (19) 230 QUADEICS : Hence The tangent planes to a central quadric along a curve, at points of which the total curvature of the surface is the same, are equally distant from the center. to the (18) we see that the umbilical points correspond The conditions (13) v. of the parameters such that u values From = show that an ellipsoid is b, and c for an hyperboloid of two sheets, whereas there are no real umbilical points for the hyperboloid of one sheet. When these this common value of u and v for values are substituted in (11), these points on the ellipsoid we have as the coordinates of \c(b-c) and on the hyperboloid of two sheets It should be noticed that these points ellipse respectively. lie on the focal hyperbola and focal 97. Fundamental quantities for the paraboloids. The equation of a paraboloid (22) 2z be replaced by = ax 2 +by* may (23) a:=V^, y=V^, z = -(au +bv l l ). Hence the paraboloids are surfaces of translation lie in ( 81) whose generating curves are parabolas which perpendicular planes. By direct calculation we find ^VS^ + ftX + l D = 0, // =-4 ^ is so that the equation of the lines of curvature a dv, b dv. . b FUNDAMENTAL QUANTITIES The general (24) 231 integral of this equation is ^ c is l , an arbitrary constant. and v 1 in (24) are given particular values, equation (24) determines two values of c, c l and 2 in general distinct. If these where When u latter values be substituted in (24) successively, we obtain in finite form the equations of the two lines of curvature through the point (., tu. / \ Tf 11 c l and ,1 cz "U 1/11 be replaced by + -f- A + au \ - ( spectively, we I have, in consequence of (23), the \ on - ) J and A + aV \ ov two equations ( re- buy2 + (25) (1 au) x 2 = u (1 + (1 an} ab -f (1 av) x2 = v + av) 2 b-a ab 1 When these equations are solved for x and y , we find that equa tion (22) can be replaced by a b b (26) / E (27) = 1b ~a 2"^- (1+aW + /i , ^ , and the parametric curves are the lines of curvature. Now we have a r b 2 (u v) a(a * b)u F = 0, u(I+au) b a G = ~7T5~ (U ~ a(a b)v 4 6 2 au}(\ + av), b] Vab and 1 V [a (a U 3 [ b) u b] [a (a v b) (a-b)(u-v) a (a-b)u-b][a(a-b)v-b] (a 1 (29) b)(u v) a(a b)u b][a(a b)v- b] v(l+av) 232 QLJADRICS (27), (28), From (30) and (29) we obtain W = ^Xx = [a (a - b)u- b]*[a(a - b)vb)u b]~ [ and = (31) [ a (a a (a-b)v- From these results we find that the ratio is W/z is constant along the curves for which the total curvature constant. We first suppose that it of (26) sign, or one b is positive and greater than a. From the follows that u and v at a real point differ in is equal to zero. both u and v are equal to zero. We consider the points at which There are two such points, and (32) their coordinates are ,-0, Evidently these points are real only on the elliptic paraboloid. From (31) it follows that p l and p z are then equal, and conse quently these are the umbilical points. than these u and v must differ in sign, Since at points other we may assume that u is always positive and v negative. Moreover, from (26) it is seen that u and v are unrestricted except in the case of the I/a. elliptic paraboloid, when v must be greater than 98. Lines of curvature (14), (27), and and asymptotic 91 we have the theorem lines on quadrics. : From The lines of curvature of a quadric surface form an isothermal system of the Liouville type. Bonnet * has shown that this property is characteristic of the quadrics. There are, however, many surfaces whose lines of curva ture form an isothermal system. faces. They are called isothermic sur The complete determination of been accomplished (cf. Ex. 3, 65). V all such surfaces has never * Meraoire sur la theorie des surfaces applicables sur une surface donne e, Journal de Ecole Poly technique, Vol. (1867), pp. 121-132. XXV ASYMPTOTIC LINES ON QUADRICS From The (17), (29), 233 and 82 follows the theorem: lines of curvature of a quadric surface form an isothermal- conjugate system, and consequently the asymptotic lines can be found by quadratures. We shall find the expressions for the coordinates in terms of the latter in another way. Equation (10) is equivalent to the pair of equations \vS or the pair (34) V&lt;y \ Vo/ \V# V&lt; For each value of u equations whose points lie on the surface. And to each point on the surface there corresponds a value of u determin ing a line through the point. Hence the surface is ruled, and it is v are where u and undetermined. (33) define a line all of nondevelopable, as seen from (18). Again, for each value of v equations (34) define a line whose points lie on the surface (10), and these lines are different from those of the other system. Hence the central quadrics are doubly ruled. necessarily the asymptotic lines. Consequently, (34) be solved for z, y, z, thus : These if lines are equations (33), x u + v V^i^r+i we have the asymptotic lines. vP^TT may y uv 1 z . v u vP ^TT the surface defined in terms of parameters referring to In like manner equation (22) ^fax or -f be replaced by i i^/by i^Tby = 2 uz, ^/ax ^Jby Vfo/ = == - , u 2 vz. V ax + Solving these, \ * &gt; V ax i v we have * I / \ """" "1 C% tJ c\ 2uv 234 QUADKICS in the preceding case, we see that the surface is doubly ruled,* in (36) refer to the asymptotic system of straight As and the parameters lines. Hence : The asymptotic lines on any quadric are straight lines. EXAMPLES 1. The focal conies of a family of confocal quadrics meet the latter in the umbilical points. 2. Find the characteristic lines on the quadrics of positive curvature. 3. The normal along which the total curvature at the point. 4. section of an ellipsoid at a point in the direction of the curve is constant is an ellipse with one of its vertices Find the equation of the form is = Md cu dv ; (cf . 79) when the corresponding surface 5. a hyperboloid of one sheet when a hyperbolic paraboloid. Find the evolute of the hyperboloid of one sheet and derive the following : properties (a) the surface is algebraic of the twelfth order ; (b) the section by a principal plane of the hyperboloid consists of a conic and the evolute of a conic ; (c) these sections are edges on the surface ; (d) the curve of intersection of the two sheets of the surface is cut by each of the principal planes in four ordinary points, four double points, and four cusps, and consequently is of the twenty-fourth order. 6. Determine for the evolute of a hyperbolic paraboloid the properties analogous 5. to those for the surface of Ex. 7. their order of Deduce the equations of the surfaces parallel to a central quadric determine and the character of the sections of the surface by the principal planes the quadric find the normal curvature of the curves corresponding to the asymp ; ; totic lines on the quadric. 99. Geodesies on quadrics. Since the quadrics are isothermic surfaces of the Liouville type, the finite equation of the geodesies can be found by quadratures * ( 93). From (VI, 74), (14) and (27), Moreover, the quadrics are the only doubly ruled surfaces. For consider such a sur and denote by a, b, c three of the generators in one system. A plane a through a meets 6 and c in unique points B and C and the line B(J meets a in a point A. The line ABC is a generator of the second system, and the only one of this system in the plane a. The other lines of this system meet a in the line a. On this account the plane a cuts the surface in two lines, a and ABC, that is, in a degenerate conic. Hence the surface is of the second degree. face, , GEODESICS ON QUADEICS it 235 follows that the geodesies on any (37) integral of the differential equation of one of the quadrics is first u is sin 2 + v cos = 2 a, a constant of integration and 6 measures the angle which a geodesic, determined by a value of a, makes with the lines where a of curvature v = const. We We recall that in equations (11) and (26) the parameter u is greater than v, except at the umbilical points, shall discuss the general case first. where they are equal. M Consider a particular point (u\ v ). According as a is given the value u or v equation (37) defines the geodesic tangent at 1 f M u , 1 to the line of curvature =u or v v respectively. It is readily seen that the other values of a, determining other geo More desies through M.\ lie in the interval between u and v . over, to each value of geodesies through with respect to the directions of the lines of curvature. this result it M r a in this domain there correspond two whose tangents are symmetrically placed From follows also that the and defined by (37), when a is all the intermediate values. whole system of geodesies is given the limiting values of u and v We (38) write equation (37) in the form (u a) sin 2 + (va) cos = 2 . 0, and consider the geodesies on a central quadric defined by this Suppose, first, that a equation when a has a particular value a is in the domain v&lt; of the values of u. Then at each point of these &gt; a have seen geodesies that these geodesies are tangent to the line of curvature u = a From (11) it follows that they lie within the zone of the surface . - a and consequently from (38) u We bounded by the two branches of the curve u = a When, now, a is positive, and con a is in the domain of the values of v, u a Hence the geodesies tangent to the sequently from (38) v a lie outside the zone bounded by the two branches of curve v . 1 &lt; . the line of curvature v Similar results are true for the parabo with the difference, as seen from (26), that the geodesies loids, a lie outside the region bounded by this curve, tangent to u whereas the curves tangent to v = a lie inside the region bounded f . a by v =a . 236 QUADKICS There remains for 100. Geodesies through the umbilical points. consideration the case where a takes the unique value which u so and v have at the umbilical points. a Q ) sin 2 Let it be denoted by , that the curves defined by (39) (u + (v a ) cos 2 = : are the umbilical geodesies. We have, at once, the theorem Through each point on a quadric with real umbilical points there pass two umbilical geodesies which are equally inclined of curvature through the point. to the lines Hence two diametrically opposite umbilical points of an ellipsoid by an infinity of geodesies, and no two geodesies through the same umbilical point meet again except at the diametrically are joined These properties are possessed also by a family of On the great circles on a sphere through two opposite points. and on each sheet of the hyperboloid of two elliptic paraboloid sheets there are two families of umbilical geodesies, but no two opposite point. of the all same family meet except ellipsoid (11) at the umbilical point common to curves of the family. For the = b and equations (VI, 72, 73) become ~~ a^_ _1 C ~db~ \ u du b 1 C \ v v)(v c) _dv v ~4J ^\(a u)(uc)u 4J N(a Similar results hold for the hyperboloid of tw sheets and the of a point P from two elliptic paraboloid. Hence the distances umbilical points (not diametrically opposite) are of the form Hence we have The : lines of curvature are geodesic ellipses on the quadrics with real umbilical points and hyperbolas with the umbilical points for foci. 101. Ellipsoid referred to a polar geodesic system. A family of umbilical geodesies and their orthogonal trajectories constitute an excellent system for polar geodesic coordinates, because the domain is unrestricted (87) except in the case of the ellipsoid, UMBILICAL GEODESICS 237 and then only the diametrically opposite point must be excluded. We consider such a system on the ellipsoid, and let denote the the other umbilical points (fig. 23). pole of the system and " , O", If we put &lt;h i =- r A I (40) _ irr du - l\ 2J N(a 2J \(a-u)(u-c) du _ 1 C u 1r I , I v *)( I v dv ~2 J \(a it is u)(u c) u b~2jv( a v)(v 00 C)V ( readily found that 1 ( j_. )(&_) By means (41) of (11) we may reduce ds 2 the linear element to the form = dp + -= In order that the coordinates be polar geodesic, ^r must be replaced by another parameter measuring the angles between the geodesies. For the is ellipsoid equation (39) (42) (u-b)s As previously seen, 6 is half of one of the angles between the two geodesies through a point M. As along approaches the geodesic joining these two M FIG. 23 points, the geodesic 1 O MO" ap &&gt;, 2 approaches proaches the section # = 0. Consequently the angle or its supplementary angle. Hence the angle MOO denoted by , we have from (43) (42) lim im u=b, = b,r=b ? ib-V\ = \U b/ We by take &&gt; in place of t/r and indicate the relation between them ^fr =/(o&gt;). From (41) we have 238 This expression second (44) is QUADRICS satisfies the first of conditions (VI, 54). The lim 1 -j If 2 -7=^ 7^-6)^-1.) u)(u - *) [&lt;* [ ^ ( &gt; 1 *&lt;H -1 find c) we make n % -yj- du -= d&lt;p \(a M -^ c)u u r use of the formulas b (III, 11) and dv - i = \(a t \\ vd(f) M -(40), we v)(v v b v j w v so th.it equation (44) reduces to lim u=,r = b m V(^M^) U-V of (43) \_\ |(c-o(uU g) + i(a-^(.-^i =1 V N J By means we pass from this to Hence the linear element has the following form due to Roberts * : siir&&gt; The second 1 of equations (40) may now 1 be put in the form 2J fjl \l(ar . HI u )(p I c) u b b 2J \(a &) - fjl HI v)(v~ -f (7, dv - b) (b - c) log tan (a \vhere this constant, denotes the constant of integration. In order to evaluate we consider the geodesic through the point (0, ft, 0). At this point the parameters have the values u = a, v = and the Hence the above equation may be angle co has a definite value C &lt;?, o&gt;. replaced by i rr ~^~ du _i rr , ~^~ ^ (-*)&lt;*-) 2 . * Journal de Mathematiques, Vol. XIII (1848), pp. 1-11. PROPERTIES OF QUADRICS 239 In like manner, for the umbilical geodesies through one of the other points (not diametrically opposite) we have i r I u u)(u c) du [ i b r c \ v dv v c) 2Ja \(a u 2j \(a v)(v (a-b)(b-c) once from formulas these that if is any point on a line of curvature u const, or v const., we have It follows at M respectively tan - -- tan - = const., tan - cot - = const. 102. Properties of quadrics. From (18) it follows that for the central quadrics Euler s equation (IV, 34) takes the form By means (47) of (19) and (37) this reduces to I? R abc In like manner, (48) we have for the paraboloids I= : _J! [& + (&_)]. Hence we have Along a geodesic or product RW* is of curvature on a central quadric the 3 s constant, and on a paraboloid the ratio fiW /z line . Consider any point P on a central quadric and a direction through P. Let a, ft, 7 be the direction-cosines of the latter. The semi-diameter of the ellipsoid (10) parallel to this direction is given by (49) a =-+ . By definition 240 QUADRICS G from and similarly for /3 and 7. When the values of #, /, 2, E, (11) and (14) are substituted, equation (49) reduces to 1 p* = cos u 2 fl sin v 2 fl By means (50) of (19) and (37) this 2 may 2 be reduced to ap W this follows the =abc. : From theorem of Joachimsthal Along a geodesic or a line of curvature on a central quadric the to the tangent product of the semi-diameter of the quadric parallel to the curve at a point P and the distance from the center to the tangent plane at P is constant. From (47) and (50) we obtain the equation for all points on the quadric. Since W is the same for all direc tions at a point, the correspond. maximum and minimum Hence we have the theorem : values of p and R In a point P principal of curvature at P.* the central section of a quadric parallel to the tangent plane at axes are parallel to the directions of the lines the EXAMPLES 1. On a hyperbolic paraboloid, of which the principal parabolas are equal, the locus of a point, the sum or difference of whose distances frotn the generators through the vertex of the paraboloid is constant, is a line of curvature. 2. Find the radii of curvature and torsion, at the extremity of the mean diam eter of an ellipsoid, of an umbilical geodesic through the pokit. 3. on an 4. Find the surfaces normal to the tangents to a family of umbilical geodesies 76). ellipsoid, and determine the complementary surface (cf. The geodesic distance of two diametrically opposite umbilical points on an one half the length of the principal section through the of the linear ellipsoid is equal to umbilical points. 5. Find the form elliptic paraboloid, when the parametric system element of the hyperboloid of two sheets or the is polar geodesic with an umbilical intersection of a geodesic through the umbilical const. , then point for pole. 6. If MI and M2 are two points of with a line of curvature v tan - point - = cot - = const. * For a to a more complete discussion of the geodesies on quadrics, the reader is memoir by v. Braunmuhl, in Math. Annalen, Vol. XX (1882), pp. 556-686. referred EQUATIONS OF A RULED SURFACE 241 7. Given a line of curvature on an ellipsoid and the geodesies tangent to it; the points of intersection of pairs of these geodesies, meeting orthogonally, lie on a sphere. 8. Given the geodesies tangent to two lines of curvature ; the points of inter section of pairs of these geodesies, meeting orthogonally, lie on a sphere. 103. Equations of a ruled surface. A surface which can be gen erated by the motion of a straight line is called a ruled surface. Developables are ruled surfaces for which the lines, called the generators, are tangent to a curve. faces do not possess this property, As and a general thing, ruled sur skew surfaces. Now we make ticularly those of the skew type, limiting our discussion to the case where the generators are real.* ruled surface is completely determined in this case they are called a direct study of ruled surfaces, par A by a curve upon the curve. Z&gt;, it and the direction of the meeting with the directrix __ generators at their points of We call the latter and the cone formed by drawing through FIG 24 M a point lines parallel to the generators the director-cone. If the coordinates of a point Q of D are # , y^ it, , from a point x is of expressed in terms of the arc v measured and Z, m, n are the direction-cosines of the gen 2 , erator through Jf (51) the equations of the surface are lu, =x If y = y +mu, Q z = z + nu, where u through the distance from . M M M Q to a point M on the generator Q Q makes with the tangent cos denotes the angle which the generator through at Jf to then Z&gt;, (52) = xJ, + y m + r z Q n, where the accent indicates differentiation with respect to v (fig. 24). From (53) (51) we ds* find for the linear 2 element the expression =du +2 cos dudv + (aV + 2 bu + 1) dv*, where we have put for the sake of brevity * We shall use the term ruled to specify the surfaces of the skew type, and developable for the others. 242 RULED SURFACES directly Since the generators are geodesies, their orthogonal trajectories arrive at this result can be found by quadratures ( 92). remarking that the equation of these trajectories is We by (HI, 26) du is + cos 6 dv = 0, and that a function of v alone. shall now con Developable surfaces. sider the quantities which determine the relative positions of the generators of a ruled surface. Let g and g be two generators determined by parametric values v and v + Sv, and let X, /*, v denote the direction-cosines of their common perpendicular. If the direction-cosines of g and g be have I + SZ, m + 8w, n + Bn respectively, we denoted I, m, n 104. Line of striction. We by ; ( l\ | + nv = 0, + \ + (m + &m) + (w + Sw) v = 0, (I + 81) nifJL A* and consequently (56) \:fji:v (54) it = (mn )* nm) : (n&l l&n) : (ISm mSl). From arid follows that 1 (mn by Taylor s - nm + (nl - ln )*+(lm - ml l+ (56) * 2 theorem, (57) = l + rftr + may gW+--. Hence equations be replaced by (58) where If e t , e 2 , e3 denote expressions of the first and higher orders in Sv. Mfa y, z) and Jf (z+8s, y + % 2 + the points of ^2) are meeting (fig. of this common 24), the length MM , and g respectively perpendicular with # denoted by A, is given by or (60) A= \&x + /x% -f LINE OF STRICTION From (51), after the 243 manner Sjc of (57), ul ) we obtain -f&lt;7, = (x -+ Bv -f IBu where cr involves the second and higher powers and similar values for By and Bz are substituted (61) of Bv. in (60), When this we have ^=p + I I , where (62) m m .n 1 n of Bv. and (52) (63) e involves (54) first and higher powers In consequence of and we have /= ^. M In order to find its As tion C, Bv approaches zero, the point approaches a limiting posi which is called the central point of the generator. Let a this point. Denote the value of u for value we remark that it follows from the equations (55) and (59) that Sx Bl By Bm Bz Bn _ ~ Bv Bv If the Sv Bv Bv 8v above expressions for these quantities be substituted in this equation, we have in the limit, as Bv approaches zero, (64) a*u +b= 0. Consequently (65) The locus of the central points is called the line of striction. Its is a necessary and parametric equation is (64). Evidently b sufficient condition that the line of striction be the directrix. From (61) is and generators (66) (63) it is seen that the distance of the second order when between near-by a2 loss sm2 6&gt; -6 2 =:0. line of striction Without of generality for directrix,, in which case we may take the we may have sin# =:0, that is, the 244 KULED SURFACES generators are tangent to the directrix. Another possibility is afforded by a 0. From (54) it is seen that the only real sur faces satisfying this condition are cylinders. Hence (cf. 4) : necessary and sufficient condition that a ruled surface, other than a cylinder, be developable is that the distance between near-by genera tors be of the second or higher orders ; in this case the edge of regres sion is the line of striction. A 105. Central plane. Parameter of distribution. plane to a ruled surface at a point It M necessarily ( The tangent contains the generator through M. has been found is opable surface this plane tangent at We 25) that for a devel points of the generator. shall see that in the case of skew all surfaces the tangent plane varies as deter moves along the generator. M We mine the character finding the at plane M by which the tangent angle makes with the tangent of this variation plane at the central point C of the gen erator through M. The tangent plane at C is called the central plane. Let g and g l be two generators, and of g draw the plane pendicular (fig. 25). Through the point normal to g it meets g^ in and the line through parallel v and The limiting positions of the planes to ff l in 2 ; MM M their common per M M M ( . M MM, as g^ approaches , and at C, the tangent planes at the limiting position of M. The angle between these planes, de is equal to MMJtt^ and the angle between g and g v noted by g, are M M^MM denoted by cr, is MMM equal to MMM 2 . By construction MM M 2 l and 2 are right angles. Hence = MM. = tan ., d&gt; = 2 MM ton a- MM (7, In the limit M is the central point v tan&lt;f&gt;=lim ,, tan&lt;f&gt; (u 2 and so we have ^ pdv -f a)da- = , (u a)a m p &i 2 ) for we have da* = lim (SI + 8m = a W. PARAMETER OF DISTRIBUTION It is 245 customary to write the above equation in the form (67) The function It is ft thus defined is called the parameter of distribution. the limit of the ratio of the shortest distance between two generators and their included angle. parameter u, we have the theorem : As it is independent of the The tangent of surface at a point distance of the angle between the tangent plane to a ruled M and the the central M from plane is proportional to the central point. From to this it follows that as M moves along a generator from Hence the tangent planes oo + oo, varies from (/&gt; Tr/2 to 7r/2. at the infinitely distant points are perpendicular to the central plane. Since /3=0 is the condition that a surface be developable, the tangent plane is the same at all points of the generator. We shall now derive equation (67) analytically. ) From (51) we find that the direction-cosines of the normal to the surface are of the form ( X=-(mz 2 nyQ ) + (mn m n) u ^ to ; the expressions for , direction-cosines JT Y and Z are similar F Z of the normal at , the above. The Q the central point are this obtained from these by replacing u by /\Q\ r\o x/\ a. From we have 2 ^T ~V~V _ 2 (mz[ which leads to ny ^f + 2 (mz[ 2 ny Q } (mn )* (aV + 2 bu + sin (a V+ m n} (u + a) -f a ua 2 ba + sin 2 )* ^2J and a 2 _. a (u 4/^2 a) From (70) ( this equation (67) we have / ._ ^~ is -_ its " 2 I m n When the surface defined by linear element, @ is thus deter mined only that this of the is to within an algebraic sign. We is shall find, however, not the case (51). when the surface defined by equations form 246 EJJLED SURFACES end we take a particular generator g for the for g we have this z-axis. To Then Let also the central plane be taken for the zz-plane and the central = 0. Since the point for the origin. From (68) it follows that y Q b= and consequently I = 0. Hence origin is the central point, the equation of the tangent plane at a point of g has the simple form (71) m u% XO T) = 0, f and rj being current coordinates. If the coordinate axes have the usual orientation, and the angle * is measured positively in the direction from the positive #-axis to the positive ^/-axis, from equation (7 2) (71) we have tan * = mu . Comparing values, with equation (67), we find for ft the value xJm In order to obtain the same value from (70) for these particular this we must take the negative sign. \ Hence we have, in general, (73) / / = -i 2 I I m m 1 n n from (72) that, as a point moves along a generator in the direction of u increasing, the motion of the tangent plane is that of a right-handed or left-handed screw, according as ft is It is seen negative or positive. EXAMPLES 1 . Show that for the ruled surface denned by 2J 2 ,. y _ = i r . ^ . -, . * ~2&gt; Cu&lt;t&gt;du where and are any functions of w, the directrix and the generators are minimal. Determine under what condition the curvature of the surface is constant. \f/ 2. Determine the condition that the directrix of a ruled surface be a geodesic. PARAMETER OF DISTRIBUTION 3. Prove, by are defined by 247 means of (62), that the lines of curvature of a surface F(x, y, z) = ^ dx dy, d_F dz dF dx &gt;*, cF dz dy a**, dy *?* dz 4. The right helicoid is the only ruled surface whose generators are the principal normals of their orthogonal trajectories. Find the parameter of distribution. 5 . Prove for the hyperboloid of revolution of one sheet that (a) (6) : the minimum circle is the line of striction is and a geodesic ; the parameter of distribution constant. 6. With every point P on a ruled surface there is associated another point P on the same generator, such that the tangent planes at these points are perpendicular. Prove that the product OP OP where denotes the central point, has the same value for all points P on the same generator. , 7. 8. The normals The is to a ruled surface along a generator form a hyperbolic paraboloid. erator 9. cross-ratio of four tangent planes to a ruled surface at points of a gen equal to the cross-ratio of the points. two ruled surfaces are symmetric with respect to a plane, the values of the parameter of distribution for homologous generators differ only in sign. If 106. Particular form of the linear element. erties of ruled surfaces are readily obtained is given a particular form, which we will Let an orthogonal trajectory of the generators be taken for the directrix. In this case (74) If A number of prop when the linear element now deduce. *,=, u f,-*. = u, vl we make the change of parameters, (75) = C a dv, I v Jo the linear element (53) (76) ds* is reducible to 2 2 = du* + [(u - a) + /3 ] dv*. is The angle 6 which given by (77) a curve v l =f(u) makes with the generators tan0 = V(w is Also the expression for the total curvature (78) JT = -- f* 248 KULED SURFACES = Hence a real ruled surface has no elliptic points. All the points are hyperbolic except along the generators for which /3 0, and at the infinitely distant points on each generator. Consequently the linear element of a developable surface (79) ds* may be put in the form = du" + (u - a) 2 dv*. face the latter has the Also, in the region of the infinitely distant points of a ruled sur character of a developable surface. As another consequence of (78) we have that, for the points of a generator the curvature is greatest in absolute value at the cen tral point, and that at points equally distant from the latter it has the same value. When the linear element is in the form (76), the Gauss equation of geodesies (VI, 56) has the form V(M - a) + @*d6 + (u-a) dv = 0. 2 1 An immediate consequence is the theorem of Bonnet : If a curve upon a ruled surface has two of the following properties, it has the third also, namely that it cut the generators under constant striction. angle, that it be a geodesic and that it be the line of formed by the family of straight lines which cut a twisted curve under constant angle and are perpen surface of this kind is A dicular to principal normals. formed of the binomials of a curve. its A particular case It is readily is the surface (73) shown from is that the parameter of distribution of this surface radius of torsion of the curve. ,- equal to the 107. Asymptotic lines. Orthogonal parametric systems. The erators are necessarily asymptotic lines on a ruled surface. gen We (68) consider now the other family of these lines. r From (51) and we find i&gt; (80) = 0, D 1 =H I m n " m z"+n"u n z +nu Hence the differential of asymp equation of the other family totic lines is of the form dv ASYMPTOTIC LINES where , 249 Jf, N are we functions of Riccati t}^pe, have, from As this is an equation of the the theorem of Serret: 14, v. The four points in which each generator of a ruled surface by four curved asymptotic lines are in constant cross-ratio. is cut From lines is 14 it follows also that when one of these asymptotic known the surface (76), the others can be found by quadratures. is When in the (81) referred to an orthogonal system and the linear element ds 2 is form written = du 2 + a 2 [(u - a) 2 + 2 /3 ] dv 2 , the expressions (80) can be given a simpler form. From (73) and (81) we have From and the equations Lx ol by = = = 0, Sz6 2 = 1, 2Z 2 = 1, (54) we obtain, differentiation, 0, Zzfceo -Ll l" ZM = , 0, Zatf J aa ZK" where t is defined by D" zr 6 == t. a2 , ZJ x6 = -b, = & -, If the expression for in (80) be multiplied hand member of (73), and the result be divided consequence of the above identities, D" by the determinant of the righta 2 /3, we have, in by its equal, =- i 2 [w (to? 2 - aa fc) + u (2 tb - aa - 66 ) +t0, & ]. If equations (74) be solved for a and 6 as functions of expressions be substituted in this equation, we have D" a and and the resulting = -~{r[(u - a) 2 + ^] + ?(u - a) + /3a }, by (75), is where the primes indicate differentiation with respect / defined by to Vi, given and r From (82) the above equations it follows that the mean curvature (cf . 52) is express ible in the form J - + * = r - ^u ~ a)2 + ^+ 2 a-) Pi Pz [(u EXAMPLES 1. When the linear element of a ruled surface is in the form (76), the direction- cosines of the limiting position of the ?&lt;L?, /3 common perpendicular to two generators are z V* + &lt; +n o/3 t aft 250 MINIMAL SURFACES with real 2. Prove that the developable surfaces are the only ruled surfaces generators whose total curvature is constant. 3. Show x _ u ^y 4. totic lines that the perpendicular upon the z-axis from any point of the cubic w a lies i n the osculating plane at the point, and lind the asymp M2 z on the ruled surface generated by this perpendicular. ? Determine the function x in the equations w, = y = un , z = 0(u), so that the osculating plane at any point M of this curve shall pass lines projection ated by the line 5. P of M on the y-axis. MP. Find the asymptotic through the on the surface gener Show 6 that the equations where z = u-fwcos0, x = M sin cos ^, y = wsinflsin^, define the most general ruled surface with a rec and ^ are functions of lines can be integrated tilinear directrix, and prove that the equation of asymptotic two quadratures. Discuss the case where is constant. by ruled surface the following are 6. Concerning the curved asymptotic lines on a t&gt;, to be proved if : one of them is an orthogonal trajectory of the generators, the determina (a) tion of the rest reduces to quadratures are curves of Bertrand if two of them are orthogonal trajectories, they (6) surface is a right helicoid. if all of them are orthogonal trajectories, the (c) be an asymptotic line, and 7. Determine the condition that the line of striction ; ; show that 8. in this case the other curved asymptotic lines can be found by quadratures. is generated by a line pass = z = 0, + y-fz = l. Show that these lines through the two lines x = 0, y ing line of striction. and the line x = 0, x + y + z = 1 are double lines. Find the Find a ruled surface of the fourth degree which ; of whose lines of curvature 9. The right helicoid is the only ruled surface each on any other ruled surface cuts the generators under constant angle however, this property. there are in general four lines of curvature which have ; In 1760 Lagrange extended to double theorems about simple integrals in the calculus integrals the Euler * of variations, and as an example he proposed the following problem 108. Minimal surfaces. : Given a closed curve C and a connected surface S bounded by curve; to the determine 8 so that the inclosed area shall be a minimum. If the surface be denned by the equation z =f(x, y), that the inte the problem requires the determination of f(x, y) so gral (cf. Ex. 1, p. 77) * CEuvres de Lagrange, Vol. I, pp. 354-357. Paris, 1867. MINIMAL SURFACES minimum. As shown by Lagrange, the condition (83) 251 extended over the portion of the surface bounded by C shall be a for this is or, in other form, (1 (84) + q )r - 2pqs + (1 + p*)t = 0. z Lagrange left the solution of the Meusnier,* sixteen years later, problem in this form, and proved that this equation is equivalent to the vanishing of the mean curvature ( 52), thus showing that the surfaces furnishing the solution of Lagrange s problem are characterized by the geometrical property which now is name usually taken as the definition of minimal surfaces; however, the indicates the connection with the definition of Lagrange. f In what follows we purpose giving a discussion of minimal sur faces from the standpoint of their definition as the surfaces whose mean curvature is zero at all the points. At each point of such a surface the principal radii differ only in sign, and so every point is a hyperbolic point and its Dupin indicatrix is an equilateral hyperbola. Consequently minimal surfaces are characterized by the property that their asymptotic lines form an orthogonal sys tem. Moreover, the tangents to the two asymptotic lines at a point bisect the angles between the lines of curvature at the point, and vice versa. We recall the formulas giving the relations between the funda mental quantities of a surface and its spherical representation (IV, 70) (85) : (o = we have at once the From these theorem : The necessary and sufficient condition that the spherical represen tation of a surface be conformal is that the surface be minimal. * Memoire sur la courbure des surfaces, Memoires des Savants Strangers, Vol. X (1785), p. 477. t For a historical sketch of the development of the theory of minimal surfaces and a complete discussion of them the reader is referred to the Lemons of Darboux (Vol. I, pp. 267 et seq.). The questions in the calculus of variations involved in the study of mini mal surfaces are treated by Riemann, Gesammelte Werke, p. 287 (Leipzig, Schwarz, Gesammelte Abhandlungen, Vol. I, pp. 223, 270 (Berlin, 1890). 1876) ; and by 252 MINIMAL SURFACES Hence isothermal orthogonal systems on the surface are repre sented by similar systems on the sphere, and conversely. All the isothermal orthogonal systems on the sphere are known ( 35, 40). Suppose that one of these systems element is * is parametric and that the linear From (86) it the general condition for minimal surfaces (IV, 77), namely &lt;D" + 3D - 2 &D = f 0, follows that in this case n 1 In consequence of this the Codazzi equations (V, 27) are reducible to (87) ?-^-0, dv du D or D we ~IQ f + ~? = du dv - By eliminating of the equation find that both D and D 1 are integrals a? Hence the most general form (88) + of ^Q = ^? * D is D = (u + iv) + ^(u- iv), and &lt;f&gt; where (89) i/r are arbitrary functions. Then from (87) we have D= ~D" =- i((j) -ty+c, To each pair of functions c is the constant of integration. there corresponds a minimal surface whose Cartesian coordi T/T nates are given by the quadratures (V, 26), namely where &lt;, ( 90) =du \\ du dv dv \ du dv and similar expressions in y and z. Evidently the surface is real and i/r are conjugate functions. only when In obtaining the preceding results we have tacitly assumed that neither D nor D is zero. We notice that either may be zero and &lt;f&gt; 1 then the other is a constant, which These results may be stated thus: is zero only for the plane. Every isothermal system on lines of curvature of lines of another a unique the sphere is the representation of the minimal surface and of the asymptotic minimal surface. LINES OF CURVATURE The converse also is true, AND ASYMPTOTIC LINES 253 namely: the The spherical representations of the lines of curvature and of asymptotic lines of a minimal surface are isothermal systems. For, if the lines of curvature are parametric, equation (86) may be replaced by D = p^ D D = _ pg &gt;, where p sign. equal to either principal radius to within its algebraic = = are substituted in the When these values and is & Codazzi equations (V, 27), we obtain so that /g=*U/V, which proves the to first part of the theorem ( 41). When the asymptotic lines are parametric, we have Z&gt;=D"=c^=0, and equations (V, 27) reduce cu (&gt;!")= from which it follows that &lt;~/^= U/V. lines. Adjoint minimal return to the consideration of equations (87) and investigate first the minimal surface with its lines of curvature represented by an isothermal system. Without loss of generality,* 109. Lines of curvature and asymptotic surfaces. We we may (91) take D= (IV, 77) it -D" = 1, &gt; =0. From follows that PiP 2 where = -- = ~X Pi = |^| = |p 2 2 , /&gt; E=G = 2 J. p, Hence we have the theorem : The parameters of ical representation the lines of curvature of a minimal surface may be so chosen that the linear elements of the surface and of its spher have the respective forms 2 2 ds 2 = p (du + dv ), dd 2 = - (du 2 + dv 2 ), P where p is the * absolute value of each principal radius. other value of the constant leads to homothetic surfaces. Any 254 In like manner (92) MINIMAL SURFACES we may take, for the solution of equations (87), I&gt;"=Q, D= find .. D = l. E=-G = p, : Again we -, J- = PiP* = -\\ Pi so that we have a result similar to the above The parameters of the asymptotic lines of a minimal surface may be so chosen that the linear elements of the surface and of 2 its spherical representation have the respective forms ds* = p (du* + di?), d&lt;r* =- (du* + dv ), where p is the absolute value of each principal radius. the symmetric form of equations (87) it follows that if one set of solutions, another set is given by (88) and (89) represent From These values are such that which is respond the condition that asymptotic lines on either surface cor to a conjugate system on the other ( 56). When this is condition satisfied by two minimal surfaces, and the tangent are parallel, the two surfaces are planes at corresponding points said to be the adjoints of one another. Hence a pair of functions &lt;, determines a pair of adjoint minimal surfaces. When, in par surface a*e parametric, the ticular, the asymptotic lines on one and on the other the values (91). functions have the values -&gt;/r (92), It follows, then, from (90), that its between the Cartesian coordinates of a minimal surface and cjx\_ relations hold: adjoint the following foi _dx dv / = fa. cu dv s du z s, and similar expressions in the and when z). the parametric curves are asymptotic on the locus of (#, #, 110. Minimal curves on a minimal surface. The lines of length When zero upon a minimal surface are of fundamental importance. of the surface the equations they are taken for parametric curves, take a simple form, which we shall now obtain. MINIMAL CURVES we have (94) 255 Since the lines of length zero, or minimal lines, are parametric, ^ = = 0. (85) it follows that the parametric lines on the sphere also are minimal lines, that is, the imaginary rectilinear generators. And from (86) we find that 1) is zero. Conversely, when the latter is zero, From and the parametric lines are minimal curves, it follows from (IV, 33) that Km is equal to zero. Hence : A necessary and sufficient condition that a surface be minimal is that the lines of length zero form a conjugate system.* In consequence of (94) and (VI, 26) the point equation of a minimal surface, referred to its minimal lines, is ducv Hence the finite equations of the surface are of the form where U^ T/2 , U s are functions of u alone, and F x, F 2, F 3 are functions of v alone, satisfying the conditions (96) U? + V? + U? = (95) ( 2 0, F{ + Fi + a 2 Fj = 0. From lation it is seen that minimal surfaces are surfaces of trans (96) 81), and from that the generators are minimal 81 curves ( 22). In consequence of the second theorem of : we may state this result thus minimal surface is the locus of points on two minimal curves. In A the mid-points of the joins of 22 we found that the Cartesian coordinates of any minimal curve are expressible in the form (97) f (1 - u*)F(u) du, i f (1 + u 2 ) F(u) du, 2 Cu F(u) du. is *This follows also from the fact that an equilateral hyperbola which the directions with angular coefficients i are conjugate. the only conic for 256 MINIMAL SURFACES Hence by the above theorem the following equations, due to Enneper *, define a minimal surface referred to its minimal lines : (98) z= 4&gt; I u F(u) du + I v&(v) dv, are any analytic functions whatever. Moreover, where F and minimal surface can be defined by equations of this form. any For, the only apparent lack of generality is due to the fact that the algebraic signs of the expressions (98) are not determined by equations (96), and consequently the signs preceding the terms in the right-hand members of equations (98) could be positive or negative. But it can be shown that by a suitable all of change of the parameters and of the functions F and these cases reduce to (98). Thus, for example, we consider the surface defined by the equations which result when the second 3&gt; terms of the right-hand members of (98) are replaced by In order that the surface thus defined can be brought into coin cidence, by a translation, with the surface (98), we must have Dividing these equations, member by member, we have from which it follows that Substituting this value in the last of the above equations, we find * Zeitschrift fur Mathematik und Physik, Vol. IX (1864), p. 107. MINIMAL CUEVES 257 and this value satisfies the other equations. Similar results fol low when another choice of signs is made. The reason for the particular choice made in (98) will be seen reality of the surfaces. when we discuss the Incidentally we have proved the theorem : When a minimal and is surface is defined by equations (98), the necessary condition that the two generating curves be congruent sufficient that (99) , )._1 ( From (98) we obtain so that the linear element is (100) ds 2 = (l + uv}*F(u)3&gt;(v)dudv. normal We find for the expressions of the direction-cosines of the 1 H- uv 1 + uv is 1 + uv and the linear element of the sphere , 2 Alt, Also we have (102) 4 dudv (1 d&lt;r + ) &gt; D= and of the asymp so that the equations of the lines of curvature totic lines are respectively (103) (104) 2F(u) du 3&gt; (v) F(u) du 2 + &lt;&gt; = 0, dv* = 0. (v) dv* : These equations are of such a form that we have the theorem When a minimal surface is referred to its minimal lines, the finite equations of the lines of curvature and asymptotic lines are given by quadratures, which are the same in both cases. In order that a surface be real be real. Consequently u and v its spherical representation must must be conjugate imaginaries, as 258 is MINIMAL SURFACES 13, seen from (101) and and the functions if F and &lt; must be denotes the real part of a conjugate imaginary. function 9, all real minimal surfaces are defined by Hence RO x = fi f (1 - u 2 ) F(u) du, y I =R ft (1 +u 2 ) F(u) du, z=R 2uF(u)du, where F(u) is any function whatever of a complex variable u. In like manner the equations of the lines of curvature may be written in the form (105) 72 / ^/F(u)du = const., 11 \ iVF(u)du = const. whether 111. Double minimal surfaces. It is natural to inquire the same minimal surface can be denned in more than one way by equations of the form (98). We assume that this is possible, and indicate by u v v^ and F^(u^ ^V^) the corresponding parameters and functions. As the parameters u^ v l refer to the lines of length zero on the surface, each is a function of either u or v. In order to determine the forms of the latter we make use of the fact that the of positive directions of the normal to the surface in the two forms parametric representation may have the same or opposite senses. When they have the same sense, the expressions (101) and similar ones in u v and v l must be equal respectively. In this case (106) %!=!*, v^v. the resulting equations If the senses are opposite, the respective expressions are equal to within algebraic signs. (107) From we find u compare equations (98) When we u l and vv we find that for the case (106) with analogous equations in we must have and for the case (107) DOUBLE MINIMAL SUBFACES Hence we have the theorem mined by (108) : 259 A necessary and sufficient condition that two minimal surfaces, deter the pairs of functions F, and Fv v be congruent is that &lt;& &lt;& ^w on one surface corresponds the point ( to the point (u, v) -- -u on the other, sense. and the normals at these points are parallel but of different In general, the functions same. If members by and l as given by (108) are not the so also are and r Ih this case the right-hand they are, of equations (98) are unaltered when u and v are replaced &lt; F F 4&gt; l/v and 1/u respectively. v) Hence the Cartesian coordinates differ at j of the points (u, and ( -- -&gt; most by constants. And so the regions of the surface about these points either coincide or can be brought into coincidence by a translation. In the latter case periodic and consequently transcendental. Suppose that it is not periodic, and consider a point -ZjJ(w V Q ). As u varies continuously from U Q to v varies from v to l/w and the point describes a closed curve on the surface by returning the surface is , l/t&gt; , , to P Q . face. But now the positive normal is on the other side of the sur Hence these surfaces have the property that a point can pass continuously from one side to the other without going through the surface. On this account they were called double minimal surfaces by Lie,* who was the first to From the third theorem of study them. 110 it follows that double minimal in both systems are congruent. surfaces are characterized by the property that the minimal curves The equations of such a surface may be written The surface is consequently the locus of the mid-points of the chords of the curve f =/,(), i =/,(), is ?=/,(), which lies upon the surface and * Math. Annalen, Vol. the envelope of the parametric (1878), pp. 345-350. curves. XIV 260 MINIMAL SURFACES EXAMPLES 1. The focal sheets of a minimal surface are applicable to one another and to the surface of revolution of the evolute of the catenary about the axis of the latter. 2. Show that there are no minimal surfaces with the minimal lines in one family straight. 3. If two minimal surfaces correspond with parallelism of tangent planes, the minimal curves on the two surfaces correspond. 4. If two minimal surfaces correspond with parallelism of tangent planes, and the joins of corresponding points be divided in the same ratio, the locus of the points of division 5. is a minimal surface. that the right helicoid is defined by F(u) constant, and that it is a double surface. 6. Show = im/2 w2 , where m is a real The surface for which F(u) = 2 is called the surface of Scherk. Find its that it is doubly periodic and that equation in the Monge form z it is a surface of translation with real generators which are in perpendicular planes. 7. = f(x, y). Show By definition is a meridian curve on a surface tion a great circle is one whose spherical representa on the unit sphere. Show that the surface of Scherk possesses two families of plane meridian curves. * remarked that 112. Algebraic minimal surfaces. Weierstrass formulas (98) can be put in a form free of all quadratures. This is done by replacing F(u) and accents indicate &lt;J&gt;(v) the differentiation, where and # by f and then integrating by "(u) "(v), parts. This gives x p. 2 + uf (u) -f(u) + "(u) ^i 4&gt;"(v) + v(v) -h iv( (109) iuf(u) + if(u) - -^- &lt;l&gt;"(v) = uf"(u) -) + v&lt;f&gt;"(v) 4&gt; (v). It is clear that the surface so denned is real when / and &lt;f&gt; are conjugate imaginary functions. In this case the above formulas may be written : (110) = R[(ly = Ri [(1 + u*)f"(u) u*)f"(u) + 2 uf (u) - 2 uf (u) + Akademie (1866), p. 619. * Monatsberichte der Berliner ALGEBRAIC MINIMAL SURFACES 261 * However, it is not necessary, as Darboux has pointed out, that f and be conjugate imaginaries in order that the surface be real. (f&gt; For, equations (109) are unaltered 2 t if /and be replaced by f (u) =f(u) + A (1 - u + Bi (1 + u*) + 2 Cu, - A(l - v + Bi ({.-{- v )- 2 Cv, ^(v) = ) 2 2 (f&gt;(v) ) where A, B, C are any constants whatever. Evidently, if / and are conjugate imaginaries, the same is not true in general of /,_ &lt;/&gt; and &lt;f&gt; l ; but the surface was real for the former and consequently for the latter also. &lt;/&gt; It is readily found that /t and x functions only in case J, J5, C are pure are conjugate imaginary is real imaginaries. surfaces. &lt; Formulas (109) are of particular value in the study of algebraic Thus, it is evident that the surface is algebraic when/ and are algebraic. Conversely, every algebraic minimal surface In proving this we is determined by algebraic functions / and &lt;/&gt;. follow the method suggested by Weierstrass.f establish first the following lemma We 4&gt; : Gttven a function *(? of , ; if in a certain 77, 4&gt; and let domain an algebraic *??) j~ + "^(f, denote the real part relation exists between M*, 77) and is an algebraic function of + irj. If the point f = 0, rj = does not lie within the domain under can be effected by a change of variables without the argument. Assuming that this has been done, we vitiating in a power series, thus develop the function consideration, this &lt;E&gt; : 4&gt; = a + #o + K + ibj (| + s irj) + (a + ib 2 2) (f + 2 irj) + is . . . , where the a and 5 s are real constants. Evidently M* given by 1 (a, - ^) (f - ^) + J (a, - i6 8) (f ii?) 2 + - - -. Let J^(^, f ?;) = denote a rational integral relation between When M* has been replaced by the above value, and the f, and 77. resulting expression is arranged in powers of | and 77, the coeffi , "SP, cient of every term is identically zero. zero when f and 77 have been replaced * Vol. I, p. They will continue to be by two complex quantities 293. f Monatsberichte tier Berliner Akademie (1867), pp. 511-518. 262 a and /3, MINIMAL SURFACES provided that the development remains convergent. The condition for the latter is that the moduli of a and /3 be each one half the modulus of f + irj. This condition is satisfied if we take Now we have - -. - K- i5 ) +- &lt; (f t irt, f + ti,] = 0, which proves the lemma. In applying this (101) that lemma to real minimal surfaces we note from l-Z ~Y~ to u l X = +v _u where u l-Z Y= _u 2i v, consequently the left-hand members of these equations are equal and v l respectively, = u^ + iv r When the surface is algebraic there exists an algebraic relation between the functions X .L Y 7 j , A. /j 7 and each of the Cartesian coordinates.* Since, then, an algebraic relation between u^ v^ and each of the coordinates given by (110), it follows from the lemma that each there is of the three expressions &lt;k( = (1 - ?/)/ = i (1 + u fa(u) M) 2 + 2 uf (u) - 2/(w), )f"(u) - 2 iuf (u) + 2 if(u), also isf(u) for, 4&gt; 9 (u) = 2uf"(u)2f(u) and so ; are algebraic functions of w, Hence we have demonstrated The necessary and algebraic surface * For. if the theorem of Weierstrass : sufficient condition that equation (110) define an is that f(u) be algebraic. defined by F(x. y, z) the surface is z. = 0, the direction-cosines of the normal are functions of x, y, Eliminating two of the latter between X _ &gt; Y F(x, y, z) = _ _&gt; and 0, we have a relation of the kind described. ASSOCIATE SURFACES 113. 263 Associate surfaces. surface S is are When the equations of a minimal written in the abbreviated form (95), the linear element This is the linear element also of a surface defined by any constant. There are an infinity of such surfaces, minimal surfaces. It is readily found that the direc tion-cosines of the normal to any one have the values (101). Hence any two associate minimal surfaces defined by (111) have their tan is where a called associate gent planes at corresponding points parallel, and are applicable. Of particular interest is the surface Sl for which a = ?r/2. Its equations are i)du - i C I (1 v 2 &lt; ) (v) dv, (112) {y = - ^ J/ & \ (1 +u 2 ) F(u) du i -I * dv. =i I uF(u) du I v&lt;& (v) In order to show that S l is the adjoint ( 109) of S, we have only to prove that the asymptotic lines on either surface correspond to the lines of curvature on the other. For S the l equations of the lines of curvature and asymptotic lines are 2 iF(u) du -i (v) dv 2 = 0, respectively. Comparing these with (103) and is satisfied. (104), we see that the desired condition From (98) and (112) we obtain the identities \ dx dx l + dy dy l + dz dz^ = 0. The latter has the following interpretation : On two adjoint minimal surfaces at points corresponding with par allelism of tangent planes the tangents to corresponding curves are perpendicular. 264 MINIMAL SURFACES (105) it From follows that if we put / u the curves curvature. + iv = ^/F(u) du, u = const, and v = const, on the surface are its lines of Moreover, for an associate surface the lines of curva ia ture are given by ttf R [e or 2 (u + iv)] = const., . R [ie u . 2 (u -h iv)] -f- = const. t u cos 22 a v sin to the a = const., a sin 22 _ v cos #, a - = const. From The this result follows the lines of curvature theorem : S correspond curves on on a minimal surface associate to a surface S which cut its lines of curvature under the constant angle a/ 2. Since equations (111) may be written xa = x cos a - + -|- x l sin (114) . ^^ycosa + ^sina, za z cos a z l sin #, the plane determined by the origin of coordinates, a point on a minimal surface and the corresponding point on its adjoint, con P on every associate minimal the locus of these points a is an ellipse with its Moreover, center at the origin. Combining this result and the first one of tains the point P a corresponding to P surface. P this section, we have *- minimal surface admits of a continuous deformation into a series of minimal surfaces, and each point of the surface describes an ellipse whose plane passes through a fixed point which is the center of the ellipse. A 114. Formulas of Schwarz. Since the tangent planes its to a minimal surface and adjoint at corresponding points are parallel, we have From this and the second of (113) we obtain the proportion dx l Zdy Ydz __ = dy l Xdz dz _ Z dx~ Y dx X dy l FORMULAS OF SCHWAKZ 265 In consequence of the first of (113) the sums of the squares of the numerators and of the denominators are equal. And so the com 1. If the expressions for the various quanti ratio is -|-1 or be substituted from (98), (101), and (112), it is found that the 1. Hence we have value is mon ties (115) dx^Ydz Zdy, dy l = Zdx Xdz, dz 1 = Xdy Ydx. From these equations and the formulas (95), (112) we have 1 =x+i * Zdy - Ydz, (116) i=y + l (xdz \ Zdx, = z-^-i = xi Ydx Xdy, and 1 \ Zdy Ydz, (117) ^z importance i \Ydx \ X dy. These equations are known as the formulas of Schwarz* Their is due to their ready applicability to the solution of : the problem To determine a minimal surface passing through a given curve and admitting at each point of the curve a given tangent plane.\ In solving this problem we let C be a curve whose coordinates #, y, z are analytic functions of a parameter f, and let JT, Y, Z be analytic functions of t satisfying the conditions X + F + Z = 1, 2 2 2 Xdx + Ydy + Zdz = 0. * t Crelle, Vol. LXXX is (1875), p. 291. : a special case of the more general one solved by Cauchy To deter mine an integral surface of a differential equation passing through a curve and admitting at each point of the curve a given tangent plane. For minimal surfaces the equation is (84). Cauchy showed that such a surface exists in general, and that it is unique unless the curve is a characteristic for the equation. His researches are inserted in Vols. XIV, XV of the Comptes Rendus. The reader may consult also Kowalewski, Theorie der partiellen Differentialgleichungen, Crelle, Vol. LXXX (1875), p. 1; and Goursat, Cours d Analyse Mathematique, Vol. II, pp. 563-567 (Paris, 1905). This problem 266 If MINIMAL SUKFACES , x uJ y u zu denote the values of x, y, z when t is replaced by a t complex variable u, by v, the equations and xv y v , , zv the values when is replaced (118) l - f"(Ydx-Xdy) Jv define a minimal surface which passes through C and admits at each point for tangent plane the plane through the point with direction-cosines X, I 7 , Z. For, these equations define C. And when u and v are replaced by the conditions (96) and , are satisfied. Furthermore, the surface defined by (118) affords the unique solution, as is seen from (116) and (117). are real, the equations of the real minimal surface, satisfying the conditions of the problem, may be ,-i put in the form When, in particular, C and t x = R\x + il (Zdy-Ydz}\, y z = R \y + i C\Xdz - Zdx)] = R \z + i r\Ydx - Xdy\\ , As an straight line. a application of these formulas, we consider minimal surfaces containing denote the angle which If we take the latter for the z-axis, and let the normal to the surface at a point of the line makes with the Y=sin&lt;t&gt;, x-axis, we have x = y = 0, z=t, JT=cos0, Z= 0. Hence the equations x of the surface are = - RiTsm &lt;f&gt;dt, y = B{J**C08^ctt, z = R(u). surface. an analytic function of t, whose form determines the character of the For two points corresponding to conjugate values of M, the z-coordinates are equal, and the x- and ^-coordinates differ in sign. Hence Here is &lt;#&gt; : Every straight line upon a minimal surface is an axis of symmetry. FORMULAS OF SCHWAKZ EXAMPLES 1. 267 The tangents If to corresponding curves on two associate minimal surfaces meet under constant angle. 2. corresponding directions on two applicable surfaces meet under constant angle, the latter are associate 3. minimal surfaces. that the catenoid and the right helicoid are adjoint surfaces and deter mine the function F(u) which defines the former. Show Let 4. C surface (a) the equations of the be a geodesic on a minimal surface S. Show that may be put in the form y = and X, /*, where f, 77, f are the coordinates of a point on C, ; v the direction-cosines of its binomial (6) if denotes the curve on the adjoint S t corresponding to C, the radii of and second curvature of C are the radii of second and first curvature of C is a plane curve, the surface is symmetric with respect to its plane. (c) if C C first ; 5. The surface for which F(u) = 1 is u4 called the surface of Henneberg ; it is a double algebraic surface of the fifteenth order and fifth class. GENERAL EXAMPLES 1. The edge of regression of the developable surface circumscribed to two confocal quadrics has for projections on the three principal planes the evolutes of the focal conies. 2. By definition a tetrahedral surface is one whose x = A (u - a) m (v - a), y = B(u- b) m (v - 6) n equations are of the form z , = C(u- c) m (v - n c) , where A, B, 0, w, n jugate, and that the asymptotic m = n, the equation of the surface III is are any constants. Show that the parametric curves are con lines can be found by quadratures also that when ; - c) + - a) + - b) = (a - b) (b 2, c) (a - c). ^)&gt; (|)"(c (0"&lt;a 3. Determine the tetrahedral surfaces, defined as in Ex. upon which the parametric curves are the lines of curvature. 4. on an 5. Find the surfaces normal to the tangents to a family of umbilical geodesies surface. elliptic paraboloid, and find the complementary At every point of a geodesic circle with center at an umbilical point on the ellipsoid (10) abc = fW &lt;i (a + c _ r ^ where r is the radius vector of the point (cf. 102). is 6. The tangent plane to the director-cone of a ruled surface along a generator distant point on the parallel to the tangent plane to the surface at the infinitely corresponding generator. 268 7. MINIMAL SURFACES Upon The the hyperboloid of one sheet, and likewise upon the hyperbolic parab oloid, the two lines of striction coincide. line of striction of a ruled surface is an orthogonal trajectory of the only in case the latter are the binormals of a curve or the surface is a generators 8. right conoid. 9. Determine for a geodesic on a developable surface the relation existing between the curvature, torsion, and angle of inclination of the geodesic with the generators. Z2 and a the angle between two lines li and about the former with a helicoidal motion of parameter a surface if a = h cot a. If a = h tan a, the 62), the locus of 1 2 is a developable (cf of the binormals of a circular helix. surface is the locus 10. If h denotes the shortest distance latter revolves , and the . 11. If the lines of curvature in one family upon a ruled surface are such that the segments of the generators between two curves of the family are of the same is constant and the line of striction is a line length, the parameter of distribution of curvature. 12. If two ruled surfaces meet one another in a generator, they are tangent to one another at two points of the generator or at every point in the latter case the central point for the common generator is the same, and the parameter of distribu tion has the same value. ; 13. If tangents be drawn to a ruled surface at points of the line of striction in directions perpendicular to the generators, these tangents form the conju line of striction as the given surface. More gate ruled surface. It has the same the normal to the surface at the central over, a generator of the given surface, and the generator of the conjugate surface through C point C of this generator, are parallel to the tangent, principal normal, and binormal of a twisted curve. and 14. Let to C normals S along be a curve on a surface S, and S the ruled surface formed by the C. Derive the following results : distance between near-by generators of S is of the first order unless C is (a) the a line of curvature denotes the distance from the central point of a generator to the point of (6) if r ; intersection with S, rS (dX) 2 Z dxd X ; is conjugate to the tangent to the surface at to C at a point (c) the tangent line of shortest distance parallel to the and minimum values of r are the principal radii of -S, pi, and (d) the maximum 2 is the where be written r = pisin 2 -f p 2 cos and the above ; M M p2 , equation may &lt;f&gt; &lt;/&gt; tf&gt;, angle which the corresponding line of shortest distance pz&gt; makes with the tangent to the line of curvature corresponding to 15. If C and (cf. 6" are two orthogonal curves on a surface, then at the point of intersection Ex. 14) 1111 4 &gt;.* rB 16. If ~tf \ + *| C and C (cf. are two conjugate curves on a surface, then at the point of 14) j j i intersection Ex. r R GEKEBAL EXAMPLES 269 17. If two surfaces are applicable, and the radii of first and second curvature of every geodesic on one surface are equal to the radii of second and first curvature of the corresponding geodesic on the other, the surfaces are minimal. 1 8. The surface for face of Enneper ; (a) it is it in (98) is constant, say 3, which possesses the following properties : F is called the minimal sur unaltered an algebraic surface of the ninth degree whose equation ; ; is when x, y, z are (6) it (c) if z respectively replaced by y, x, meets the plane z = in two orthogonal straight lines we put u = a i/3, the equations of the surface are x = 3a + 3 ap? - a3 , , y = 3 ft + 3 a2 ft -ft 3 , z = 3 a2 ; 3 2 /3 , and the curves a const. ft = const, are the lines of curvature (d) the lines of curvature are rectifiable unicursal curves of the third order and they are plane curves, the equations of the planes being x (e) + az -3a-2a = 0, 3 y - ftz - 3ft - 2 ^ = 0; of circles to by a double family whose planes form two pencils with perpendicular axes which are tangent the sphere at the same point ; the lines of curvature are represented on the unit sphere (/) the asymptotic lines are twisted cubics (g) the sections of the surface by the planes ; are double curves on the surface x= and y = are cubics, which and the locus of the double points of the lines of curvature ; (h) the associate through the angle (i) minimal surfaces are positions of the original surface rotated 113 a/2, about the z-axis, where a has the same meaning as in ; the envelope of the plane normal, at the mid-point, to the join of any two points, one on each of the focal parabolas is the surface X = 4 cr, y = 0, z - 2 a2 - 1 ; x - 0, y = 4 ft, z = 1-2 ft2 - the planes normal to the two parabolas at the extremities of the join are the planes of the lines of curvature through the point of contact of the first plane. 19. Find the equations of Schwarz of a minimal surface when the given curve an asymptotic line. 20. is Let S and S be two surfaces, and ; let the points at which the normals are parallel correspond for convenience let S and S be ; referred to their common con jugate system. Show that if the correspondence is conformal, either S and S are homothetic or both are minimal surfaces or the parametric curves are the lines of curvature on both surfaces, and form an isothermal system. ; Find the coordinates of the surface which corresponds to the ellipsoid after 20. Show that the surface is periodic, and investigate the points corresponding to the umbilical points on the ellipsoid. 21. the manner of Ex. the equations of an ellipsoid are in the form (11), the curves u + v = on spheres whose centers coincide with the origin and at all points of such a curve the product pW is constant ( 102). 22. When const, lie ; CHAPTER VIII SURFACES OF CONSTANT TOTAL CURVATURE. W-SURFACES. SURFACES WITH PLANE OR SPHERICAL LINES OF CURVATURE 115. Spherical surfaces of revolution. Surfaces whose total cur vature K ( is the same at all points are called surfaces of constant surfaces of this kind are called curvature. When this constant value is zero, the surface is devel opable 64). The nondevelopable is positive or negative. spherical or pseudospherical, according as consider these two kinds and begin our study of them with K We the determination of surfaces of revolution of constant curvature. When upon a surface of revolution the curves v = const, are is the meridians and u = const, the parallels, the linear element reducible to the form (1) d8*=du*+Gdif, where G is a function of u alone ( 46). is In this case the expres sion for the total curvature (V, 12) (2) K= , 2 For spherical surfaces we have 7f=l/a where a is a real constant. Substituting this value in equation (2) and integrating, we have (3) constants of integration. From (1) it is seen that a change in b means simply a different choice of the parallel u = 0. If we take 6 0, the linear element is where b and c are (4) ds 2 =du + c 2 2 cos 2 -^ a . 2 . From (5) (III, 99, 100) r= it follows that the equations of the meridian curve are u cos-&gt; z a it = C \1 J\ / c 2 . a2 jsma 9 u 270 SPHERICAL SURFACES OF REVOLUTION and that v 271 measures the angle between the meridian planes. There are three cases to be considered, according as c is equal greater than, or less than, a. to, CASE r I. c = a. Now z = a cos-) a u a sin a . &gt; and consequently the surface is a sphere. CASE for z it II. c &gt; a. From 2 the expression follows that sin &gt; a &lt; 1 and con- 0. Hence the surface is sequently r made up of zones bounded by minimum parallels whose radii are equal to the ?/ FIG. 26 minimum value of cos ; and the greatest parallel of each zone is of radius c as in &lt; fig. 26, where the curves represent geodesies. from to c, CASE III. c a. is any odd integer. At these mcnr/2, where ing to the value u on the axis the meridians meet the latter under the angle points v? Now r varies = is the former correspond m 1 sin" -. a 27). Hence the surface made up of a series of spindles z (fig. For the cases II and III the expression for can be integrated in terms of elliptic functions.* It is readily found that these two surfaces are applicable to the sphere with the meridians and parallels of each in correspondence. Thus, if we write the linear element of the sphere in the form ds it 2 2 du 2 4- a 2 cos 2 - dv , a follows from (4) that the equations u FIG. 27 = u. It is evident that for values of b other determine the correspondence desired. than zero we should be results. brought to the same However, I, for the sake of future *Cf. Bianchi, Vol. p. 233. 272 reference SURFACES OF CONSTANT CURVATURE we write down when b = 7r/2 and (i) the expressions for the linear element Tr/4 together with (4), thus : ds 2 =du* (6) (ii) ds =du*2 2 (iii) ds =du* cos u TT\ --a , }dv\ 4/ Let S be a surface with the linear element (6, i), and consider the zone between the parallels u = const, and rt 1 = const. A point of the zone is determined by values of u and v such that The parametric are such that values of the corresponding point on the sphere 9 _ the given zone on S does not cover the zone on the sphere between the parallels M O = const, and u^ = const. a it not only covers it, but there is an overlapping. but when c Hence when c &lt; , ; &gt; pseudospherical 116. Pseudospherical surfaces of revolution. In order to find the 2 in (2) by surfaces of revolution we replace I/a K and integrate. This gives V5 = where ct c. cosh a + &lt;?_ sinh - &gt; a and c 2 are constants of integration. We consider first the particular forms of the linear element arising when either of these constants is zero or both are equal. They may be written ds 2 (i) = = du ?/ a (ii) +c oU 2 sinh i a (iii) ds^dtf+fe" dv*. Any case other than these may be obtained by taking for either of the values cosh - or sinh(- where b is a constant. PSEUDOSPHEKICAL SURFACES OF REVOLUTION By 273 a change of the parameter u the corresponding linear elements are reducible to (i) or (ii). Hence the forms (7) are the most general. The corresponding meridian curves are defined C by = c cosh r &gt; 2 = \ sum2 - aw . , U , ; (8) (ii) v w = tfsinha 2= C \ 1 JN I &lt;? a 2 .u -costf-du; a (iii) r = cea . z = We and consider these three cases in detail. I. CASE 0. c. The maximum and minimum values of sinh 2 - are a 2 /e 2 a Hence the maximum and minimum values of r are Va + c 2 2 and At points of a maximum parallel the tangents to the merid ians are perpendicular to the axis, and at points of a minimum parallel they are par allel to the axis. Hence the former is a cus pidal edge, and the latter a circle of gorge, so that the surface is made up of spool-like sections. It is represented by fig. 28, upon which the closed curves are geodesic circles and the other curves are geodesies. These pseudospherical surfaces are said to be of the hyperbolic type.* CASE c 2 II. In order that the surface be real , 2 cannot be greater than a a restriction not necessary in either of the other cases. we put e = asino:,f the maximum and minimum values of cosh2 are cosec 2 o; and 1, and If FIG. 28 a the correspond- ing values of r are a cos a and 0. The tangents to the meridians at points of the former circle are perpendicular to the axis, and at the points for which r is zero they meet the axis under the angle a. Hence the surface . is made up I, p. of a series of parts similar in shape f Cf. * Cf Bianchi, Vol. 223. Bianchi, Vol. I, p. 220. 274 SURFACES OF CONSTANT CURVATURE is to hour-glasses. the curves Fig. 29 represents one half of such a part one of an asymptotic line and the others are parallel geodesies. ; The surface is called a pseudospherical surface of the elliptic type. CASE III. equations of the In the preceding cases the meridian curve can be expressed without the quadrature sign by means of elliptic functions.* In this case the same can be done by means if of trigonometric functions. sin d) For, we put = a ea. equations FIG. 29 (9) (iii) of (8) 2 become cos(/&gt;). r = asin&lt;, = a (log tan^-f We find that point makes curve. Since the length of the segment of a tangent between the point of contact and the intersection with the axis is r cosec or a, the angle which the tangent to a meridian at a with the axis. Hence the axis is an asymptote to the is c/&gt; c/&gt; the length of the segment is independent of the point of contact. Therefore the meridian curve is a tractrix. The surface is of revolution of a tractrix about its asymptote sphere, or the pseudospherical surface of the parabolic type. The surface is shown in fig. called the pseudo- 30, which also pictures a family of line. parallel geodesies and an asymptotic If the integral (3) be written in the form = the cases (i), c, cos u a -f- c sin a 1 of (6) are seen to correspond to the similar cases of (7). shall find other marks of similarity between (ii), (iii) We these cases, but now we desire to call at FIG. 30 tention to differences. of the three forms (7) determines a particular kind of in value pseudospherical surface of revolution, and c is restricted Each *Cf. Bianchi, Vol. I, pp. 226-228. APPLICABILITY only for the second case. 275 On the contrary each of the three forms (6) serves to define any of the three types of spherical surfaces of revolution according to the magnitude of c. (IV, 51) we find that the geodesic curvature of the par allels on the surfaces with the linear elements (7) is measured by From the expressions -. 1 a , - tann . * i a 1 ., - cotn M a a -, 1 - -, a Since no two of these expressions can be transformed into the other if u be replaced by u plus any constant, it follows that two pseudospherical surfaces of revolution of different types are not applicable to one another with meridians in correspondence. show that of Applicability. Now we shall that in corresponding cases of (6) and (7) the parametric geodesic systems are of the same kind, and then we shall prove 117. Geodesic parametric systems. when such a geodesic system is chosen for any surface constant curvature, not necessarily one of revolution, the linear element can be brought to the corresponding form of (6) or (T). place we recall that when on any surface the curves const, their orthogonal trajectories, are geodesies, and u the linear element is reducible to the form (1), where G is, in In the first v = const, = general, a function of both u and v ; and the geodesic curvature of the curves u const, is given by (IV, 51), namely p ff When, is in particular, the curvature of the surface is constant, 2 given by equation (2) in which may by replaced by l/a K . Hence, (11) for spherical surfaces, the general form of V& is a and for pseudospherical surfaces &lt;/&gt; V& = VG = (v) cos - + A/T (v) sin a , (12) &lt; (v) cosh ci + i/r (v) sinh a , where &lt; and i/r are, at most, functions of (7). v. We consider now the three cases of (6) and 276 SURFACES OF CONSTANT CUKVATUKE I. CASE From cv. the forms (i) of (6) and (7), and from its (10), it follows that the curve u = is a geodesic and that arc is Moreover, a necessary and sufficient condition u = on any surface with the linear element (1) that the curve satisfy these conditions is measured by =o. Applying these conditions to (11) and forms (i) of (6) and (7) respectively. (12), we are brought to the CASE II. The forms (ii) of (6) and (7) satisfy the conditions = 0, which are necessary and geodesic polar, in which system be measures angles (cf. VI, 54). When these conditions are applied to (11) and (12), we obtain (ii) of (6) sufficient that the parametric cv and of (7) respectively. III. For (iii) of (6) the curve u = has constant geodesic curvature I/a, and for (iii) of (7) all of the curves u = const, have the same geodesic curvature I/a. Conversely, we find from is satisfied on any sur (11) and (12) that when this condition CASE face of constant curvature the linear element is reducible to one of the forms the theorem : (iii). We gather these results together into The linear element of any surface of constant curvature to the is reducible forms (i), (ii), (iii) of (6) or (7) according as the parametric a point, or are geodesies are orthogonal to a geodesic, pass through to a curve of constant geodesic curvature. orthogonal the linear element of a surface of constant curvature is in one of the forms (i), (ii), (iii) of (6) and (7), it is said to be of the hyperbolic, elliptic, When The above theorem may be or parabolic type accordingly. stated as follows : is applicable to a sphere spherical surface of curvature l/a that to a family of great circles with of radius a in such a way z Any the same diameter there correspond the geodesies orthogonal to a APPLICABILITY given geodesic 277 point of curvature I/ a. it, on the surface, or all the geodesios through any or those which are orthogonal to a curve of geodesic surface of curvature Any pseudo spherical I/a pseudospherical surface of revolution of any according as the latter surface is applicable to a of the three types ; is 2 of the hyperbolic, elliptic, or par abolic type, to its meridians correspond on the given surface geodesies which are orthogonal orthogonal to a geodesic, or pass through a point, or are a curve of geodesic curvature I/a. to In the case of spherical surfaces one system of geodesies can satisfy all three conditions circles for in the case of the sphere the great with the same diameter are orthogonal to the equator, pass through both poles, and are orthogonal to two small circles of ; radius a/V2, whose geodesic curvature is I/a. But on a pseudospherical surface a geodesic system can satisfy only one of these conditions. Otherwise it would be possible to apply two surfaces of revolution of different types in parallels correspond. such a way that meridians and From the foregoing theorems it follows that, in order to carry out the applicability of a surface of constant curvature upon any one of the surfaces of revolution, it is only necessary to find the geodesies on the given surface. set forth in the The nature of this problem is theorem : The determination of the geodesic lines on a surface of constant curvature requires the solution of a Riccati equation. In proving this theorem we consider first defined in terms of any parametric system. It a sphere of the same curvature with center a spherical surface is applicable to at the origin. The tion lines coordinates of u, v, this sphere, expressed as functions of the parameters ( can be found by the solution of a Riccati equa great circles on the sphere correspond geodesic hence the finite equation of ; 65). To is on the spherical surface ax the geodesies constants. + by is + cz = 0, where a, b, c are arbitrary When the surface pseudospherical we use an imaginary is sphere of the same curvature, and the analysis similar. 278 SURFACES OF CONSTANT CURVATURE Let a spherical surface of 2 118. Transformation of Hazzidakis. curvature I/a ters. be defined in terms of isothermal-conjugate parame Then * D D" 1 and the Codazzi equations (V, 13 ) reduce to 1 dE T dG _ ^dF / - - II dv dv du : From The these equations follows the theorem lines of curvature of a spherical surface form an isothermal- conjugate system. For, a solution of these equations is E G = const., When (15) F0. G= cosh 2 a sinh 2 a&gt;. this constant is zero the surface is a sphere because of (13). this case, Excluding we 2 replace the above by a&gt;, E= a 2 cosh F= 0, Now (16) D D n = a sinh a&gt; a&gt;. When these values are substituted in the Gauss equation (V, 12), namely 2 it is # I a^ L^ ^ H ^ HE o&gt; 2 _ + du \ du ULE jv R found that must a ft&gt; satisfy the equation 2 /18} a H--- + smh dv du o) 2 - a) cosh o&gt; A = 0. 2 the quantities (15) Conversely, for each solution of this equation and (16) determine a spherical surface. and v respec If equations (14) be differentiated with respect to u be added, we have tively, and the resulting equations (19) ^ + 0^ "" du 2 dv 2 a change of sign gives a surface * The ambiguity of sign may be neglected, as metrical with respect to the origin. sym TRANSFORMATION OF HAZZIDAKIS In consequence of (14) equation (17) 4 is 279 reducible to H 4 ( \\du] \dv / J L#ti v dv du Equations (14) are unaltered sign of if E and G is F be interchanged and the be changed. The same : true of (17) because of (19) and (20). Hence we have If the linear element of a spherical surface referred conjugate system of parameters be ds 2 to an isothermal- = E du 2 +2F dudv + G dv 2F dudv -f E dv 2 , there exists a second spherical surface of the same curvature referred to a similar parametric system with the linear element ds 2 = Gdu 2 2 , surface moreover, the lines of curvature correspond on the two surfaces. and with the same second quadratic form as the given ; The latter fact is evident from the equation of the lines of curva ture (IV, 26), which reduces to Fdu 2 + dv 2 = 0. (G E) dudv From (IV, 69) it is seen that the linear elements of the spherical representation of the respective surfaces are - -F da 2 = da 2 = -(Gdu CL 2 -2F dudv + E dv + ZFdudv : 2 ), (E du a/ 2 -f- G dv ). 2 In particular we have co 2 the theorem Each solution of curvature I/ a ; of equation (18) determines two spherical surfaces the linear elements of the surfaces are 2 2 ds ds =a =a 2 2 (cosh 2 co 2 (sinh co du 2 + sinh 2 co dv 2 ), du 2 -f cosh 2 co dv-), and of , their spherical representations { 9 -, v j da 2 = sinh 2 co du 2 -f- cosh 2 co dv\ 2 cosh co du 2 + sinh 2 co dv 2 d&lt;r*= ; moreover, their principal radii are respectively pl = a coth = a tanh p[ &), co, p2 p2 = a tanh a coth co, co. 280 SURFACES OF CONSTANT CURVATURE Bianchi * has given the name Hazzidakis transformation to the relation between these two surfaces. It is evident that the former theorem defines this transformation in a more general way. 119. Transformation of Bianchi. spherical surface of curvature conjugate parameters. We We consider now a pseudo2 I/a defined in terms of isothermalhave , H~ D__ #_ H~~ to _1 a 4 and the Codazzi equations reduce ^+ du -2^=0, du dv ^+-2^=0. du dv dv These equations are (22) &&gt; satisfied 2 by the values E= a- cos w, ^=0, G = a2 sin of the 2 to, where is a function which, because must satisfy the equation /c . Gauss equation (V, 12), n , (23) ___ = o2 &lt;"2 &&gt; (0 8in coB. Conversely, every solution of this equation determines a pseudospherical surface whose fundamental quantities are given by (22) (24) and by D= dois 2 D" = 2 a sin w cos to. Moreover, the linear element of the spherical representation (25) is =sin 2 o&gt;c^ +cos 2 a&gt;dv 2 . f There ilar to the not a transformation for pseudospherical surfaces sim Hazzidakis transformation of spherical surfaces, but there are transformations of other kinds which are of great im portance. One of these is involved in the following theorem of Ribaucour : If in the tangent planes 2 I/ a circles of radius a surfaces of curvature * Vol. t a pseudo spherical surface of curvature be described with centers at the points of to contact, these circles are the orthogonal trajectories of an infinity of 1J a made 2 . II, p. 437. is This choice of sign ary form. so that the following formulas may have the custom TKANSFOKMATION OF BIANCHI 281 In proving this theorem we imagine the given surface S referred to its lines of curvature, and we associate with it the moving trihe dral whose axes rt are tangent to the parametric lines. From (22) and (V, 75, 76) it follows that P P\ = cos w = a cos ft), = sin t] l , t =s 0, n r = d(0 cv 77 , r1 = 3(0 vU =a sin &&gt;, fx == 0. In the tangent z^-plane we draw from the origin a segment of length #, and let 6 denote its angle of inclination with the #-axis. The coordinates of the other extremity 1 with respect to these M M axes are a cos 0, a sin 0, 0, a displacement of M l as M and the projections upon these axes of moves over S are, by (V, 51), a L sin 6 dO -f cos oadui \dv du rfw -\ du dv } sin 6 L I J a cos 6 d6 -f sin L &) c?y +( &) + \cv cos du dv } cos / J , a [cos &&gt; sin c?v sin du\. We seek line now l the conditions which must l MM be tangent to the locus of M denoted by S^ and that the tangent plane to S at M be perpendicular to the tangent plane to l satisfy in order that the 1 S at M. Under these conditions the direction-cosines of plane to S with reference to the moving trihedral are l the tangent (26) sin0, -COS0, 0, and since the tangent to the above displacement must be in plane, we have (27) this dO this +( \dv - sin cos co] / du+(+ cos \cu all sin a&gt;\ dv = 0. ) Jtf", As equation must hold for displacements of it is These equations satisfy the condition of integrability in conse is a solution of equation (23), as is quence of (23). Moreover, seen by differentiating equations (28) with respect to u and v respectively and subtracting. 282 SURFACES OF CONSTANT CURVATURE of (28) the above expressions for the projections of a of M^ can be put in the form displacement By means a cos a sin a (cos (cos (cos &) ft) o&gt; + sin cos 9 du + sin cos 6 du &&gt; sin 6 dv), sin 6 dv), o&gt; sin # c?t&gt; sin &) cos du). From these it follows that the linear element of S l is =a ds? In order to prove that its lines 2 (cos 9 is 2 du 2 + sin 2 2 cfrr ). S l of curvature, it remains for us to a pseudospherical surface referred to show that the spherical representation of these curves forms an orthogonal system. obtain this representation with the aid of a trihedral whose vertex is We fixed, and which rotates so that its axes are always parallel to the corresponding axes of the trihedral for S. The point whose coordinates with reference to the new trihedral are given by (26) serves for the spherical representation of Sr The projections upon these axes of a displacement of this point are reducible, by means of (28), to cog e ^ cos sin #(cos sin &) m &) sin sin du du _ sin a cos sin ft) &lt;w dv ^ dv), cos sin du cos cos 9 dv, is from which it follows that the linear element Since is curvature a solution of (23), the surface Sl is pseudospherical, of 2 and the lines of curvature are parametric. To 1/ , each solution 9 of equations (28) there corresponds a surface Sr Darboux * has called this process of finding S1 the transformation the complete integral of equations (28) involves an as remarked arbitrary constant, there are an infinity of surfaces of Bianchi. &gt;S\, As by Ribaucour. (29) Moreover, if we put *-tan|. &lt;. these equations are of the Riccati type in Hence, by 14, one transform of Bianchi of a pseudospherical surface is known, the determination of the others requires only quadratures. * Vol. Ill, p. 422. When TRANSFORMATION OF BIANCHI From ! are (III, 24) it 283 follows that the differential equation of the curves to which the lines joining corresponding points on S and tangent is (30) cos co smddu sin o&gt; cosddv = 0. Hence, along such a curve, equation (27) reduces to 7/1 d6-\ -- du H-- dv da) , d(o i f. 0. dv du But from geodesies have the values (22). (VI, 56) it is seen that this is the Gauss equation of upon a surface whose first fundamental coefficients Hence : The curves on S to which the lines joining corresponding points on S and S l are tangent are geodesies. trajectories of the curves (30) are defined The orthogonal (31) by coswcostfdtt + sinw sinflcto = 0. In consequence of (28) the left-hand member of this equation is an exact differential. d the quantity = a (cos w cos 0du + sin w sin 6dv), (30). e~& a is an integrating factor of the left-hand member of rj Conse quently we may define a function drj thus : = ae~ /a (cos w sin 6 du sin w cos 6 dv) . In terms of (32) and i\ the linear element of 2 &lt;Zs S is expressible in the parabolic form cfys. (7), = d 2 + e^A Equation (31) defines also the orthogonal trajectories of the curves on Si which the lines MMi are tangent, and the equation of the latter curves is sin to w cos 6 du cos w sin 6 dv = 0. The quantity e* /a is an integrating factor of d this equation, and dv) , if we put accordingly = ae /a (sin w cos 6 du cos a; sin the linear element of Si (33) may be expressed in the parabolic form ) dsf = dp + e-*/ a dp. form of the linear element of a surface of by quadratures. Hence : As the expressions (32) and (33) are of the revolution, the finite equations of the geodesies can be found When a Bianchi transformation is known for a surface, the finite equation of its geodesies can be found by quadratures. 117. This follows also from the preceding theorem and the last one of 284 SURFACES OF CONSTANT CURVATURE The transformation of Bianchi 120. Transformation of Backlund. is only a particular case of a transformation discovered by Backlund,* by means of which from one pseudospherical surface S another S^ of the same curvature, can be found. Moreover, on these two sur faces the lines of curvature correspond, the join of corresponding and is of constant points is tangent at these points to the surfaces meet under length, and the tangent planes at corresponding points constant angle. We case, refer S to the same moving trihedral o&gt;axis. and the angle X and 6 denote the length of l The coordinates of 3/x are which the latter makes with the X cos 0, X sin 0, 0, and the projections of a displacement of l are and let MM as in the preceding M X (34) sin d0 -f a cos wdu \ sin 6 ( \0t&gt; du-\ (?M dv } / , \cosOdO X (cos ft) -f a sin&xi*; sin + o&gt; X cos#( \dv . du -\ dv du ), I sin 6 dv cos 6 du) S denotes the constant angle between the tangent planes tP and Jft respectively, since these planes are to inter and Sl at If cr M sect in MMv the direction-cosines of the normal to sin S l are & sin 0, sin a cos 0, cos a. Hence must satisfy the condition a- X sin dB -f a sin or (cos G&gt; sin 6 du 7 sin &&gt; cos dv} X sin &lt;r du H--- dv , \dv 4- cu &) X cos cr (sin cos 6du cos &) sin 0dv) = 0. it is Since this condition must be satisfied for every displacement, equivalent to X X sin a ( \dw /Q /I --[-) = # sin fltf/ a- cos &) sin 6 X cos a sin &) cos 0, sin &lt;r ( --h v Q \ ) = a sin &lt;r sin w cos + X cos a cos &) sin 6. cu *Om (1883). of New Universitets Arsskrift, Vol. XIX ytor med konstant negativ krokning, Lunds Miss Emily Coddington English translation of this memoir has been made by York, and privately printed. An TRANSFORMATION OF BACKLUKD If these equations 285 be differentiated with respect to v and u respect ively, and the resulting equations be subtracted, we have a sin 2 2 cr-X2 =0, a constant. from which erality it follows that X is Without loss of gen we take X = a sin dco\ ( cr. If this value be substituted in the above equations, . we have f sin cr (d6 \du = sin a cos co . cos a & cos 6 . sin o&gt;, dv/ 1 (35) smcr ( ) = cos sin &lt;w + cos er sin cos co, \dv du/ and these equations satisfy the condition of integrability. If they be differentiated with respect to u and v respectively, and the is a solution resulting equations be subtracted, it is found that of (23). In consequence of (35) the expressions (34) reduce to a cos (cos &) cos -f cos a sin &&gt; sin 0) &&gt; du cos a cos C?M &&gt; + a cos (cos co a sin cos cr (sin cos sin 0) dv, sin sin &) cos 0) &) + a sin cr(cosft) sin0o?v &lt;* sin #(sin sin 6 + cos cr cos &) cos sin CD cosOdu), and the linear element of ^ 2 is 2 d** =a (cos &lt;9 dw 2 -f- sin 2 In a manner similar to that of 119 it can be shown that the spherical representation of the parametric curves is orthogonal, and consequently these curves are the lines of curvature on S^ Equations (35) are reducible to the Riccati form by the change of variable (29). Moreover, the general solution of these equations involves two constants, namely cr and the constant of integration. Hence we have the theorem : integration of a Riccati equation a double infinity of pseudospherical surfaces can be obtained from a given surface of this kind. By the We it refer to this as the transformation of Backlund, and indicate by Bv , thus putting in evidence the constant cr. 286 121. SURFACES OF CONSTANT CURVATURE Theorem of permutability. , Let Sl be a transform of S by is means of Sj, of the functions (0 X o^). Since conversely S a transform and the equations for the latter similar to (35) are reducible to the Riccati type, all the transforms of Sl can be found by quad ratures. But even these quadratures can be dispensed with because of the following theorem of permutability due to Bianehi*: If S and S 1 2 are transforms of of functions (0 1? a^ and is l (0 2 , &lt;r 2 ), S by means of the respective pairs a function can be found without &lt;f&gt; quadratures which (&lt;, such that by means of the pairs 2 f &lt;r ((/&gt;, 2) and o-j) the surfaces pseudospherical S and S surface S . respectively are transformable into a By hypothesis sin o-J is &lt;/&gt; a solution of the equations * \H I sin &lt;T 9( 2 */)= /p -^- 4- -^ = ^/l \ ) -- + sin cos l cos cr 2 cos (/&gt; sin 0^ cos 6 sin 0. + cos cr 9 sin 6 cos ^, and also of the equations pi sin &lt;7, p l* 4- - - = sn = d&gt; cos 9 cos a. cos 6 sn (37) cos &lt;&gt; sn + coso-Sn) cos projections of the line If^Tf on the tangents to the lines of and are correspond curvature of Sl and on its normal, where l The M M 1 ing points on (38) and S l , are a sin &lt;r 2 cos &lt;/&gt;, a sin &lt;r 2 sin (#&gt;, ; 0. of The direction-cosines of the tangents to the lines of curvature S with respect to the line JOf1? the line MQ^ perpendicular to the latter and in the tangent plane at J/, and the normal to S are l cos sin to, cos &lt;r 1 sin o&gt;, sin sin &lt;T I sin &), a), cos it cr 1 cos w, o-j cos w. From these and (38) follows that the coordinates of to &lt;r M with respect to MM^ MQ^ l and the normal (j&gt; S 2 are o-j a [sin a- + sin cr 2 cos [sin o1 a))], a [sin &lt;r cos sin (&lt;f&gt; &))], sin 2 sin (&lt;/&gt; w)]. * Vol. II, p. 418. THEOREM OF PEEMUTABILITY Hence the coordinates of 287 M 1 with respect to the axes of the moving trihedral for S are x = cos 0j sin = sin = sin cr o- l 4- cos &lt;r l sin & 2 cos (&lt; (&lt; &&gt;) sin 0, sin 2 cos o^ sin &lt;w), (39) l sin cr 1 + sin &lt;7 6 l sin cr 2 cos ((/&gt; + cos 0j sin sm & a If 2 cos o^ sin ( &lt;w), sin (9 2 the coordinates be transformed by means of a l and the same function c, of the resulting surface can be obtained x", z" y", from by interchanging the subscripts 1 and 2. Evidently z are equal. necessary and sufficient condition that x\ y be equal to respectively is (39) f and z A #", y" cos 1 (d r x") cos If the [sin 2 (x x") + sin B^(y + sin (y z y") 0, r = y") 0. above values be substituted in these equations, we obtain ((/&gt; a l cos (# 2 0^ sin o- 2 ] cos sin a l cos cr 2 sin (# 2 0^ sin &lt;7 co) = sin a (^&gt; &&gt;) l sin cr 2 cos (Q n #J, [sin -f- 2 sin cos (^ 2 cos 2 &lt;7 0j) cr 1 siu crj cos 2 (&lt;/&gt; o&gt;) sin(^ ^)sin(^) w) = sin cr z sin a^ cos(# 9 ) 6^). Solving these equations with respect to sin and cos (&lt;/&gt; (&lt;/&gt; &&gt;), we get sin ,. cos (&lt;f&gt; (0) sin =- sin -^ o-, o-j sin sin &lt;7 2 cos (^ 2 o-- + (cos cos (0 2 /i &lt;r. coscr 9 ^ - cr 1 sin &lt;r 2 c/j) - -^ 2 X) + cos cos o-j cos o- 1 l)cos(^ 9 " ^.) 4- cos r 1 cr 2 These two expressions satisfy the condition that the sum of their satisfies equations (36) and squares be unity, and the function (j&gt; (37). Hence our hypotheses is are consistent and the theorem of permutability demonstrated. We may replace the above equations by 288 SURFACES OF CONSTANT CURVATURE result The preceding When may be expressed in the following form : all the transformations of the the transforms of a given pseudospherical surface are known, former can be effected by algebraic processes and differentiation. Thus, suppose that the complete integral of equations (35) (41) is =/(w, is v, &lt;r, c), and that a particular integral ^i=/( v &gt; *v c i)i of the constants, and let corresponding to particular values denote the transform of S by means of ^ and r All the trans where and formations of S are determined by the functions &lt;7 ^ &lt;r, l &lt;f&gt; cr has the value r For all values Exceptional cases arise when = to + WTT, where m is an of c other than c l formula (42) gives odd integer. When this is substituted in equations (36) they re duce to (35). In this case S coincides with S. We consider now the remaining case where c has the value c 1? In is indeterminate. whereupon the right-hand member of (42) order to handle this case we consider c in (41) to be a function of &lt;r o-, reducing to cl for &lt;r = a-^ L If the function tan for to Ism ~ we apply * the ordinary methods to which becomes indeterminate a &lt;r, = o- numerator and denominator with respect v differentiating we have or tan /6-w\ = sin ^ . ^ /a/ 4- c V . / - , where c is an arbitrary constant. satisfies the * It is necessary to verify that this is value of ^ equations (36), which Cf Bianchi, Vol. . easily done.* II, p. 418. TRANSFORMATION OF LIE 122. Transformation of Lie. 289 of pseudo- Another transformation is analytical in character was spherical surfaces which, however, Lie.* It is immediate when the surface is referred discovered by to its asymptotic lines, or to any isothermal-conjugate system of lines. Since the parameters in terms of which the surface is defined in 119 are isothermal-conjugate, the parameters of the asymptotic lines may be given by In terms of these curvilinear coordinates the linear elements of the surface and its spherical representation have the forms ds da2 2 = a (da + 2 cos = da 2 cos 2 2 2 2 ft) &) dad/3 -f- + d/3 2 ), 2 dad/3 and equation (23) takes the form sin &) cos ft). dad/3 the form of this equation a solution, so also is co l = From it is evident that 9 if &&gt; = &lt;(#, ft) be constant. ft/m) where m is any Hence from one pseudospherical surface we can obtain an infinity of others by the transformation of Lie. It should be remarked, (f&gt;(am, however, that only the fundamental quantities of the new surfaces are thus given, and that the determination of the coordinates re the solution of a Riccati equation which may be different quires from that for the given surface. Lie has called attention to the fact that every Biicklund trans formation is a combination of transformations of Lie and Bianchi.f In order to prove this equations (35) we effect the change of parameters (43) upon and obtain d , n da (44) d (0 v 1 + cos a Q sin (6 + co) = - sin . . . , a x d3 (6 V (w) = 1 : sin tr &)), cos &lt;r er Q sin (9 . + x &lt;). *Archivfor Mathematik og Naturvidenskab, Vol. IV (1879), t Cf. Bianchi, Vol. II, p. 434; Darboux, Vol. Ill, p. 432. p. 150. 290 . SUEFACES OF CONSTANT CUEVATUEE In particular, for a transformation of Bianchi we have (B 4ccc o)) = sin (B o&gt;), dp (6 &&gt;) = sin (6 + &&gt; o&gt;). Suppose that we have a pair of functions 6 and satisfying these equations, and that we effect upon them the Lie transforma tion for which has the value (1 + cos cr)/sin a-. This gives m 1 4- cos _ + a a 1 , cos sin cr cr sin or /1 \ s s in cr sm cr As Ba . these functions satisfy (44), they determine a transformation But O l may be obtained from o^ by effecting upon the latter 1 Z" , an inverse Lie transformation, denoted by upon this result a B n/2 and then a direct Lie transformation, Bianchi transformation, , Z a Hence we may . write symbolically which may be expressed thus : Backlund transformation B is the transform of a Bianchi * transformation ly means of a Lie transformation L a ff A EXAMPLES 1. The asymptotic lines on a pseudospherical surface are curves of constant lines are of the 1. torsion. 2. Every surface whose asymptotic is same length as the curves their spherical images 3. a pseudospherical surface of curvature that on the pseudosphere, defined Show by (9), = where 4. 0, 6 is a constant, are geodesies, and find the radius of curvature of these curves. the linear element of a pseudospherical surface is When (iii) in the parabolic form of (7), the surface defined by y a dy z z x is = x . a 76) dx du pseudospherical (cf * ; y it is a dz cu du a Bianchi transform of the given surface. The Spherical surfaces admit of transformations similar to those of Lie and Backlund. such combinations of them can be made that the resulting surface is real. For a complete discussion of these the reader is referred to chap. v. of the Lezioni of Bianchi. latter are imaginary, but TF-SURFACES 5. 291 The X helicoids = U COS V. fc y = u sin u, z= f */ J \a Idu k~u z u2 + hv, where 6. a, A, are constants, are spherical surfaces. helicoid whose meridian curve is the tractrix is called the surface of Dini. equations when sin denotes the helicoidal parameter and cos the con stant length of the segment of the tangent between the curve and its axis. Show that the surface is pseudospherical. The Find its &lt;r &lt;r 7. The curves tangent to the joins of corresponding points on a pseudospherical surface and on a Backlund transform are geodesies only when = ir/2. &lt;r 8. Let S be a pseudospherical surface and Si a Bianchi transform by means of ( a function d 119). Show that X{ cosw(cos0X1 -f -f sin0JT2 ) sin0JT2 ) sinwJT, X X where .Xi, 2 , % = = sin w (cos&lt;? Xi + coswJT, X X are direction-cosines, with respect to the x-axis, on S and of the normal to S, to the lines of curvature and JT{, XX of the tangents are the %, similar functions for Si. 123. W-surfaces. and surfaces great Fundamental quantities. Minimal surfaces of constant curvature possess, in common with a many cipal radii is a function of the other. first other surfaces, the property that each of the prin Surfaces of this kind were studied in detail by Weingarten, * and, in consequence, are called Weingarten surfaces, or simply W-surfaces. Since the prin cipal radii of surfaces of revolution and of the general helicoids are functions of a single parameter ( 46, 62), these are TF-surfaces. shall find other surfaces of this kind, but now we consider We the properties which are common to TF-surfaces. When a surface S is referred to its lines of curvature, the Codazzi equations may be given the form (45) glogV^ = dv 1 dp^ Pi dv P2 d If a relation exists between p l and /&gt; 2, as the integration of equations (45) r dpi is reducible to quadratures, thus : =Ue J *-*, Crelle, Vol. V^= Ve r J Pl &lt;/p 2 ~ P2 , LXII (1863), pp. 160-173. 292 JF-SUKFACES and V are functions of u and v respectively. Without changing the parametric lines the parameters can be so chosen that the above expressions reduce to where U / A T\ re and r dp, I a o r^ f_dpj I expressible as functions of p l or /3 2 , and conse are functions of one another. This relation becomes quently they more clear when we introduce an additional parameter K defined by (48) Thus ^ are * =* / &lt;*pi *-* we have a By the elimination of p 2 from this equation and (46) j\( \ relation of the form When this value is substituted in (48) we obtain where the accent indicates differentiation with respect (47) it to K. From follows that -. V^=, K -, ^=T, &lt;/&gt; When these values are substituted in the Gauss equation for the sphere (V, 24), the latter becomes 1/* du \ &lt;/&gt;" M du) , jL/* dv \tc* aY dv) _1. = K&lt;f&gt; but This equation places a restriction upon the forms of K and it is the only restriction, for the Codazzi equations (45) are &lt;(), satisfied. Hence we have the theorem of Weingarten * : When one has an orthogonal system on the unit sphere for which the linear element is reducible to the form there exists a W-surface whose lines of curvature are represented by this system and whose principal radii are expressed by ft (50) =*(*), P2 = *(*)- * (*) Z.c., p. 163. FUNDAMENTAL QUANTITIES If the 293 functions of coordinates of the sphere, namely X, Y, Z, are known u and v, the determination of the JF-surface with this For, from the formulas representation reduces to quadratures. of Kodrigues (IV, 32) we have x = = r dX Pi du du 7 , J y / cu +p + dX 2 , dv, dv C pi dY , dY /? 2 , ^r~ dv, cz = ~ rft ~^~ ^ + P* a^ dv , - J / du v" dv The right-hand members of these equations are exact differentials, A", since the Codazzi equations (45) have been satisfied. If F, Z are not known, their determination requires the solution of a Riccati equation. is The relation between the radii of the form (46) obtained by eliminating K from equations (50). find readily that the fundamental quantities for the sur We face have the values (51) And from (52) (48), (50), and (51) we obtain t &lt; pi Ve = p^ "- ft , vG = p,e _ r Jf Pi -p &gt;. Consider the quadratic form (53) H [(EJJ -FD) du . 1 + (El)"- GD) dudv + (FD"- GD is ) dv*], which when equated to zero defines the lines of curvature. When these lines are parametric, this quadratic form means of (IV, 74) to reducible by But consequence of (47) this is further reducible for JF-surfaces to Since the curvature of this latter form is zero, the curvature of (53) also is zero, and consequently ( 135) the form (53) is redu cible by quadratures to dudv. Hence we have the theorem of Lie in dudv. : The lines of curvature of a W-surface can be found by quadratures. 294 JF-SURFACES The evolute of a JF-surface pos 124. Evolute of a W-surface. sesses several properties results of which are characteristic. Referring to the 75, by means of (52) the linear elements of the sheets of the evolute of a JF-surface are reducible to the form we see that or, in terms of K, (55) From these results and the remarks of : 46 we obtain at once the following theorem of Weingarten Each surface of center of a W-surface is applicable to a surface of revolution whose meridian curve is determined by the relation between the radii of the given surface. We have also the converse theorem, likewise due to Weingarten : If a surface Sl be applicable to a surface of revolution, the tan meridians of the gents to the geodesies on S^ corresponding to the surface of revolution are normal to a family of parallel W-surfaces; the relation between the if Sl be deformed in any manner whatever, radii of these W-surfaces is unaltered. In proving this theorem linear element of Sl be i we apply i*** the results of r ? t 76. If the the principal radii of /tM . S are given by V p^u, (56) ft--^7U alone, Since both are functions of a single parameter, a relation exists between them which depends upon unaltered in the deformation of and consequently is Sr trihedral for 8^ (V, 99) the projections upon the moving of a displacement of a point on the complementary surface 2 are From (___), 0, (qdu-, ai U ~ /ir EVOLUTE OF A JF-SUBFACE is 295 In consequence of formulas (V, 48, 75) the expression U(q du + q t dv) an exact differential, which will be denoted by dw. Hence the 2 linear element of (57) is dl = it l from which revolution.* follows that Sz also is applicable to a surface of The last theorem of 75 may be stated thus : A necessary and sufficient condition that the asymptotic lines on correspond is that S be a W-surface ; in this case to every conjugate system on Sl or S2 there corresponds a conjugate system on the other. the surfaces of center S^ S2 of a surface S From (58) . (V, 98, 98 ) it follows that when S is a TF-surface, and only in this case, we have . . ^-E^bsame kind. is Hence at corresponding points the curvature is of the afforded by the case where (46) one or both of the principal radii is constant. For the plane both radii are infinite for a circular cylinder one is infinite and the other ; An exceptional form of equation has a finite constant value. The sphere if is the only surface with both For, p r and p 2 are different constants, from (45) it follows that and ^ are functions of u and v respec tively, which is true only of developable surfaces. When one of the radii is infinite, the surface is developable. There remains the case radii finite and constant. where one has a finite constant value ; then S is a canal surface In considering the last case then, from (48), ( 29). we take we have is and the linear element of the sphere do* = ~+ K dv\ Conversely, when the linear element of the sphere is reducible to this form, the curves on the sphere represent the lines of curvature on an infinity of parallel canal surfaces. * Cf. Darboux, Vol. Ill, p. 329. 296 TF-SURFACES mean curvature. For surfaces of con 125. Surfaces of constant stant total curvature the relation (46) may be written where c denotes a constant. When this value is substituted in (48) we have, by integration, (59) P is so that the linear element of the sphere (60) Conversely, when we have an orthogonal system on is which the linear element the sphere for reducible to the form (60), it serves for the representation of the lines of curvature of a surface of constant curvature, and of an infinity of parallel surfaces. When c is positive, two of these parallel surfaces have constant mean curvature, as follows from the theorem of Bonnet fact, the radii of these surfaces tifcT (61) If (73). In pl =^/ K *+cy/~c J p9 = -=L== vK ~r~ V~ C . c we put c (62) =a 2 , ic =a csch &&gt;, and replace u by au, the linear da-- element (60) becomes 2 = sinlr co du + cosh 2 o&gt; dv 2 . r In like manner, (63) if we c replace u by iau, v by iv, and take = a 2 , K = 2 ai sech CD, the linear element of the sphere da2 2 is = cosh w du + we sinh 2 &) dv*. For the values (62) have, from (61), and the linear elements a of the corresponding surfaces are a (65) &lt;f* =ffV aw (dw SURFACES OF CONSTANT MEAN CURVATURE Moreover, for the values (63) the radii have the values 297 cosh co sinh (65). co but the linear elements are the same curvature is In each case the mean l/. We state these results in the following form: upon a surface of constant mean curvature an isothermic system, the parameters of which can be chosen form so that the linear element has one of the forms (65), where co is a The lines of curvature solution of the equation l (67) du ^ 2 4- dv ^ 42 sinh co cosh co = 0. Conversely, each solution of this equation determines two pair* of l/a, whose lines applicable surfaces of constant mean curvature of curvature correspond, and for which the radii p^ p 2 of one surface are equal to the radii of p 2 p^ of the applicable surface. , It can be shown that is co if co = (u, v sin cr, v) is a solution of equation (67), so also (68) 1 = cf)(u cos cr u sin &lt;r + v cos cr), where cr is any constant whatever. Hence there exists for spherical surfaces a transformation analogous to the Lie transformation of pseudospherical surfaces. This transformation can be given a geo metrical interpretation if it is considered in connection with the sur faces of constant mean curvature parallel to the spherical surfaces. Let Sl denote the surface with the linear element (69) If ds 2 = aV w cr, 2 &gt; (du + dv =u 2 ). we put u v (70) =u cos cr v sin co l v1 l, sin cr + v cos cr, the solution (68) becomes = cf)(u v^), and (69) reduces to if we make a point (u, v) on S with the linear element (65), which the positive sign is taken, correspond to the point (u v vj on 8^ the surfaces are applicable, and to the lines of curvature u = const., v = const, on S correspond on Sl the curves Hence in u cos cr v sin a = const., u sin cr -f- v cos cr = const. 298 ^-SURFACES latter cut the lines of curvature But the l u = const., v = const, on S under the angle a-. Moreover, the corresponding principal radii of S and S are equal at corresponding points. Hence we have tha l * following theorem of Bonnet : A surface of constant mean curvature admits an and infinity of appli cable surfaces of the same kind with preservation of the principal radii at corresponding points, the lines of curvature on one the lires surface correspond to lines on the other which cut curvature under constant angle. of of Weingarten has considered the IF-surfaces whose lines curvature are represented on the sphere by geodesic ellipses and hyperbolas. In this case the linear element of the sphere is reducible to the form ( 90) do* = sm Comparing this CS *2 *2 with /c (49), we have ., &lt; = .to sin-&gt; . =cosft) ft) from which it follows that to -f- sin 4 Hence &) -f sin ft) ft) sin and the relation between the radii is found, by the elimination of w, to be (72) 2(^-^)=sin2(^+/) , 2 ).t * Memoire sur la theorie des surfaces applicables sur une surface donnce, Journal de solves com VEcole Poly technique, Cahier 42 (1867) pp. 72 et seq. In this memoir Bonnet surfaces with corresponding principal radii equal. pletely the problem of finding applicable When a surface possesses an infinity of applicable surfaces of this kind, its lines of curv ature form an isothermal system. follows: tDarboux (Vol. Ill, p. 373) proves that these surfaces may be generated as locus of the Let C and Ci be two curves of constant torsion, differing only in sign. The of translation. of the join of any points P and PI of these curves is a surface mid-points of the osculating planes of C and If a line be drawn through parallel to the intersection above type for all positions of M. Ci at P and Pi, this line is normal to a IP-surface of the M M RULED JF-SURFACES 126. 299 Ruled W-surfaces. We conclude the present study of Tr-surfaces with the solution of the problem : To determine the W-surfaces which are ruled. This problem was proposed and solved simultaneously by Beltrami* and Dini.f We follow the method of the latter. In 106, 107 we found that when the linear element of a ruled surface is in the form 2 ds 2 = du + [(u - a) + /3 2 2 ] dv\ are the expressions for the total and 2 mean curvatures ~ where r is /3 = a function of v at most, and /=(tt~a) -h^. In order that a relation exist between the principal radii necessary and sufficient that the equation it is a 1* a* du dv dv *jr.-l*:-o du above values be substituted, the be satisfied identically. If the resulting equation reduces to 2u a d rr 2 +/3 ! u-a a l \ As this is it case an identical equation, reduces to /3 =0. Hence r (u it is /3 is 2 true when u = a, in which a constant and the above equation becomes of + r ft + /3a" = 0. Since this equation must be true independently of the value of w, both r and are zero. Therefore we have a" (73) a=cv + &lt;?, d, P= e, r = k, where d, e, k are constants. is The linear element ds * 2 = du2 + [(t* - cv - d) + e 2 t 2 ] dv 2 . Annali, Vol. VII (1865), pp. 13&-150. Annali, Vol. VII (1865), pp. 205-210. 300 SURFACES WITH PLANE LINES OF CURVATURE for In order to interpret this result we calculate the expression the tangent of the angle which the generators v = const, make with the line of striction u cv d = 0. From (III, 24) we have tan d =c ; 6 the param consequently the angle is constant. Conversely, if and eter of distribution j3 be constant, a has the form (73). Hence we have the theorem : be a necessary and sufficient condition that a ruled surface distribution be constant and that is that the of W-surface A parameter is the generators be inclined at a constant angle to the line of stric tion, which consequently a geodesic. EXAMPLES 1. Show that the helicoids are ^surfaces. 2. Find the form of equation (49), when the surface is minimal, and show that each conformal representation of the sphere upon the plane determines a minimal surface. 3. Show the linear element that the tangents to the curves v = const, on a spherical surface with of (6) are normal to a TT-surface for which (i) P-2 - PI = COt - 4. The const, helicoids are the only &gt;F-surfaces Pi = 5. meet the lines of curvature lines which are such that the curves under constant angle (cf. Ex. 23, p. 188). The asymptotic Pz ; Pl + const, correspond to the of the surface and, when /&gt;i on the surfaces of center of a surface for which minimal lines on the spherical representation on the sphere. p 2 = const., to a rectangular system 127. Spherical representation of surfaces with plane lines of curvature in both systems. Surfaces whose lines of curvature in one or both systems are plane curves have been an object of study to a line of curvature and by many geometers. Since the tangents to its spherical representation at corresponding points are parallel, a plane line of curvature is represented on the sphere by a plane is plane curve, that is, a circle and conversely, a line of curvature ; when its spherical representation is a circle. SPHERICAL REPRESENTATION 301 We lines consider first the of curvature in determination of surfaces with plane both systems from the point of view of their spherical representation.* To this end we must find orthog onal systems of circles on the sphere. If two circles cut one another orthogonally, the plane of each must pass through the pole of the plane of the other. Hence the planes of the circles of one system pass through a point in the plane of each circle of the second system, and consequently the planes of each family form a pencil, the two axes being polar reciprocal with respect to the sphere.f consider separately the two cases I, when one axis is tan gent to the sphere, and therefore the other is tangent at the same We : point and perpendicular to it ; II, when neither is tangent. CASE x- I. We and ?/-axes parallel to the take the center of the unit sphere for origin 0, the axes of the pencils, and let the coor dinates of the point of contact be (0, 0, 1). pencils of planes may be put in the form (74) The equations of the- x v + u(z are 1)=0, y + v(z 1) = 0, where u and the parameters of the respective families. If these equations be solved simultaneously with the equation 7 of the sphere, and, as usual, X, I Z denote coordinates of the , latter, we have v Now (T6) - ^v r7 _u?~ the linear element of the sphere is ^= JtXl?- CASE II. As in the preceding case, we take for the z-axis the common perpendicular to the axes of the pencils, and for the xand ?/-axes we take lines through parallel to the axes of the coordinates of the points of meeting of the latter with the z-axis are of the form (0, 0, a), (0, 0, I/a). The equa tions of the two pencils of planes could be written in forms pencils. * t The Bianchi, Vol. II, p. 256; Darboux, Vol. I, p. 128, and Vol. IV, p. 180. (1853), pp. 136, 137. Bonnet, Journal de I Ecole Poly technique, Vol. XX 302 SURFACES WITH PLANE LINES OF CURVATURE similar to (74), but the expressions for X, Y, Z will be found to be of a more suitable form if the equations of the families of planes be written tanw atanhv Proceeding as in Case I, we find Vl cosh v (77) a sin u 2 + a cos u a sinh v -f 2 1 Y=- cosh v a cos u Z= and the linear element (78) is cos u -\- a cosh v cosh v -f- a cos w (cosh v + a cos w) we have tacitly excluded the sys and parallels. As before, the planes of the two families of circles form pencils, but now the axis of one pencil passes through the center of the sphere and the other is at infinity. From the preceding discussion tem of meridians Hence fact, if this case corresponds to the value zero for a in Case II. In we put a = referred to a system _ ( in (77), the resulting equations define a sphere of meridians and parallels, namely JL I Q V ) JL -sinw - t sinhv -- - &gt; Z/ -cosw - cosh v cosh v cosh v Since the planes of the lines of curvature on a surface are parallel const, on a to the planes of their spherical images, the curves v surface with the representation (79) lie in parallel planes, and the planes of the curves u = const, envelop a cylinder. These surfaces shall consider them later. are called the molding surfaces.* We 128. Surfaces with plane choice of lines of curvature in both systems. By a suitable coordinate axes and parameters the expressions for the direction-cosines of the normal to a surface with plane lines of curvature in both systems can be given one * These surfaces were first studied trie, by Monge, Application de L Analyse a la Geomt- 17. Paris, 1849. IN of the forms (75) or (77). BOTH SYSTEMS 303 surfaces of this kind it For the complete determination of all remains then for us to find the expres sion for the other tangential coordinate W, that is, the distance from the origin to the tangent plane. The linear element of the sphere in both cases is of the form d(T = 7 2 -g du 2 + dv 2 -&gt; where \ (80) is such that -^- = 0. cudv (VI, 39) From we see that the equation satisfied gfl by W is cucv dv du log du X d6 dv _Q In consequence of (80), if we change the unknown function in accordance with B l =\0 the equation in 6 l is of the form (80). &gt; Hence the most general value *for W is where U and V are arbitrary functions of u and v respectively. Hence any surface with plane lines of curvature in both systems is the envelope of a family of planes whose equation is of the form (81) 2 ux + 2 vy + (u*+ v -l)z = 2 (U+V), 2 or (82) Vl a" sin ux Vl a sinh vy 2 . 2 + (cos u + a cosh v) z = (U+ F)Vl-a The expressions for the Cartesian coordinates of these surfaces 67. Thus, can be found without quadrature by the methods of for the surface envelope of (81) we have to solve for x, y, z equa tion (81) (83) and its derivatives with respect to u and v. The latter are x + uz = Z7 , y + vz = V\ where the accents indicate differentiation. We shall not carry out this solution, but remark that as each of these equations contains a single parameter they define the planes of the lines of curvature. 304 SURFACES WITH PLANE LINES OF CURVATURE the form of (83) it is From seen that these planes in each sys tem envelop a these two cylin cylinder, and that the axes of This fact was remarked by Darboux, ders are perpendicular. who also observed that equation (81) defines the radical plane of the two spheres These are the equations whose centers lie on the of two one-parameter families of spheres, focal parabolas -U, and whose radii are 2/1=0, determined by the arbitrary functions its U and V. The characteristics of each famity are defined by the corresponding equation of the pair (83). Consequently the orig inal surface is the locus of the point of intersection of the planes of these characteristics and the radical planes of the spheres. Similar results follow for the equation (82), which defines the radical planes of two families of spheres whose centers are on the focal ellipse equation and and hyperbola (86) a; 2 =0, 2/ 2 = = 0, When in particular a these curves of center are a circle and r i its axis. From the foregoing results it follows that these surfaces to may be : generated by the following geometrical method due Darboux * be Every surface with plane lines of curvature in two systems can obtained from two singly infinite families of spheres whose centers lie on focal conies and whose radii vary according to an arbitrary law. The surface belonging infinitely to is the S and 2, envelope of the radical plane of two spheres two different families. If one associate with S and 2 two S and 2 f , near spheres the radical center of these the radical four spheres describes the surface ; and of 2 and 2 are the planes of the lines of curvature. * Vol. i, and planes of S and S p. 132. SURFACES OF MONGE 129. Surfaces with plane lines of 305 curvature in one system. the lines of curvature in one system Surfaces of Monge. are plane, the curves on the sphere are a family of circles and this When and conversely. Every system of ; be obtained from a system of circles and their may orthogonal trajectories in a plane by a stereographic projection. their orthogonal trajectories kind The determination of such a system in the plane reduces to the integration of a Riccati equation (Ex. 11, p. 50). Since the circles are curves of constant geodesic curvature we have, in consequence of the first theorem of 84, the all the theorem : The determination of quadratures. surfaces with plane lines of curva ture in one system requires the solution of a Riccati equation and lines of curvature in with plane one system, and begin with the case where these curves are geodesies. They are consequently normal sections of the surface. We shall discuss at length several kinds of surfaces Their planes envelop a developable surface, called the director-developable, and the lines of curvature in the other sys tem are the orthogonal trajectories of these planes. Conversely, the locus of any simple infinity of the orthogonal trajectories of a one-parameter system of planes is a surface of the kind sought. For, the planes cut the surface orthogonally, and consequently they are lines of curvature and geodesies ( 59). Since these planes are the osculating planes of the edge of regression of the developable, the orthogonal trajectories can be found by quadratures ( 17). Suppose that we have such a surface, and that C denotes one of the orthogonal trajectories of the family of plane lines of curvature. Let the coordinates of C be expressed in terms of the arc of the curve from a point of it, which will be denoted by v plane of each plane line of curvature F is normal to C at . As its the point of meeting with the latter, reference to the moving the coordinates of a point trihedral of C are 0, 77, f. Since P of F with P describes 82) an orthogonal trajectory of the planes, we must have (I, dv 306 SURFACES WITH PLANE LINES OF CURVATURE C. where r denotes the radius of torsion of parameter of If we change the C in accordance with the equation the above equations become The (88) general integral of these equations ?; is = U^ cos v l U 2 sin v^ f = U^ sin v l -f ?72 cos v lt where C^ and ?72 are functions of the parameter u of points of F. l When v = we have v = 0, and so the curve F in the plane through = U^ ?= Z7 Hence the of C has the equations the point v = character of the functions U^ and U is determined by the form of the curve and conversely, the functions U and U determine the 77 2 . 2 ; } 2 character of the curve. By definition (87) out in the plane normal to the function v t measures the angle swept C by the binormal of the latter, as this = to any other point. Hence equations (88) plane moves from v define the same curve, in this moving plane, for each value of v^ but it is the angle v r defined with respect to axes which have rotated through Hence we have the theorem : surface whose lines of curvature in one system are geodesies can be generated by a plane curve whose plane rolls, without slipping, Any over a developable surface. These surfaces are called the surfaces of Monge, by whom they were first studied. He proposed the problem of finding a surface with one sheet of the e volute a developable. It is evident that the above surfaces satisfy this condition. only solution. ment lie in the plane tangent along this element, and if these tangents are normals to a surface, the latter is cut normally by this plane, and consequently the curve of intersection is a line of curvature. of Moreover, they furnish the the tangents to a developable along an ele For, Monge In particular, a molding surface ( 127) with a cylindrical director-developable. is a surface Since every curve in the moving plane of the lines of curva ture generates a surface of Monge, a straight line in this plane MOLDING SURFACES 307 generates a developable surface of Monge. For, all the normals to the surface along a generator lie in a plane ( 25). Hence: necessary and sufficient condition that a curve F in a plane normal to a curve C at a point Q generate a surface of Monge as A plane moves, remaining normal to the curve, is that the joining a point of T to Q generate a developable. the . line the orthogonal trajectory C is a curves F are perpendicular to the plane curve, the planes of the plane of C, and consequently the director-developable is a cylinder whose right section is the plane evolute of C. The surface is a 130. Molding surfaces. When molding surface ( 127), and all the lines of curvature of the sec ond system involutes of the right section of the cylinder. Hence a molding surface may be generated by a plane curve whose plane rolls without slipping over a cylinder. are plane curves, We shall apply the preceding formulas to this particular case. are Since 1/r is equal to zero, it follows from (88) that ?; and If all functions of u alone. u be taken as a measure of the arc of the curve F, we have, in ?; generality, = U C U, f = I Vl U 2 du, If where the function plane of the curve determines the form of F. for 2 we take the = 0, and XQ yQ denote the coordinates , of a point of C, the equations of the surface may 2 be written / x = x + U cos v, Since y = 2/o + u sin v ^ i = Vl to U 2 du, where v denotes the angle which the principal normal the a&gt;axis. C makes with ^x^ = sin v ( , = C, cos v, if V denote the radius of curvature of * : then ds Q = V dv, and the equations of the surface can be put in the following form, given by Darboux ( r v -fI Fsin J I v dv, (89) = U sin v V cos v dv, * Vol. I, p. 105. 308 SURFACES WITH PLANE LINES OF CURVATURE of the right section of the cylinder are The equations x = X + V cos v = Q I V cos v dv, V sin v dv. y In passing, faces, =y Q -f- V sin v = I we remark whose 0. that surfaces of revolution are molding sur this corresponds to the director-cylinder is a line ; case V EXAMPLES a surface is 1. When the spherical representation of the lines of curvature of isothermal and the curves in one family on the sphere are circles, the curves in the other family also are circles. 2. If the lines of curvature in one system on a minimal surface are plane, those in the other 3. system also are plane. that the surface _|_ Show x au sin u cosh v, its lines y = v + a cos u sinh v, z V 1 a 2 cos u cosh v, is minimal and that of curvature are plane. Find the spherical representa tion of these curves 4. and determine the form of the curves. Show that the surface of Ex. 3 and the Enneper surface (Ex. 18, p. 209) are the only minimal surfaces with plane lines of curvature. 5. is When the lines of curvature in one system lie in parallel planes, the surface of the molding type. 6. A necessary and sufficient condition that the lines of curvature in one system on a surface be represented on the unit sphere by great circles is that it be a sur face of Monge. 7. Derive the expressions for the point coordinates of a molding surface by the of 67. method 131. Surfaces of Joachimsthal. Another interesting class of surfaces with plane lines of curvature in one system are those for which all the planes pass through a straight line. Let one of these lines of curvature be denoted by F, and one of the other system by C. The developable enveloping the surface along the latter has for its elements the tangents to the curves F at their points of intersection with 0. Since these elements lie in the planes of the curves F, the developable is a cone with its vertex on the line to the through which all these planes pass. This cone is tangent surface along (7, and its elements are orthogonal to the latter. Con Z&gt;, sequently C is the intersection of the surface and a sphere with SURFACES OF JOACHIMSTHAL 309 center at the vertex of the cone which cuts the surface orthogo * nally. Hence we have the following result, due to Joachimsthal : the lines of curvature in one system lie in planes passing a line D, the lines of curvature in the second system lie on through and which cut the surface orthogonally. spheres whose centers are on When D Such surfaces curves of the are called surfaces of Joachimsthal. Each of the circles first system is an orthogonal trajectory of the in which the spheres are cut by its plane. Therefore, in order to derive the equations of such a surface, we consider first the orthog onal trajectories of a family of circles whose centers are on a line. If the latter be taken for the f ?;-axis, the circles are defined by = r sin 0, 77 = r cos 6 + u, where r denotes the radius, 6 the angle which the latter makes with the ?;-axis, and u the distance of the center from the origin. Now r is a function of u, and 6 is independent of u. In order that these same equations may define an orthogonal trajectory of the circles, 6 must be such a function of u that cos or r 0^- sin 0^ = du cu 0, f^_ sin = du tan| . By integration we have (90) = r F&lt;/ , where V denotes the constant of integration. Since each section of a surface of Joachimsthal by a plane through its axis is an orthogonal trajectory of a family of circles whose centers are on this axis, the equations of the most general surface of this kind are of the form x = r sin 6 cos v, y = r sin 6 sin v, z u -f r cos #, where v denotes the angle the axis makes with the plane which now V is a function of v. * Crelle, Vol. which the plane through a point and y 0, and 6 is given by (90), in LIV (1857), pp. 181-192. 310 SURFACES WITH PLANE LINES OF CURVATURE constant is When V is is a function of u alone, and the surface For other forms of Vihe geometrical genera tion of the surfaces is given by the theorem one of revolution. : Given ters lie the orthogonal trajectories of a line family of circles whose cen on a right D; to ferent angles, according surface of Joachimsthal. through dif a given law, the locus of the curves is a if they be rotated about D 132. Surfaces with circular lines of curvature. We consider next surfaces whose lines of curvature in one Let o- denote the constant angle between system are circles. the plane of the circle the tangent planes to the surface along C (cf. 59), p the radius of normal curvature in the direction of C, and r the radius C and of the latter. (91) Now equation (IV, 17) r may be written = p sin a. the theorem : As an immediate consequence we have A necessary be the and all sufficient condition that be a circle is that the a plane line of curvature normal curvature of the surface in its direction points. same at of its Since the normals to the surface along C are inclined to its plane under constant angle, they form a right circular cone whose vertex is on the axis of C. Moreover, the cone cuts the surface at right and center at the angles, and consequently the sphere of radius p vertex of the cone surface is is tangent to the surface along C. Hence the the envelope of a family of spheres pf variable or con stant radius, whose centers lie on a curve. Conversely, we have seen in the family of spheres 29 that the characteristics of where x, y, z are the coordinates of a curve expressed in terms of its of radius arc, and 11 is a function of the same parameter, are circles (92) r whose axes are tangent have the coordinates (93) to the curve of centers and whose centers xl = x - aRR , y CIRCULAR LIKES OF CURVATURE where a, ft, 311 indicates differentiation. characteristic 7 are the direction-cosines of the axis, and the accent The normals to the envelope along a form a cone, and consequently these circles are lines of curvature upon and it. Hence : of curvature in one family be circles is that the surface be the envelope of a single infinity of spheres, the locus of whose centers is a curve, the radii being determined by an arbitrary law. A necessary sufficient condition that the lines From surfaces equations (91), (92) it follows that R cos a. is, Hence the for canal circles are geodesies only ( when R is is constant, that 29). In this case, as seen from (92), all the circles are equal. The circles are likewise of equal radius a when where s is the arc of the curve of centers and c is a constant of integration. Now (s equations (93) a. become (s ^=x ( + c) y l =^y also of + c}IB, : zl =z (s + c)y, which are the equations 21). an involute of the curve of centers This result be may be stated thus* extremity M generates be If a string a curve in such a way that its moving a circle an involute of the curve, and if at constructed whose center is and whose plane is normal to the unwound from M M string, then as the string is unwound this circle generates a surface with a family of equal circles for lines of curvature. locus of the centers of the spheres enveloped by a surface is evidently one sheet of the evolute of the surface, and the radius of the sphere is the radius of of The normal curvature in the direction this the circle. Consequently radius is a function 75, of the that parameter of the spheres. when 2 is a curve H= 2 0, Conversely, from and consequently we have Cf. Bianchi, Vol. II, p. 272. 312 SURFACES WITH PLANE LINES OF CURVATURE Excluding the case of the sphere, we have that p.2 is a function of u alone. From the formulas of Rodrigues (IV, 32), dX _ d^~~~ Pz ~^ 2x dy 1 _ dY dz fo~~ p2 ~fo Tv~~ pz ^v _ a we have, by integration, Hence the points of the surface lie a 2 ) on the spheres , - U, + (y - tg + (z - P,) = ft (x and the spheres are tangent to the surface. Since the normals to a surface along a circular line of curvature form a cone of revolution, the second sheet of the e volute is the envelope of a family of such cones. The characteristics of such a family are conies. Hence we have the theorem : necessary and sufficient condition that one sheet of the evolute of a surface be a curve is that the surface be the envelope of a single infinity of spheres ; the second focal sheet is the locus of a family of conies. 133. Cyclides of Dupin. A that it is if the preceding theorem it results also the second sheet of the evolute of a surface be a curve, From a conic, and then the first sheet also is a conic. Moreover, these conies are so placed that the cone formed by joining any point on one conic to all the points of the other is a cone of revolution. pair of focal conies is characterized by this property. And so A we have the theorem : A both families be circles necessary and sufficient condition that the lines of curvature in is that the sheets of the evolute be a pair of focal conies.* These surfaces are called the cy elides of Dupin. They are the envelopes of two one-parameter families of spheres, and all such sphere of one family touches envelopes are cyclides of Dupin. each sphere of the other family. Consequently the spheres of which the cyclide is the envelope are tangent to three spheres. A We is shall prove the converse theorem of Dupin f to : The envelope of a family of spheres tangent a cyclide. * Cf. Ex. 19, p. 188. t three fixed spheres Applications de geomttrie et de mechanique, pp. 200-210. Paris, 1822. CYCLIDES OF DUPIN 313 The plane determined by the centers of the three spheres cuts the latter in three circles. If any point on the circumference (7, orthogonal to these circles, be taken for the pole of a transforma tion by reciprocal radii (cf. 80), C is transformed into a straight line L. Since angles are preserved in this transformation, the three fixed spheres are changed into three spheres whose centers are on L. Evidently the envelope of a family of spheres tangent to these three is spheres a tore with tore. L as axis. Hence the given envelope is trans However, the latter surface is the envelope of a second family of spheres whose centers lie on L. Therefore, if the above transformation be reversed, we have a second family of spheres tangent to the envelope, and so the latter is a cyclide of Dupin. We shall now find the equations of these surfaces. Let (x^ y^ zj and (#2 y 2 z 2 ) denote the coordinates of the points on the focal conies which are the curves of centers of the , , formed into a and (94) jR 1? E 2 the radii of the spheres. ( Xi The spheres, condition of tangency is -x the case where the evolute curves are the focal We consider first parabolas defined by (85). Now equation (94) reduces to Since is 2i l and R z are functions of u and v respectively, this equation equivalent to where a surfaces. is an arbitrary constant whose variation gives parallel By the method of 132 we find that the coordinates (f, 77, f) of the centers of the circular lines of curvature u const, and the radius p are 9-0, 314 SURFACES WITH PLANE LINES OF CURVATURE if be a point on the circle and 6 denote the angle which the radius to P makes with the positive direction of the normal to Hence P the parabola (85), the coordinates of P are 2 x = fH ^ cos 2 v1 +u is 0, ^ = p sin 0, = ? -- cos if 9. Vl + This surface If algebraic and of the third order. the evolute curves are the focal ellipse and hyperbola (86), we have (96) R = - (a cos u + *), l -! A = - (cosh v 2 - /c), where /c is an arbitrary constant whose variation gives parallel surfaces. This cyclide of Dupin is of the fourth degree. When in particular the constanta is zero, the surface is the ordinary tore, or anchor ring.* with spherical lines of curvature in one system. Surfaces with circular lines of curvature in one system belong evi dently to the general class of surfaces with spherical lines of curva 134. Surfaces consider now surfaces of the latter kind. ture in one system. S be such a surface referred to its lines of curvature, and Let const, be spherical. The coordinates in particular let the lines v of the centers of the spheres as well as their radii are functions of v alone. We = They will be denoted ( thal s theorem by (V^ F2 F3 ) and It. By Joachimseach sphere cuts the surface under the same 59) , angle at all its points. Hence for the family of spheres the expres sion for the angle is a function of v alone ; AVC call it V. Since the direction-cosines of the tangent to a curve u = const, are dX when the do2 3Y 1 dZ = (odu linear element of the spherical representation 2 -}dv\ the coordinates of S are of the form is written R sin VdX = VA---=+XR , , cos F, T_ /07\ (97) y =F + 2 Tr + YR _ cos F, 7 sin F dZ * to the article in the For other geomotrical constructions of the cyclides of Dupin the reader is referred Encyklopadie der Math. Wissenschaflen, Vol. Ill, 3, p. 290. SPHERICAL LINES OF CURVATURE By r hypothesis A, l , 315 Z are the direction-cosines of the normal to S ; consequently we must have YA-- = O, ^ du If the values of the derivatives VA-- = O. ^ dv means of (V, 22), and the obtained from (97) be reduced by results substituted in the above equa tions, the first vanishes identically and the second reduces to (98) XV[ + YVt + ZVZ + (R when cos V) R sin FvV = 0, to v. where the primes indicate differentiation with respect versely, this condition is satisfied, Con : equations (97) define a surface on which the curves v = const, are spherical. Hence A necessary and can Ie sufficient condition that the curves v = const, of an orthogonal system on vature upon a surface the unit sphere represent spherical lines of is cur R, V, found which that five functions of v, namely Vr F2 , satisfy the corresponding equation (98). F 3, We only note that FF 1? 2, F to within additive constants. for the first three gives a R cos V are determined by (98) A change of these constants translation of the surface. If R cos V be 3, and increased by a constant, other one. Hence * : we have a new surface parallel to the the If the same lines of curvature in one system is upon a surface be spherical, true of the corresponding system on each parallel surface. , Since equation (98) is homogeneous in the quantities F/, F^, F3 (R cos F) R sin F, the latter are determined only to within a factor which may be a function of v. This function may be chosen so , that all the spheres pass through a point. have the theorem of Dobriner f : From these results we With each surface with spherical lines of curvature in one system there is associated an infinity of nonparallel surfaces of the same kind with the same spherical representation of these lines of curvature. Among these surfaces there is at least one for which all the spheres pass through a point. At corresponding points of the loci of the cen ters of spheres of two surfaces of the family the tangents are parallel. * Cf. Bianchi, Vol. II, p. 303. f Crelle, Vol. XCIV (1883), pp. 118, 125. 316 SURFACES WITH PLANE LINES OF CURVATURE from (97) be substituted in the formulas If the values of x, y, z of Rodrigues (IV, 32), dx &lt; dx "* dx 99) for a^-^ means dx of (V, 22), and similarly y and z, we obtain by =R cos Conversely, when for a surface referred to its lines of curvature is the principal radius p l of the form -.-*." where (^ and 2 are any functions whatever v = const, are spherical. For, by (V, 22), of v, the curves dv V~ ov du is Consequently, from the value, first of (99), in which p l given the above we obtain by integration where V l is a function of v alone. As v= these expressions are of the form (97), Similar results follow for y and we have the theorem z. : A necessary and sufficient condition that the lines of curvature be spherical is that p l be of the const, form (100). EXAMPLES 1. If the lines of curvature in one system are plane and one is a circle, all are circles. 2. When the lines of curvature in one family on a surface are circles, their indispherical images are circles whose spherical centers constitute the spherical catrix of the tangents to the curve of centers of the spheres which are enveloped surface. Show also that each one-parameter system of circles on the the by given unit sphere represents the circular lines of curvature on an infinity of surfaces, for one of which the circles are equal. EXAMPLES 3. 31T If the lines of curvature of a surface are parametric, and the curves u j = const. are spherical, we have j cot Pi F Pgu B sin F the radii of geodesic curvature and normal curvature in the denotes the angle under const, and of the sphere respectively, and direction v which the sphere cuts the surface. where pgu , /&gt;i, E denote F 4. When a line of curvature is spherical, the developable circumscribing the surface along this line of curvature also circumscribes a sphere and conversely, if such a developable circumscribes a sphere, the line of curvature lies on a sphere ; concentric with the latter 5. (cf. Ex. 7, p. 149). Let S be a pseudospherical surface with the spherical representation of curvature. its lines Show that a necessary a/ 1 and (25) of sufficient condition that the curves v = const, be plane is a&\ _ du \sin w dv/ show also that in this case w is given by COS 0) = V -U , where V and V are functions of u and U * = + (a - 2) C72 + 6, 4 U" v respectively, 2 which 2 satisfy the conditions (a -f b F = F* + aF -f - 1), a and "V b being constants, and the accent indicating differentiation, unless U or is zero. 6. When the lines of curvature v is const, upon a pseudospherical surface are plane, the linear element reducible to the form _ ~ where A, B, 7. a 2 sech 2 (u 4- v} dv 2 a2 tanh 2 (u + v) dw2 C -A cosh 2 u -f B sinh 2 u G + A cosh 2 v + B sinh 2 v - 1 C are constants. Find the expressions for the principal radii. the lines of curvature v is When = v) const, on a spherical surface are plane, the a2 esc 2 (u linear element reducible to _ ~ where J. a 2 cot2 (u + dw2 I + v} d i? 2 .A sin 2 w -f B - A sin 2 u - 7? of Exs. 5 and J5 are constants. The surfaces and 6 are called the surfaces of Enneper of constant curvature. GENERAL EXAMPLES 1. The lines of curvature and the asymptotic lines on a surface of constant curvature can be found by quadratures. 2. the equations x = cw, y upon the plane, which the plane this axis. When the linear element of a pseudospherical surface is in the form (iii) of (7), M = ae~a determine a conformal representation of the surface is by a circle with such that any geodesic on the surface is represented on its center on the ic-axis, or by a line perpendicular to 318 TF-SURFACES 3. When the linear elements of a developable surface, a spherical surface, and a pseudospherical surface are in the respective forms ds 2 = a?(du 2 + sin 2 wdu 2 ), ds 2 = 2 (dw 2 + sinh^udw 2 ), ds~ = du 2 + u-dv 2 , the finite equations of the geodesies are respectively Au cos v -f Bu sin v -f C= 0, A + .B tan u cos v A Z&gt;, tanh u cos v ; tanh u sin v C are constants if the coefficients of where A, to x and y, the resulting equations define a correspondence between the surface and the plane such that geodesies on the former correspond to straight lines on the latter. Find the expression for each linear element in terms of x and y as parameters. 4. B tan u sin v + C = 0, + C = 0, A and B are in any case equated -f- Each surf ace of center of a pseudospherical surf ace is applicable to the catenoid. 5. The asymptotic lines curvature correspond to 6. on the surfaces of center of a surface of constant mean the minimal lines on the latter. u and if = 7. const. Surfaces of constant mean curvature are characterized by the property that is a function of u alone v = const, are the minimal curves, then , D D" of v alone. (23) Equation admits the solution w = 0, in which case the surface degen erates into a curve. (35) is tan 0/2 Show M+ r cos Bin&lt;r that the general integral of the corresponding equations &lt;r = Ce ; take for S the line x = 0, y - 0, z = an and derive the equations of the transforms of 122), or a pseudosphere. (Ex. C, -S; shc^w that the latter are surfaces of Dini 8. Show that the Backlund transforms of the surfaces of Dini and of the pseudosphere can be found without integration, and that if the pseudosphere be trans formed by the transformation of Bianchi, the resulting surface may be defined by x = 2 a cosh u V 2 a cosh u . (sinu ucosu), y sinh u cosh u\ (cosv V + vsm v), (2 U cosh 2 w + v2 / Show 9. that the lines of curvature v const, lie in planes through the 2-axis. The tangents to a family of geodesies of the elliptic or hyperbolic type ; on a pseudospherical surface are normal to a W-surface are respectively P\ + c Pl PO the relations between the radii = . a tanh (cf. , a 7(3). pl p2 = ., a coth Pi 4- c , a where a and 10. c are constants Show that the linear elements of the second surfaces of center of the are reducible to the respective forms 2 du" &gt;F-surfaces of Ex. ds.? = tanh 4 a + sech 2 - dv 2 a , ds.? - coth 4 U a du 2 + csch 2 ~ du 2 a , and that consequently these surfaces are applicable whose meridians are defined by to surfaces of revolution where K denotes a constant. GENERAL EXAMPLES 11. 319 of the curves Determine the particular form of the linear element (49), and the nature upon the surface to which the asymptotic lines on the sheets of the evolute correspond, (a) when p. = const; (6) 11 -- = const. Pi Pz Pz a JF-surface is of the type (72), the surfaces of center are applicable to one another and to an imaginary paraboloid of revolution. 12. When 13. When a IF-surface is has the form (VI, GO), the curves of the type (72) and the linear element of the sphere u+v const, and u const, on the spherical y representation are geodesic parallels whose orthogonal trajectories correspond to the asymptotic lines on the surfaces of center hence on each sheet there is a family of geodesies such that the tangents at their points of meeting with an asymptotic ; line are parallel to a plane, which varies in general with the asymptotic line. 14. Show that the equations -+ Jf Vs m-dv. a a y = aUsina C J where a denotes an arbitrary constant, define a family of applicable molding surfaces. 15. When is lines of the latter the lines of curvature in one system on a surface are plane, and the second system lie on spheres which cut the surface orthogonally, the a surface of Joachimsthal. 16. The spherical lines of curvature on a surface of Joachimsthal have constant geodesic curvature, the radius of geodesic curvature being the radius of the sphere on which a curve lies. 17. When the lines of curvature in one system on a surface is lie on concentric its spheres, it is a surface of Monge, whose director-developable vertex at the center of the spheres and conversely. ; a cone with 18. The sheets of the evolute of a surface of Monge are the director-developable and a second surface of Monge, which has the same director-developable and whose generating curve is the evolute of the generating curve of the given surface. 19. If the lines of curvature in one system on a surface are plane, the second system are plane, then all in the latter system are plane. and two in 20. A A surface with plane lines of curvature in both systems, in one of which circles, is they are (a) (5) surface of Joachimsthal. locus of the orthogonal trajectories of a family of spheres, with centers The on a straight line, which pass through a circle on one of the spheres. (c) The envelope of a family of spheres whose centers lie on a plane curve C, and whose radii are proportional to the distances of these centers from a straight line fixed in the plane of C. 21. If an arbitrary curve C be drawn in a plane, and the plane be made to move way that a fixed line of it envelop an arbitrary space curve T, and at the same time the plane be always normal to the principal normal to T, the curve C in such a describes a surface of Monge. 320 TF-SUKFACES S are surfaces of Enneper 23. 22. If all the Bianchi transforms of a pseudospherical surface 134), S is a surface of revolution. (cf. Ex. 5, When u has the value in Ex. 5, 134, the surfaces with the spherical representation (25), and with the linear element ds* = (HI cos w + Ydu 2 + U? sin 2 u du 2 , where U\ is an arbitrary function of M, are surfaces of Joachimsthal. same spherical representation surface, 24. If the lines of curvature in both systems be plane for a surface S with the of its lines of curvature as for a pseudospherical S is a molding surface. 25. If its lines S is of curvature, a pseudospherical surface with the spherical representation (25) of and the curves v = const, are plane, the function 6, given by c2w sin 6 a 2 + aw aw cos . h sin w aw dv aw dv = 0, . determines a transformation of Bianchi of S into a surface Si for which the lines of curvature v = const, are plane. 26. necessary and sufficient condition that the lines of curvature v= const, on a pseudospherical surface with the representation (25) of its lines of curvature be spherical is that -\r a., A cot w = y1 + J / , sin w dv that where V and V\ are functions of and of i aw sin 2 w aw a/ i v alone. a Show i when w is a solution of (23) aw\ aw\ aw \sinwatv aw\ a2 / i aM 2 \sinwau/ au\sin 2 wau/ the curves v = const, are plane or spherical, and that in the latter case V and V\ can be found directly. 27. Show that when w is a solution of (23) and of dv aucv 2 a 2 awau du \au/ and du \cos w dv/ ( ) ^. 0, the lines of curvature u = const, are spherical ; on the pseudois spherical surface with the spherical representation (25) a function, upon the surfaces with the linear element and that when w such /aw\ 2 or / r)w\ 2 ) \dv/ I (sin \ w +F du 2 + cos w -f V +V former dv/ where is a function of alone, the curves case the spheres cut the surface orthogonally. t&gt; F t&gt; = const, are spherical; in the CHAPTER IX DEFORMATION OF SURFACES Problem of Minding. Surfaces of constant curvature. Ac 43 two surfaces are applicable when a one-to-one cording to 135. correspondence can be established between them which is of such a nature that in the neighborhood of corresponding points corresponding figures are congruent or symmetric. It was seen that two surfaces with the same linear element are applicable, the parametric curves on the two surfaces being in correspon dence. But the fact that the linear elements of two surfaces are unlike is in evidence of this not a sufficient condition that they are not applicable we have merely to recall the effect of a change ; by Minding of parameters, to say nothing of a change of parametric lines. Hence we are brought to the following problem, first proposed * : To find a necessary and applicable. sufficient condition that two surfaces be From the second theorem of is condition follows that a necessary that the total curvature of the two surfaces at corre it 64 sponding points be the same. is We shall show all that this condition sufficient for surfaces of constant curvature. In 64 we found that when K is zero at the surface is applicable to the plane. the system of straight lines parallel to the rectangular axes, points of a surface, If the plane be referred to its linear element is ds 797272 =dx + 2 2 dy*. Hence the surface analytical problem of the application of a developable upon the plane reduces to the determination of orthogonal systems of geodesies such that when these curves are parametric the linear element takes the above form. * Crelle, Vol. XIX (1839), pp. 371-387. 321 322 DEFORMATION OF SURFACES 39, Referring to the results of factor tt v we see that in this case the unity. Consequently we must find a function 6 such that the left-hand members of the equations must equal du + + -.dv = d(x + iy}, du \ which case these equations give x and y by quadratures. Hence we must have are exact differentials, in du\ -^E to 99 which are equivalent du~Hu dv ZEHdu F d_E_ dd _ J_ d_G_ _ ~ 2 H dv ^H ~du 2 EH dv From (V, 12) K= 0. seen that these equations are consistent when In this case 6, and consequently x and y, can be found by it is quadratures. The additive constants of integration are of such a character that if ar , y are a particular set of solutions, the most general are x =x cos a yQ sin a + a, y =X Q sin a +y Q cos a + /&gt;, where of a, #, 5 are arbitrary constants. effect the isometric representation In the above manner we can any developable surface upon the plane, and consequently upon itself or any other developable. These results may be stated thus : A developable surface is applicable to itself, or to any other develop able, in a triple infinity of ways, and the complete determination of the applicability requires quadratures only. Incidentally we have the two theorems: be foundby quadratures. The geodesies upon a developable surface can If the total curvature of a quadratic form be zero, the quadratic form is reducible by quadratures to dad&. SUEFACES OF CONSTANT CURVATURE 323 Suppose now that the total curvature of two surfaces S, Sl is 2 I/a where a is a real constant. Let P and 7? be points on S and /S^ respectively, C and C geodesies through these respective and C and C for the points, and take P and I[ for the poles of a polar geodesic system on these surfaces. The curves v = , l l linear elements are accordingly (VIII, 6) d** = du + sin 2 2 - dv\ u ds 2 v = du* + sin 2 v ^ dv 2 . Hence the equations u = to determine an isometric representation of one surface upon the other, in which P and C correspond P in and Cl respectively. the second equation According as the upper or lower sign is used, corresponding figures are equal or symmetric. Similar results obtain for pseudospherical surfaces. Hence we have: Any two surfaces of constant curvature, different from zero, are in two ways applicable so that a given point and geodesic through it on one surface correspond to a given point and geodesic through it on the other. itself so that a In particular, a surface of constant curvature can be applied to given point shall go into any other point and a geodesic through the former into one through the latter. Combin 117, we have: ing these results with the last theorem of nondevelopable surface of constant curvature can be applied to in a triple infinity of itself, or to any surface of the same curvature, A ways, and the complete realization of the applicability requires the solution of a Iliccati equation. 136. Solution of the problem of Minding. proceed to the determination of a necessary and sufficient condition that two sur faces S, 8 of variable curvature be applicable. Let their linear We elements be ds 2 = E du + 2 Fdudv + G dv 2 2 , ds 2 =E if du 2 + 2F du dv +G dv 2 . By (1) definition S and S are applicable there exist two independ ent equations (/&gt; (U, V) = (U J , V 1 ), ^ (U, V) = ^&lt;(U , V ), establishing a one-to-one correspondence between the surfaces of such a nature that by means of (1) either of the above quadratic forms can be transformed into the other. 324 DEF011MATION OF SURFACES two surfaces are applicable, the differen formed with respect to the two linear elements are parameters It is evident that if the tial equal. (2) Hence a necessary condition A;&lt;/&gt; is A^ = , A^, f)=A!(&lt;#&gt; , f), A f=A;t 1 . , where the primes indicate functions pertaining to S These con ditions are likewise sufficient that the transformation (1) change either of the above quadratic forms into the other. For, if the curves (/&gt; = const., ^r const. ; &lt;/&gt; = const., ^ = const, f be taken for the parametric curves on S and S linear elements may be written (cf. , respectively, the respective 37) Hence when equations The next step is Since the curvature of two applicable surfaces at corresponding points is the same, one such equation is afforded by the necessary condition (3) the surfaces are applicable. the determination of equations of the form (1). (1) (2) hold, and K(u,v) first = K (u ,v . ). The (4) of equations (2) is A^A;* K 37), Both members this case the curves ( of this equation cannot vanish identically. const, would be const, and For, in K= minimal are and consequently imaginary. is, If these two equations independent of one another, that they establish a correspondence, and the condition that metric is, as seen from (2), it be iso If, however, (5) \K for the second of (1) we may take (6) unless (7) A 2J fiT PROBLEM OF MINDING If this condition 325 (3), (6) be not satisfied, the conditions that define an isometric correspondence are AA 1 a JST = Finally, we consider the case where both (5) and (7) hold. Since the ratio of and A 2 is a function of JC, the curves const. \K K K= form an isothermal sys trajectories the function t can be found 41). Moreover, by quadratures, and the linear element is reducible to t and their orthogonal tem of lines on S ( = const, (8) ds 2 =2 2 J( K (dK ) +e J &lt; A &gt; dt 2 ). When in particular A ^T= 0, the linear element is In like manner the linear element of S is reducible to or, in the particular case A^ = 0, to In either case the equations K=K where a surfaces. is i t = t +a, an arbitrary constant, define the applicability of the all We have thus treated possible cases and found that it can be determined without quadrature whether two surfaces are appli cable. Moreover, in the first two cases the equations defining the correspondence follow directly, but in the last case the determina tion requires a quadrature. The last case differs also in this respect the application can be effected in an infinity of ways, whereas in : the first two cases it is unique. &lt;r * If the surface be referred to the curves where a that Vol. AI(&lt;T, J ~vf(K) &&lt;&lt;?) C= = = const, and their orthogonal trajectories, , equation , A%&lt;r (6) may be replaced by A2 tr = A^ , and it can be shown Cf. Ai(&lt;r ) is a consequence of the other conditions. Darboux, Ill, p. 227. 326 DEFORMATION OF SURFACES (8) Furthermore, we notice from face that in the third case the sur S is applicable to a surface of revolution, the parallels of the latter corresponding to the curves K= const, of the former. Con versely, the linear element of every surface applicable to a surface of revolution can be put in the form (8). For, a necessary and sufficient condition that a surface be applicable to a surface of revolution is that its linear element be reducible to where U is a function of u alone ( 46). Now Cdu A lW = l, From the X= --, U" second it follows that u F(K), and consequently F^K). equations, (9) When these values are substituted in the above we have, in consequence of Ex. 5, p. 91, A 1 /C=/(A"), A,JiT =*&lt;*). Hence we have the theorem Equations (9) constitute to : surface be applicable a necessary and sufficient condition that a a surface of revolution. The equations K=K, Therefore t = t +a : define an isometric representation of a surface with the linear ele ment (8) upon itself. we have K = const, is Every surface applicable to a surface of revolution admits of a continuous deformation into itself in such a way that each curve slides over itself. itself in Conversely, every surface applicable to an infinity of applicable to a surface of revolution. For, if the curvature ways is constant, the surface is applicable to a surface of revolution 135), and the only case in which two surfaces of variable curva ture are applicable in an infinity of ways is that for which condi tions (5) and (7) are satisfied. ( DEFORMATION OF MINIMAL SURFACES 137. Deformation of 327 means of determining the minimal surfaces. These results suggest a minimal surfaces* applicable to a surface of revolution. In the first place we inquire under what conditions two minimal surfaces are applicable. The latter problem reduces to the determination of two pairs of parameters, w, v and u^ v v and and Ffa^, ^ 1 (f ), which satisfy two pairs of functions, F(u), &gt; &lt;f&gt;(v)* 1 the condition (10) (1 + uvfF(u)(v) dudv = (1 + u^)* f\(uj 4^) du^dv r it From the nature of this equation serve to establish the correspondence between the either of the form (11) follows that the equations which two surfaces are ^=0(M), ,= *(), u and l V 1= ^(V), or (12) , = *() and if If either set of values for v l be substituted in (10), removing the common factor dudv we take the logarithmic derivative with respect to u and v, we obtain after (1 + u^Y du dv l l (1 + uvf dudv 2 As ,.. this may be written ~~ o (i + uft)* (I +^^y the spherical images of corresponding parts on the two surfaces are equal or symmetric according as (11) or (12) obtains ( 47). The latter case reduces to the former when the sense of the normal to either surface is changed. When this has been done, corresponding spherical images are equal and can be made to coincide by a rota tion of the unit sphere about a diameter. Hence one surface can be so displaced in space that corresponding normals become parallel, in which case the two surfaces have the same representation, that is, Wj = u, is vx = v. Now equation (10) is which equivalent to F (u)=cF(u), l 3 * no. 328 DEFORMATION OF SURFACES i&lt;x where c denotes a constant. If the surfaces are real, c must be of the form e Hence, in consequence of 113, we have the theorem . : minimal surface admits of a continuous deformation into an which are either associate to it or can infinity of minimal surfaces, be A made such by a suitable displacement. pass to the determination a continuous deformation into of We of a minimal surface which admits and consequently is appli In consequence of the interpre cable to a surface of revolution. tation of equation (13) it follows that if a minimal surface be deformed continuously into itself, a point p on the sphere tends to itself, move in the direction of the small circle the momentary small circles moves over axis of rotation, itself. through p, whose axis is and consequently each of these 47 it From follows that if the axis of rotation be taken for the 2-axis, these small circles are the curves uv = const. In the deformation each point of the surface moves along the curve tion of uv. through From (VII, 100, 102) we have K= const, it. Hence K is a func A= ___^l_ uF The common denoted by K, ; function of uv, and hence consequently F(u)Q(v) must be a (u) __ v&lt;& (v) F(u) value of these 4&gt;(v) two terms is a constant. K , If it be we have where c and c1 are constants. Hence from (VII, to 98) we have : Any minimal surface applicable a surface of revolution can be the defined by equations of form &lt;?j * c f (1 _ u *) u du + | f (1 - v 1 ) v*dv, (14) y -c Cfl + u^u du 2 /r J { ^Ci 1(1 &lt;+i 2V iv, u ^du + c^Ji constants. ivhere c, c., and K are arbitrary DEFOKMATION OF MINIMAL SUEFACES Since the curves 329 on the sphere by the the z-axis, in each finite deformation of the surface into itself, as well as in a very small one, the unit sphere undergoes a rotation about this axis. In 47 it was seen that such ia ia a rotation is equivalent to ve~ where a replacing u, v by ue denotes the angle of rotation. Hence the continuous deformation const, are represented is K small circles whose axis , , of a surface (14) is defined by the equations resulting from the substitution in (14) of ue ia ve~ for u, v respectively. An important property of the surfaces (14) is discovered when i&lt;z , z-axis. its we submit such a surface to a Let S denote the surface equations in the rotation of angle a about the in its new position, and write form f (1 - tf and similarly for y and z. Between the parameters ta , u, v and u, v the following relations hold: u and we have x also = ue v = ve~ tar , = x cos a y sin a, ~y = x sin a -f y cos (14), 5&gt; a:, z = z. Combining these equations with we (v) find F (u) = is cu e~ K ia(K + 2) , = K c { v e ia(K + 2) . Hence, for the correspondence defined by u = ?/, v = v, the surface S an associate of S, unless K -f- 2 = 0, in which case it is the same surface. We are (cf. consider the latter case, and remark that its equations 110) be replaced by ue ia ve~ ia and the resulting expressions be denoted by x v y^ z x we have If u, v , , , (15) x l = xcosa ysina 1 y^ = x sin a + y cos a, zl = z + lR(iac}. c Hence, in a continuous deformation, the surface slides over itself with a helicoidal motion. Consequently it is a helicoid. Moreover, it is the minimal helicoid. For, every helicoid is applicable only 330 DEFORMATION OF SURFACES and each minimal surface applicable to to a surface of revolution, a surface of revolution with the z-axis for the axis of revolution 2 will the of the sphere is defined by (14). But only when ic = ia a set of equations such as (15). ve~ substitution of we give 1 ", Hence we have : The helicoidal minimal surfaces are defined by the Weierstrass formulas when F(u)=c/u 2 . And we may state the other results thus : applicable to a surface of revolution, be rotated through any angle about the axis of const. the unit sphere whose small circles represent the curves If any nonhelicoidal minimal surface, which is K on the surface, and a correspondence with parallelism of tangent are associate ; con planes be established between the surfaces, they a minimal surface are supcrposable. sequently the associates of such EXAMPLES 1 . Find under what conditions the surfaces, whose equations are z\ F(r) + av, can be brought into a one-to-one correspondence, so that the total curvature at surfaces corresponding points is the same. Determine under what condition the are applicable. tangent planes to two applicable surfaces at corresponding points are the surfaces are associate minimal surfaces. parallel, 2. If the 3. Show that the equations x = ea u t y = e- a v, z aea u z -f b&lt;y~ av2 , where a is a real parameter, and a and 6 are constants, define a family of parab oloids which have the same total curvature at points with the same curvilinear coordinates. 4. Are these surfaces applicable to one another Find the geodesies on a surface with the linear element d u z _ 4 y dudv + 4 u dv 2 2 ds =. 2 4(w-t&gt; ) -6 in ? Show 5 that the surface is the applicable to a surface of revolution, and determine latter. form of a meridian of the . Determine the values of the constants a and ds* = du 2 + [(u + au) 2 + 6 2 ] du 2 , so that a surface with this linear element shall be applicable to (a) (&) the right helicoid. the ellipsoid of revolution. SECOND GENERAL PROBLEM 6. 331 and sufficient condition that a surface be applicable to a surface that each curve of a family of geodesic parallels have constant geodesic curvature. of revolution is A necessary 7. Show that the helicoidal minimal surfaces are applicable to the catenoid and to the right helicoid. have seen that 138. Second general problem of deformation. can always be determined whether or not two given surfaces are applicable to one another. The solution of this problem was an it We important contribution to the theory of deformation. An equally important problem, but a more difficult one, is the following : To determine all the surfaces applicable to a given one. This problem was proposed by the French Academy in 1859, and has been studied by the most distinguished geometers ever since. Although it has not been solved in the general case, its profound study has led to If many interesting results, some of which we shall derive. the linear element of the given surface be 2 c?s = Edu? + 2 F dudv it is -f- G dv*, this every surface applicable to determined by 2 , form and by a dudv + D"dv whose coefficients satisfy second, namely Ddu*+ 2 the Gauss and Codazzi equations ( 64). Conversely, every set of of these equations defines a surface applicable solutions D, , D D D" to the given one, and the determination of the Cartesian coordinates of the corresponding surface requires the solution of a Riccati equa But neither the Codazzi equations, nor a Riccati equation, can be integrated in the general case with our present knowledge of differential equations. Later we shall make use of this method in tion. the study of particular cases, but for the present we proceed to the exposition of another means of attacking the general problem. n obtained from the Gauss equations When the values of D, DD , (V, 7) are substituted in the equation H K=J)D"D 2 2 , the result 2 ing equation is reducible, inconsequence of the identity (cf. Ex. 6, p. 120), to (16) A^ =1 A dtf " \du dv~\ 1J ^ I 2 J dv 332 DEFORMATION OF SURFACES is This equation, which E, F, satisfied also by y and 2, involves only its G and their derivatives, and consequently , integration complete solution of the problem. It is linear in 2 / c x Yl &x tfx tfx , , ^, \tfxtfx , , ^ri^l^r-sr) J dir 1 T-T- Tl and therefore is of the form oTi cv 0tr \dudv / \_du studied by Ampere. Hence we have the theorem: will give the ^&lt;7v the integration of The determination of all surfaces applicable to a given one requires a partial differential equation of the second order of the Ampere type. In consequence of (16) and (V, 36) of a surface with the linear element (17) we have that the coordinates ds 2 = Edu + 2 Fdudv + G dv 2 2 are integrals of (18) A 22 = (l 1 A^JST, the differential parameters being formed with respect to (17). shall find that when one of these coordinates is known the other We two can be found by quadratures. Our general problem may be G-iven three functions E, F, x, y, stated thus : G of u and v ; to find all functions z of u and dx 2 v which 1 satisfy the equation 2 + dy + dz = E du* + 2 Fdudv + G dv\ may be chosen arbitrarily. where du and dv Darboux (19) * observed that as the equation may 2 be written dx 2 + dy* = Edu +2Fdudv + Gdv - dz\ 2 whose left-hand member is the linear element of the plane, or of a developable surface, the total curvature of the quadratic (20 ) form \E- p?Y"U + 2\F- dudv\ dudv **\ L dv\ L WJ ( * must be zero known, 64). this, In order to find the condition for we assume that z is and take for parametric lines the curves z = const, and *L.c., Vol. Ill, p. 253. their SECOND GENERAL PROBLEM orthogonal trajectories for ters the right-hand v 333 = const. With this choice of 2 parame The of (19) reduces to (E I)dz condition that the curvature of this form be zero is member + Gdv*. tr where K denotes the curvature of the surface. But 2 . this is the condition also that z be a solution of (18) when the differential parameters are formed with respect to Edz 2 -}-Gdv However, the ; members z is of equation (18) are differential parameters consequently a solution of this equation whatever be the parametric curves. By reversing the above steps z is we prove the theorem : When any integral of the equation (18), the quadratic form (20) has zero curvature. is known we can find by quadratures two functions x, y such that the quadratic form (20) is (cf. 135) z equal to dx* + dy provided that When such a solution , that is, Ajg &lt; 1. Hence we have the theorem : If z be a solution of A 22 (1 A X 0) K such that A^ &lt; 1, it is one a surface with the given linear ele the other two coordinates can be obtained by quadratures. ment, and of the rectangular coordinates of 139. Deformations which change a curve on the surface into a given curve in space. We consider the problem : Can a surface be deformed in such a manner that a given curve C upon it comes into coincidence with a given curve F in Let the surface be referred to a family of curves orthogonal to C and to their orthogonal trajectories, C being the curve v = 0, and its arc being the parameter u, so that conditions hold for F on the deform. E \ for v = 0. The same 334 DEFORMATION OF SURFACES ( Since the geodesic curvature of C is unaltered in the deformation for the new surface, 58), it follows from the equation (IV, 47) namely (21) p is = p g amw, if that the deformation impossible, the curvature of F at any point point. to, is than the geodesic curvature of C at the corresponding Since both p and p g are known, equation (21) determines less and consequently the direction of the normal to the new surface along F is fixed. to the curves u = This being the case, the direction of the tangents const, on the new surface at points of F can be . found, and so as well as we have &gt; 1 V&lt;? dx - 1 c&gt;f 1 the values of to r-i = cz \IG co vV; w tor v = U, for v = 0, cu cu du the latter being the direction-cosines of the tangent to F. respect to u, for we obtain the values ot If these expressions , tfx c~y ^ 3u" cu" be differentiated with 2 2 2 cz cx dz tfy T r- 7777, cudv ducv cucv ; &gt; cu" v=0. Since F=Q are tfx and E=1 SEdx to to for v = 0, the Gauss equations (V, 7) for v = = J. " 1 du 2 2G I/ r JJJL+ Jjy t/*f J- *y*-* f**- 2 "a^ a^ 2G 2 du to dv* 2 cu du 6r 00 dv All the terms of the first two equations have been determined hence the latter are given by these equations. except D and D 1 ; Since the total curvature known /&gt;/&gt;" /&gt; and p =p unaltered by the deformation, it is = at all points of F; consequently // is given by H*K 2 is zero, in which case F is an asymptotic line , unless 2 dx from is found we can obtain the value of When 5 A" is D . /)" (J the last of equations (22). From the method of derivation of equa tion (16) it follows that the above process is equivalent to finding the value of ^ from to equation (16), which is possible unless D= 0. Excluding this exceptional case, we remark that if equations (22) PARTICULAR DEFORMATIONS be differentiated with respect to u, 335 all we obtain the values of the derivatives of x of the third order for v = except - The latter be obtained from the equation which results from the differ entiation of equation (16) with respect to v. By continuing this of the derivatives of x of all process we obtain the values for v may orders, and likewise of y and z. If we , indicate the values of functions, when u = UQ n 2\#ir/o v = 0, by subscript null the expansions = z + fdx\ \dufy idx \dvfy \dudvh as of : and similar expansions for y and 2, are convergent in general, Cauchy has shown,* and x, y, z thus defined are the solutions equation (16) which for can be v = satisfy the given conditions. Hence A it surface S deformed in such a manner that a curve C upon curvature of of comes into coincidence with a given curve F, provided that the F at each point is greater than the geodesic curvature C at the corresponding point. ormation There remains the exceptional case p is possible, F is an asymptotic =p . a If the desired def line on the deform, and is consequently, by Enneper s theorem ( must satisfy the condition r 2 = 1/JC. is 59), its radius of torsion Hence when C D" given, F determined, If F to be an asymptotic line. satisfies these conditions, the value of if it is for v = is arbi trary, as we have seen. But when it it has been chosen, the further determination of the values of the derivatives of order for v = is the general case. of these surfaces, depending upon an arbitrary function. For all .of these surfaces the directions of the tangent planes at each #, y, z of higher the same as that pursued in unique, being Hence equation (16) admits as solution a family point of F are the same. Hence we have the theorem : Criven a curve curve tion of F with 8 in an to C upon a surface 8 ; there exists in space a unique which C can be brought into coincidence by a deforma infinity of ways ; moreover, all the new surfaces are tangent . one another along F. aux derivees partielles du second ordre, chap. * Cf Goursat, Lemons surT integration des Equations ii. Paris, 1896. 336 If DEFOKMATION OF SURFACES C is an asymptotic line on S, it may be taken for F; hence : A surface may be subjected to which a given asymptotic line is be an asymptotic line on each deform. a continuous deformation during unaltered in form and continues to This result suggests the problem : Can a surface be subjected to a continuous deformation in which a curve other than an asymptotic line is unaltered? By hypothesis the curvature is not changed and the geodesic curvature is necessarily invariant; hence from (21) we have that sin o&gt; must have the same value for all surfaces, for all the surfaces. If o&gt; is the the tangent plane is the same, and consequently same the expansions (23) are the same. Hence all the surfaces coincide in this case. However, there are always two values of W for which sin o&gt; has the same value, unless w is a right angle. Hence it is have two applicable surfaces passing through a curve whose points are self-correspondent, but not an infinity of such possible to surfaces. Therefore : An asymptotic line is the only curve on a surface which can remain unaltered in a continuous deformation. 140. Lines of curvature in correspondence. We C upon in such a manner become a line of curvature on the new surface. may Suppose it is possible, and let F denote this line of curvature. The radii of curvature and torsion of F must satisfy (21) and dto/ds 1/T (cf. 59), where p g is the same for F as for C. If we choose for w any function whatever, the functions p and r are a surface S can be deformed inquire whether that a given curve it ; thus determined, and F is unique. Since o&gt; fixes the direction of the tangent plane to the new surface along F, there is only one deform of S of the kind desired for each choice of w (cf. 139). Hence : surface can be deformed in an infinity of ways so that a given curve upon it becomes a line of curvature on the deform. A This result suggests the following problem of Bonnet*: To determine the surfaces which can be deformed with preservation of their lines of curvature. I * Memoire sur la theorie des surfaces de applicables sur une surface donnee, Journal Ecole Poly technique, Cahier 42 (1867), p. 58. LINES OF CUBVATUBE IN COBBESPONDENCE 337 We follow the method of Bonnet in making use of the funda mental equations in the form (V, 48, 55). We assume that the lines In this case these equations reduce to of curvature are parametric. (&gt;i _ dv dc t-~~ Sr ^ (24) From these equations it follows that if and S are two applicable surfaces referred to corresponding lines of curvature, the functions r and r l have the same value for both surfaces, and consequently the same is true of the product qp r Hence our problem reduces ; to the determination of the above equations. (25) p v q p[, q In consequence of the identity p(q two sets of functions , satisfying =P& we have from the first two of (24) *&lt;f&gt;(u) &lt;S?-S&lt;f+ &lt;#&gt;(M), 2 2 where f(v) of which the integrals are p i = p?+f(v), q = and are functions of v and u respectively. The parameters w, v may be chosen so that these functions become constants #, /3, and consequently 2 (27) If these equations ;&gt;I = K+ 2 2 =&lt;Z ? +is be multiplied together, the resulting equation reducible by means of (25) to either of the forms (28) pi ft + (fa + a(3 = 0, p(*P + z *a-a& = b. the first we see that a and fi cannot both be positive if S is real, and from the second that they cannot both be negative. We assume that a is negative and j3 positive, and without loss of generality write From (29) rf-rf-li ^-tf+lq* The first of (28) reduces to pi we introduce a function thus o&gt;, = l. In conformity with this p = cosh l &&gt;, q = sinh o&gt;. Then equations (29) may be replaced by ft), = sinh p[ ^ = cosh o&gt;. 338 DEFORMATION OF SURFACES Moreover, the fundamental equations (24) reduce to _ dv a 2 o&gt; du V] , o&gt; du , : a H c -- = o&gt; o2 smh . , cosh o&gt;. dt* 0v- Comparing these results with 118, we see that the spherical f representation of lines of curvature of the surfaces S and S respec tively is the same as of the lines of curvature of a spherical surface that every surface of this kind admits of an applicable surface with lines of its and of Hazzidakis transform. Conversely, we have curvature in correspondence. The preceding the first investigation rested on the hypothesis that neither nor second of equations (24) vanishes identically. Suppose that the second vanishes then q is a function of u alone, say (u). Since the product p^q differs from the total curvature only by a ; factor (cf. 70), is Equation (25) nated from this equation and the also is p cannot be zero now of the form p l ; therefore r l &lt;f&gt;(u)=p[&lt;t&gt; l = and q ^(u). (u). If p[ be elimi first of (27), it is found that p l a function of u alone. Hence the curves v = const, on the sphere are great circles with a is a molding surface that find we may take q -h #, common diameter, and therefore S The parameter u may be chosen so 130). ( = 1 and p^=U\ then from (27) and (25) we where a is = Vf/ /&gt;( 2 q = U/^/U* + a, : an arbitrary constant. Hence we have the theorem necessary and sufficient condition that a surface admit of an is that applicable surface with lines of curvature in correspondence its lines of cur the surface have the same spherical representation of vature as a spherical surface 2, or be a molding surface ; in the first case there is one applicable surface, and the spherical representation A of its lines of curvature is the of 2 ; in the second case there is same as of the Hazzidakis transform an infinity of applicable surfaces.* 141. Conjugate systems in correspondence. When two surfaces are applicable to one another, there is a system of corresponding lines which is conjugate for both surfaces (cf. 56). The results of 140 show that for a given conjugate system * Cf EX. 14, p. 319. . on a surface S CONJUGATE SYSTEMS IN COEEESPONDENCE there is 339 not in general a surface S l applicable to S with the corre sponding system conjugate. We inquire under what conditions a given conjugate system of S possesses this property. Let S be referred to the given conjugate system. If the corre sponding system on an applicable surface Sl is conjugate, we have D = D[= 0, for the total curvature of the this equation Dip? = DD" is ; two surfaces the same. We replace by the two D! - tanh 6 D, D[ = coth 6 . D", thus defining a function 6. The Codazzi equations for S are Since these equations must be satisfied by 30 D l and Z D", we have 22 lz The condition D , c6 fll of integrability of (30) is reducible to 2 D m As to the two roots of this equation differ only in sign, and thus lead symmetric surfaces, we need consider only one. If it be substi tuted in (30), we are necessary in order that obtain two conditions upon E, F, G 7), which S admit of an applicable surface of the ; D", Hence in general there is no solution of the problem. the two expressions in the brackets of (31) vanish identically, the conditions of integrability of equations (30) are completely satisfied, and S admits of an infinity of applicable sur kind sought. However, if faces upon which the coordinate curves form a conjugate system. : Consequently we have the theorem If a conjugate system on a surface S corresponds to a conjugate system on more than one surface applicable to S, it corresponds to a conjugate system on an infinity of surfaces applicable to S. 340 DEFORMATION OF SURFACES shall give this result another interpretation by considering the spherical representation of S. From (VI, 38) we have We ii/^ = la/ {rtV ment &gt; 1-22-1 .D f!2V are firi-D". /12V la/3" ~li/ ele formed with respect to the linear If of the spherical representation of S. values in (30), we get we substitute these d6 = sf!2\ f 12J tanh d6 0, ... /12V coth 0, =\ \ a^ 0* 11 J and the condition that these equations have an integral involving a parameter becomes a ri2V_ a ri2V Sii/~Sla./ The py ri2V "-iJisr first of these equations is the condition that the curves the sphere represent the asymptotic lines upon a certain sur upon denotes the total curvature of face 2 (cf. Moreover, if 78). K S, and we put (34) K= 2 l//&gt; we have 2 du Now equations (33) are equivalent to (34), and & log p r,&lt;2 which reduces is to = cucv ), 0. As and the general integral of this equation p = cf)(u) 4- ^(i v respectively, we are arbitrary functions of u and have the following theorem due to Bianchi* where (/&gt; ^ : necessary and sufficient condition that a surface S admit a con tinuous deformation in which a conjugate system remains conjugate is A totic lines that the spherical representation of this system be that of the asymp of a surface whose total curvature, expressed in terms of to parameters referring these lines, is of the form * Annali, Ser. 2, Vol. XVIII (1890), p. 320; also Lezioni, Vol. II, p. 83. CONJUGATE SYSTEMS IN COKKESPONDENCE The pseudospherical 341 K of this form. In this case ("12") surfaces afford an example of surfaces with and ^r are constants, so that equa&lt;/&gt; tions (34) reduce to f 12"| 1 \ ~\ a ** &gt; | ^ which, in consequence of f {11"| = 1f22 ^, tions that the parametric curves a conjugate system of geodesies state these results thus : f=0. But these are the condi on S be geodesies. A surface with is | called a surface of /r l oss. We of l^oss admits of a continuous deformation in which the geodesic conjugate system is preserved ; consequently all the new sur faces are of the same kind. A surface EXAMPLES 1. Show that every integral of the equation Ai0 (18). = 1 is an integral of the funda mental equation 2. On a right helicoid the helices are asymptotic lines. Find the surfaces appli cable to the helicoid in such a way that one of the helices is unaltered in form and continues to be an asymptotic 3. line. A on the two surfaces 4. surface applicable to a surface of revolution with the lines of curvature in correspondence is a surface of revolution. that the equations Show X=KTCOS-, K y = train-, K z = /Vl J /c 2r 2 c?M, define a family of applicable surfaces of revolution with lines of curvature in corre spondence. Discuss the effect of a variation of the parameter K. 5. applicable to Let S denote a surface parallel to a spherical surface S with preservation of the lines of curvature. S. Find the surface 6. It Si and S2 be applicable surfaces referred to the common conjugate sys tem, their coordinates &i, y\,z\\ 2 ?/ 2 2 are solutions of the same point equation (cf. VI, 26), and the function xf -f y? + zf (x| + y.| + z|) also is a solution. , , 7. Show corresponding points on the surfaces Si and parametric lines form a conjugate system. applicable to Si and 8. /S 2 that the locus of a point which divides in constant ratio the join of 2 of Ex. 6 is a surface upon which the &lt;S Under what condition is this surface ? The tetrahedral surface x = A(a + u)*(a + )*, y admits of an infinity of deforms The curves u = B(b + u)*(b + v)*, z = C(c + w)*(c + f v) , = v upon these surfaces are congruent, and consequently each it. is an asymptotic line on the surface through 342 9. DEFORMATION OF SURFACES If the equations of a surface are of the form x the equations = U1 Vll y = UiV!, z=V* t sin 0, where h denotes a constant, define a family of applicable surfaces upon which the parametric lines form a conjugate system. 10. Show that the equations of the quadrics can be put in the form of Ex. results to this case. 9, and apply the 142. Asymptotic lines in correspondence. Deformation of a ruled 139) that a surface can be subjected to a continuous deformation in which an asymptotic line remains surface. ( We have seen ask whether two surfaces are applicable with asymptotic. the asymptotic lines in one system corresponding to asymptotic lines of the other. We We assume that there are two such surfaces, S, S and we lt v = take the corresponding asymptotic lines for the curves const, and their orthogonal trajectories for u = const. In con sequence of this choice and the fact that the total curvature of the two surfaces is the same, we have (36) J9 = D =0, 1 JF=0, ) D = D[. to The Codazzi equations &lt; (V, 13 for S reduce Q Because of (36) the Codazzi equation for S1 analogous to the first of (37) will differ from the latter only in the last term. Hence we = must have either or Ef(u}. In the former case the sur faces S and Sl are congruent. Hence we are brought to the second, which is the condition that the curves v = const, be geodesies. As the latter are asymptotic lines also, they are straight, and conse quently 8 must be a ruled surface. By changing the parameter w, we have J5? = l, and equations (37) reduce to D" Z&gt;", EULED SURFACES By a suitable choice of the parameter v may be replaced by JX=1/V5, and the the first 343 of these equations second becomes = ra/i ~(I J 9*\& These results establish the fol : is an arbitrary function. theorem of Bonnet lowing where &lt;/&gt; sufficient condition that a surface admit an with the asymptotic lines in one system on each applicable surface surface corresponding is that the surface be ruled; moreover, a A necessary and ruled surface admits of a generators remain straight. continuous deformation in which the To this may be added the theorem : the asymptotic lines in both on each surface are in correspondence, the surfaces are con systems If two surfaces are applicable and gruent, or symmetric. This is readily proved when the asymptotic lines are taken as parametric. We shall establish the second part of the above theorem in another manner. ruled surface in (38) For this purpose we take the equations of the the form ( 103) Q x , =x its +lu, and y = y +mu, Q z = z +nu, functions of C expressed as m, n are the direction-cosines of the generators, also functions of v. They satisfy the conditions where XQ y^, z are the coordinates of the directrix arc v, I, (39) aJ 1 +jtf-K -!. 2 * +w +n = l, a a where the accents indicate differentiation with respect to Furthermore, the linear element is (40) v. ds*= du 2 + 2 cos - dudv n + (aV+ b 2 2 = l x Hence if we have problem of finding a ruled surface applicable to a ruled surface with the linear element (40), the it, with the gener ators of the nation of six functions of two surfaces corresponding, reduces to the determi v, namely X Q y z I, m, n, satisfying , , ; 344 DEFORMATION OF SURFACES the five conditions (39), (41). From this it follows that there is an arbitrary function of v involved in the problem, and consequently there is an infinity of ruled surfaces with the linear element (40). There are two general ways in which the choice of this arbi either as determining the form of trary function may be made, the director-cone of the required surface, or by a property of the consider these two cases. directrix. We 143. Method of Minding. /, The first case was studied by Mind n ing.* (42) He / took m, n cos in the i/r, form &lt; = cos &lt; m = cos sin i/r, = sin &lt;/&gt;, which evidently satisfy the second of reduces to 2 (39). The first of (41) (43) If &lt;J&gt; +^ 2 cos 2 =a a . solve equations (39) and (41) for x^ y Q z expressions are reducible by means of (VII, 63) to , we , the resulting (44) Q = I cos 6 Q +^ [I b f (mn - m n) VV sin 2 -6 if &lt; 2 ], and analogous expressions for y[ and z[. trary function of v, and ^ be given by (46) Hence, be an arbi +=(^f^ J COS (/&gt; the functions # with # z obtained from (44) by quadratures, together m, n from (42), determine a ruled surface with the linear element (40). , , , /, Each choice of gives a different director-cone, which is deter mined by the curve in which the cone cuts the unit sphere, whose center is at the vertex of the cone. Such a curve is defined by a = 0, so that instead of choosing arbitrarily we relation (/&gt; /(c/&gt;, -&lt;fr) may /(&lt;, take ifr) / 0, by combining equations (43) and and ^r as functions we obtain the expressions for as arbitrary; for, &lt; of v. Hence : A ruled surface may be deformed in such a way that the director- cone takes an arbitrary form. * Crelle, Vol. XVIII (1838); pp. 297-302. RULED SURFACES When the given ruled surface is 345 nondevelopable, the radicand in (44) is different from zero, and consequently there are two dif ferent sets of functions XQJ yQ1 Z Q Hence there are two applicable . ruled surfaces with the same director-cone. distribution of these are If the parameters of found : to differ only in sign. two surfaces be calculated by (VII, 73), they Hence we have the theorem of Beltrami * A ruled surface admits of an applicable ruled surface such that corresponding generators are parallel, and the parameters of distri bution differ only in sign. 144. Particular deformations of ruled surfaces. By means of the preceding results we prove the theorem : ruled surface may be deformed in an infinity of ways so that a given curve becomes plane. A surface. Let the given curve be taken for the directrix of the original Assuming that a deform of the kind desired exists, we its take plane for the zy-plane. a n cos 2 From I (44) we have 2 + bn f f (lm m) a sin 2 6 2 = 2 0, which, in consequence of (42) arid b cosc/&gt;.( (43), 2 reduces to 2 &lt;// +a 2 sine/) cos# cosc^Va Va 2 sin b2 = 0. The integral of this equation involves an arbitrary constant, is and thus the theorem proved. to the class of problems The preceding example belongs general statement is whose as follows: into a ruled surface in such a To deform a ruled surface the way that deform of a given curve C on the original surface shall possess a certain property on the resulting surface. We consider this general problem. Let the deform of C be the ; , , ; , J TW O n X /* , v directrix of the required surface, and let , /3 , 7 denote the direction-cosines of its tangent, principal normal, and If denotes the angle between the osculating plane to the curve and the tangent plane to the surface, we have binormal. &lt;r (46) I =# cos # * -f sin (/ cos &lt;r +X sin &lt;r), Annali, Vol. VII (1865), p. 115. 346 DEFORMATION OF SURFACES and similar expressions for m and n. When these values are sub stituted in the first two of equations (41), the resulting equations are reducible, by means of the Frenet formulas (I, 50), to coscr /-, b , P ~cos0 n (47) (cos cr . sin n . ,, . sincr sin# sin cr sin 2 ~] ) H + T/ (sin cr sin aM Q) COS cr sin ft"] \ab. I2 ; 2 These are two equations of condition on cr, /o, T, as functions of v. Each set of solutions determines a solution of the problem for, the directrix is determined by expressions for p and r, and equa tions (46) give the direction-cosines of the generators. leave it to the reader to prove the above theorem We by this means, and we proceed to the proof of the theorem: A curve ruled surface may be deformed in such a manner that a given C becomes an asymptotic line on the new ruled surface. On the deform we must have a = or a = TT, so that from (47) p the sign being fixed by the fact that p second of (47) reduces to 2 6&gt; is necessarily positive. The sin If the curve with these intrinsic equations be constructed, and in the osculating plane at each point the line be drawn which makes with the tangent, the locus of these lines is a ruled the angle surface satisfying the given conditions. When the curve C is an orthogonal trajectory of the generators, the same is true of surface its deform. be Hence : A ruled ators become the deformed in such a way that all the gener one of their principal normals of the deform of any may orthogonal trajectories. in Having thus considered the deformation of ruled surfaces which the generators remain straight, we inquire whether two RULED SURFACES 347 ruled surfaces are applicable with the generators of each corre sponding to curves on the other. Assume that it is possible, and let v = const, be the generators of on S corresponding to the S and u = const, generators of Sr From the curves (V, 13) it follows that the conditions for this are respectively where K= \/p\ But equations (48) are the necessary and applicable to sufficient conditions that there be a surface 2 S and Sv upon which the asymptotic lines are parametric (cf. VI, 3). But the curves v = const, and u = const, are geodesies on S and 8^ and consequently on 2. Therefore 2 is doubly ruled. Hence : If two ruled surfaces S and Sl are applicable to one another, the generators correspond unless the surfaces are applicable to a quadric with the generators of S and Sl corresponding to the two different systems of generators of the quadric. EXAMPLES 1. A A ruled surface can be deformed into another ruled surface in such a way that a geodesic becomes a straight line. 2. a right conoid the latter Prove the converse also. ; ruled surface formed by the binomials of a curve C can be deformed into is the right helicoid when the torsion of C is constant. 3. On the hyperboloid of revolution, defined xwu.v = + c by v c cos c sin c , y c A , = u . sin -- A cos c v , z -= u d . A the generators where A 2 = c 2 + d 2 the under the anle cos4. circle of gorge is a geodesic, which is met by Show that the ruled surface which results from the deformation of the 3, in hyperboloid of Ex. which the cos u , circle of gorge becomes z straight, v. is given by x 5. = ud Ad d2 . ud y Ad . sin - v , = -A uc \- Show x c with parallelism of corresponding generators v = u cos - H that the ruled surface to which the hyperboloid of Ex. 3 is the helicoid is applicable A c -c2 C2 + d2 v sin -i c y c = u sin v . A c --+ c2 d2 cos - v , z - = u I 2 --c - c2 d2 c d A A2 v, and that the 6. circle of gorge of the former corresponds to a helix upon the latter. is When the directrix Bin * .*; a geodesic, equations 6 (47) reduce to + = 0, 348 7. DEFORMATION OF SURFACES When an hyperboloid of revolution of one sheet is deformed into another ruled surface, the circle of gorge becomes a Bertraml curve and the generators are parallel to the corresponding bmormals of the conjugate Bertrand curve. 8. A ruled surface can be deformed in such a of arbitrary radius. way that a given curve is made to lie upon a sphere 9. When a ruled surface admits a continuous deformation into is itself the total curvature of the surface constant along the line of striction, the generators meet the latter under constant angle, and the parameter of distribution is constant (cf. 126). 10. Two applicable ruled surfaces whose corresponding generators are parallel cannot be obtained from one another by a continuous deformation. GENERAL EXAMPLES 1. Determine the systems of coordinate lines in the plane such that the linear element of the plane is ^ U 2 _j_ ^2 = where 2. 3. U and V are functions of u and u respectively. 1. Solve for the sphere the problem similar to Ex. Determine the functions 0(w) and ^ x (u) so that the helicoids, defined by = a\/U 2 6 2 cos-, y = shall be applicable to the surface whose equations are where 4. U is any Apply function of u. the method of Ex. 3 to find helicoids applicable to the pseudosphere ; to the catenoid. 5. The equations x = a V2 u 2 cos - , a define a paraboloid of revolution. y = a V2 it 2 sin -, z = a (u 2 it 1) Show /* that surfaces applicable to are defined by , X - id r /3 02 ~/203 + J (fzdfz -fsdfz) - /* J (02^03 ~ 03 dfa) -? 2 = -I /201 /102 where a /3 is a real constant, and the respectively such that / s and s are functions of a parameter a and 6. Investigate the special case of Ex. 5 for which functions, and 2+ /2 = l fl = 7=- a and /3 are conjugate imaginary a-2a* -2V2a .2-a-2a* 2V2a , -the/ / =* s. and the s are functions conjugate imaginary to GENERAL EXAMPLES 7. 349 Show that the surface of translation x is a(cosw + cosv), y a(sinw + sinv), z = c(u + v) applicable to a surface of revolution. 8. Show that the minimal surfaces applicable to a spiral surface (Ex. 22, p. 151) = cio m ~ in , and that the asso are determined by the functions F(u) = cu m + in one. ciate surfaces are similar to the given , 4&gt;(u) 9. If the coefficients E, F, 6? of the linear element of a surface are homogeneous functions of u and v of order 2, the surface is applicable to a surface of revolution. 10. If z, y, z are the coordinates of a surface S referred to a conjugate system, the equations ctf__ ~ dx aw if ~ ~ W__pdy_ ^i_p^:. ^L-Q^L ^-Q^y. aw au aw aw~ aw au au au aw cv ~ = Q~ au are integrable P and Q satisfy the conditions Show where the Christoffel symbols are formed with respect to the linear element of S. that on the surface S whose coordinates are x y z the parametric curves form a conjugate system, and that the normals to S and S at corresponding points , , , , are parallel. 11. Show that for the surface x - f\fi(u)du is + 0i(w), y = f A/2 (w)dit + u, 2 (u), z = , j \f 3 (u)du + 3 (u), where \ any function u respectively, the parametric curves of Ex. 10 to this u and /2 /3 0i, 2 form a conjugate system. Apply the results of v. surface, and discuss the case for which X is independent of u and and/!, , ; 3 *are functions of 12. If S and Si are two applicable surfaces, and S{ denotes the surface corre the same sponding to Si in the same manner as S to S in Ex. 10 and by means of functions P and Q, then S and S{ are applicable surfaces. 13. If x, ?/, z and i, 2/1, z\ are the coordinates of a pair of applicable surfaces S and Si, a second pair of applicable surfaces S and S{ is denned by - h(z + zi) + k(y + x = x + h(z + zi) - k(y + T/J), x[ = x l 2/1), y z = y + k(x + xi)-g(z + *i), = z + g(y + yi) - h(x + x ^-, yi zi x ), = = z/i zt - k (x + xi) + g(z + - g (y + z/i) + h(x + ; zi), KI), where and fc are constants. Show that the line segments joining correspond S and S are equal and parallel to those for Si and S{ that the lines for S and S{ and joining corresponding points on S and Si meet the similar lines that the common conjugate system on S and Si corresponds to the common conju gate system on S and Si. #, ing points of ; 14. Apply the results of Ex. 13 to the surfaces of translation -|- x =w 2 v2 2 av, y 2 = 2 w2 -I- v2 - 2 au - 2* V& 2 + 3 w2 dw, z = 2 6u, 2 u2 - 2 au - Z! = 2 1, fa 2 - 3u 2 dv. S is Show that when g = h = 0, k =- the surface an elliptic paraboloid. 350 15. DEFORMATION OF SURFACES Show that the equations " " 2 "a y ~ J where the accent indicates differentiation with respect to the argument, define a family of applicable surfaces of translation. Apply the results of Ex. 12 to this case. 16. Show that when S and their generating curves correspond, the Si in Exs. 12 and 13 are surfaces of translation, and same is true of S / and S{. 17. If lines be drawn through points of a Bertrand curve parallel to the binormals of the conjugate curve, their locus is applicable to a surface of revolution. 18. If a real ruled surface is to the right helicoid or to a hyperboloid of revolution of applicable to a surface of revolution, it is applicable one sheet (cf. Ex. 9, 144). 19. A ruled surface can be deformed in an infinity of ways so that a curve not orthogonal to the generators shall be a line of curvature on the new ruled surface, unless the given curve is a geodesic in the latter case the deformation is unique ; and the line of curvature is plane. 20. Let P be any point of a twisted curve C, and MI, / { M 2 points on the principal normal to C such that = - PM2 = is / /f a sin ( H where a, 6 are constants and p the radius of curvature of C. The loci of the lines through 21. M\ and 3f2 parallel to the tangent to C at P are 00 applicable ruled surfaces. On the surface whose equations are x = M, y =f(u)&lt;f&gt; (v) + i//(v), z = /(u)[0(i&gt;) (u)] + t(o)- fl^ (u), the parametric curves form a conjugate system, the curves u = const, lie in planes const, in planes parallel to the x-axis parallel to the yz-plane, and the curves v ; hence the tangents to the curves u curve v = const, are parallel. = const, at their points of intersection with a 22. Investigate the character of the surfaces of Ex. 21 in the following cases = Vv2 l (b), (u) = const. (c), t(v) = Q; (d), f(u) = au + b. (v) (a), : + ; ; 23. If the equations of Ex. 21 be written the most general applicable surfaces of the same kind with parametric curves cor responding are defined by where AC is a parameter, and the functions 4&gt;i, 2, ^i, ^2 satisfy the conditions 2 + &lt;I&gt;| = 02 + 0| - K, 4&gt;i* + &lt;J&gt; 2 2 = 0{2 + 02, ~ 1(0212 Show also that the determination of 4&gt;i and 3&gt; 2 requires only a quadrature. CHAPTER X DEFORMATION OF SURFACES. THE METHOD OF WEINGARTEN 145. Reduced form of the linear element. Weingarten has re of all surfaces appli marked that when we reduce the determination cable to a given one to the solution of the equation (IX, 18), (1) namely J = (\-\6)K, we make no use of our knowledge of the given surface, and in reality are trying to solve the problem of finding all the surfaces with an assigned linear element. In his celebrated memoir, Sur la deformation des surfaces,* which was awarded the grand prize of the French Academy in 1894, Weingarten showed that by taking by another which can be solved chapter is account of the given surface the above equation can be replaced in several important cases. This begin by determining a particular moving trihedral for the given surface. It follows from (VII, 64) that the necessary and sufficient con dition that the directrix of a ruled surface be the line of striction is (2) devoted to the exposition of this method. We 6 = a# +#X+*X=0. We (2) The functions // m/ n are proportional to the direction-cosines of the curve in which the director-cone of the surface meets the unit sphere with center at the vertex of the cone. the spherical indicatrix of the surface. From ll call this curve and the identity +mm + nn = seen that the tangent to the spherical indicatrix is perpen dicular to the tangent plane to the surface at the corresponding point of the line of striction. This fact is going to enable us it is under what conditions a ruled surface 2, tangent to a curved surface S along a curve C, admits the latter for to determine its line of striction. *Acta Mathematica, Vol. 351 XX (1896), pp. 159-200. 352 DEFORMATION OF SURFACES suppose that the parameters is We w, v are any whatever, and that the surface referred to a moving trihedral. We consider the ruled surface formed by the z-axis of the trihedral as the origin of the latter describes the curve C. The point (1, 0, 0) of a second trihedral parallel to this one, but with origin fixed, describes the spherical indicatrix of 2. From equations (V, 51) we find that the components of a displacement of 0, this point are r du 4- r^v, (qdu + q^dv). In order that the displacement be perpendicular to the tangent plane to 2 at the corresponding point of (7, that is, perpendicular to the zy-plane of the moving trihedral, we must have (3) rdu if + r l dv = Q. Hence the a trihedral ner, as the vertex of 2&gt;axis T be associated with a surface S in any man T describes an integral curve of equation (3), a ruled surface of T generates whose line of striction is this curve. When the parametric lines on a&gt;axis S which the of T makes with the are are given, and also the angle tangent to the curve v const., U the functions r and r l are completely determined, as follows from (V, 52, 55). They Hll Hence if cU U be given the value C121 /// &lt;f&gt; It") *+*&lt;&gt; where (u) denotes an arbitrary function of M, the function r is zero, and as the vertex of the trihedral describes a curve u = const., the z-axis describes a ruled surface whose line of striction is this curve. t = Suppose now that the trihedral is such that r x 0. From (V, 48, 64) it follows that (6) consequently (1) r= C ty is where an arbitrary function of u. PARTICULAR TRIHEDRAL 353 Let the right-hand member of (7) be denoted by f(u, v), and change the parameters of the surface in accordance with the equations u =u, l v^ = f(u, v). From 32 and equation (7) it follows that idv unaltered by the transformation, in terms of the new coordinates is equal to V unity, and hence from (6) we have Since is K HK Therefore the coordinate curves and the moving trihedral of a surface can be chosen in such a way tnat r (8) = vr ^=0, we It should be r = v, HK=l. In this case reduced form. say that the linear element of the surface is in its remarked that for surfaces of negative curvature the parameters are imaginary. 146. General formulas. If X^ Y^ Z^ X, r, Z denote A;, F2 the direction-cosines of the axes of the moving trihedral with ; , Zj&gt; respect to fixed axes, we have, from (V, 47), du (9) il dv " _ Xa qi q&lt;&gt; - dv i satisfy dv The rotations p,p^ equations (V, 48) in the reduced form dv du x, y, z dv of du reference to these fixed axes are The coordinates c S with given by / (11) y 2 = = f(^i + ^2) ^ + (f 1^1 + where and (13) ^ 2) ^N dv du " 354 DEFORMATION OF SURFACES s Weingarten f* 77, , method consists in replacing the coefficients of of u f t rj l in the last of equations (13) by differential parameters formed with respect to the linear element of the spherical representation of the z-axis of the moving is 2 ) trihedral. By means (14) of (9) this linear element reducible to da 2 = dX + dY + dZ = (v + q 2 2 2 2 du2 + 2 qq^dudv +q 2 dv 2 . The differential parameters of u, formed with respect to this * form, have the values I ^ (15) _4(y2 + v *ti 2 2 ) A (u* A u} = Aq v^q l Aoit = q v ql p vq l Because of the identity (V, 38) we have (16) also A^= 2 -^-by q lt and the values of and (16) be substituted, we have If the last of equations (13) be divided i Pi/2i obtained from (15) 22 3 2 4 t i 2v In consequence of the (18) first of equations (15), written v = -L=, VAjt* in (17) are expressible in terms of to (14), as was the coefficients of f, TJ, fv , rj l differential parameters of u formed with respect to be proved. 0. Under this condition An exceptional case is that in which the spherical representation of the z-axis reduces to a curve, as is seen from (14). q^ * Previously we have indicated to the linear element of the spherical representation. regard this practice in this chapter. by a prime differential parameters formed with respect For the sake of simplicity we dis THEOREM OF WE1NGAKTEN By means (19) 355 of (9) we find that A, (A,, tO = ^. M^, (11) ) = ^f. \(Z ) = f and consequently equations (20) may du be written x =[^ + W\ (X *&gt;)] + [f ^ + v,v\ (X u)] dv, and similarly for y and 2. 147. The theorem of Weingarten. Equation (17) is the equation which Weingarten has suggested as a substitute for equation (1). We notice that f, ?;, fx , T/ I are known functions of u and v when of (18) equation (17) can be given. a form which involves only u and differential parameters given of u formed with respect to (14). On account of the invariant is the surface S By means of these differential parameters this linear element be expressed in terms of any parameters, say u and v may We shall show that each solution of equation (17) determines a surface applicable to S. We formulate the theorem of Wein . character garten as follows : Let S (21 ) be a surface whose linear element in the reduced form ) is ds* %, = (? + 2 ?; 1 2 T? du* + 2 (^ + wj dudv + (tf + u and v such that then rj, fj, are functions of Z^ le the coordinates of a point on the unit in terms of any two parameters u and v , the linear sphere, expressed Furthermore, let Xv Yv 2 1 element of the sphere being (23) da integral 1 = & du * +2& du dv + &gt;dv \ Any (24) u of l the equation Ju, L A M u - Ju, -l= - u, A,w t )= 0, 356 the differential DEFORMATION OF SURFACES parameters being formed with respect to (23), renders the following expression and similar ones in y and z total differentials: (25) where f Ae surface whose coordinates are the functions has the linear element (21). x, y, z thus defined and Before proving this theorem we remark that the parameters u v may be chosen either as known functions of u and v, or in that the linear element (14) shall have a particular In the former case X^ v Z^ are known as functions of u such a form. way Y f and v , and in the second their determination requires the solution of a Riccati equation. However, r f in what follows we assume that are known. l Suppose now that u and v are any parameters whatever, and that we have a solution u^ of equation (24), where the differential Xv Yv Z parameters are formed with respect to (23). quantity (A^)"*. Let v^ denote the v , Both u^ and v l are functions of u and and consequently the latter are expressible as functions of the former. We express X^ Y^ Z{ as functions of u and v l and determine the l corresponding linear element of the unit sphere, which (26) l we write dffl = (; dul + l 2 ^ dujvt + ^ c(v*. In terms of u and v we have , i From these expressions it follows that if we put we have ( 27 ) METHOD OF WEINGARTEN Hence if 357 we put x = Y& - Z,Y the functions A^, tions (V, 47). Y = z& - x&, . z =x Y - r^, 1 a Yv , Z satisfy a set of equations similar to equa In consequence of (27) the corresponding rotations have the values dX .dx r1= It is readily 0. dX shown that these functions satisfy equations similar to (10). Since the functions f, in (21), equations similar to the first sarily satisfied. Hence same form in (25) as of equations (13) are neces the only other equation to be satisfied, in 77, , ^ are of the two order that the expressions (25) be exact differentials, is But it can be shown that the coefficients of (26) are expressible in the form so that g _ v *\ ^ by means &lt;% __ ^_ -2 of differential parameters of u^ formed with respect to (26) the equation (28) can be given the form (17). Hence all the conditions are satisfied, and the theorem of Wein- garten has been established. 148. Other forms of the theorem of Weingarten. It is readily found that equations (22) are satisfied by the expressions dv (29) du dv dv v. where (/&gt; is any function of u and ,7 Since now (30) + ^=0, equation (17) reduces to 358 DEFOKMATION OF SURFACES of This equation will be simplified still more by the introduction two new parameters which are suggested by the following considerations. As previously defined, the functions X^ v Z^ are the directioncosines of lines tangent to the given surface S in such a way that the ruled surface formed by these tangents at points of a curve u = const, has this curve for its line of striction. Moreover, from the theorem of Weingarten it follows that the functions X^ Yv Zl have the same significance for the surface applicable to S which Y corresponds to a particular solution of equation (17). But v Yv Zl may be taken also as the direction-cosines of the X normals to a large group of surfaces, as shown in 67. In partic ular, we consider the surface S which is the envelope of the plane + Each solution Zj, = u. of equation (17) determines such a surface. If x, y, z denote the coordinates of the point of contact of this plane with S, we have from (32) (V, 32) x = uX +\(u,X l 1 ), which, in consequence of (19), (32 may v be written ) i = wX-f-X. of contact of S lies in the plane through the origin to the tangent plane to S at the corresponding point. parallel If the square of the distance of the point of contact from the Hence the point origin be denoted by 2 a ^, and the distance from the origin a a to the tangent plane by p,* (33) we have 2g = s (V, 35, 37) +ya +i =w +^. p=u . From it follows that the principal radii of 2 are given by (34) * The reader will observe that the functions the rotations designated by the same letters. the treatment of the theorem of Weingarten, risk of a confusion of notation. p and q thus defirfed As this notation is are different from it in generally employed has seemed best to retain it, even at the METHOD OF WEINGARTEN where the differential parameters are 359 (14). formed with respect to From (35) these equations we have We shall now effect a change of parameters, using ones. ^*r o_ defined by (33) as the y^r new *^r ^ I By ^ direct calculation ^*r ^ v *r -.;? p and q we obtain . du o- - _ -i dp dq dv . - o v* dq dif +P dp* 2 a/?a^ z. /^ ^ T 2 + ~ a^ (36) _i cudv vdpcq Q^,2-r ^-1^ ~ dv 2 1 tf v 6 2 dq By means of the equations (33) and (36) the fundamental equa tion (31) can be reduced to (37) This is the form in whicli the fundamental equation was first con sidered by Weingarten.* The method of 146, 147 was a subse quent development. In terms of the parameters p and q the formulas (29) become dpdq (38) dq If these values and the expression it is for \(u, X^) given by (32) be substituted in (20), reducible to " dp* * dpdq/ \ "tip tiq cq Comptes Rendus, Vol. CXII (1891), p. 607. 360 DEFORMATION OF SURFACES for Hence the equations S may be written (39) and consequently the linear element of S is of the form from those which figure Since these various expressions and equations differ only in form in the theorem of Weingarten, the latter is remark also that the rightjust as true for these new equations. hand member of (40) depends only upon the form of c. Hence we have the theorem of Weingarten in the form : We a definite function of p and q, this with the same spherical equation defines a large group of surfaces the functions p l and p 2 denoting the principal radii, representation, When (j) in equation (37) is and p and 2q the distance from the origin to the tangent plane and surface the square of the distance to the point of contact. Each 2 a surface with the satisfying this condition gives by quadratures (39) each surface with this linear element linear element Conversely, (40). stands in such relation equation (37). to some surface satisfying the corresponding As a corollary to the preceding results, we have the theorem : The linear element of any surface S (41) di~ is reducible to the form = du* + 2 ^ dudv + ^ dv\ du 2 dv v. where ^r is a function of u and that the linear element of any surface is reducible to the form (40). If, then, we change the parameters by For, we have seen means of the equations we have (43) ds* = du*+2p dudv + 2q dv\ METHOD OF WEINGARTEN From (42) it follows that 2 2 2 , 4&gt; 361 &lt;fu =a ^ 1 dp + , 3 -i&lt;t&gt; , dt&gt; = tf i- , &lt;fy, dp + , d dp ejpdg dp 3 a/ ^ rfg, and consequently A (44) A where 2 2 = dp dq \dp dq the inverse From ~(44) it is seen that dv =-^j and consequently cu of equations (42) are of the fAfii form (45) = -X, du (43) is of the W q * = ^L. dv (41), as W Hence equation form was to be proved. Moreover, equations (44) reduce to (46) 2 = A -= (#?/ ---- A dpdq w, v In terms of these parameters (47) equations (39) reduce to dz dx = X^du + ^c?v, of = I^c^tt + ^^v, are given = Z^du -f ^c?v. Hence the coordinates AQ (48) 2 by J _ X ^C = T^ dv y = -dv _ 3v z = ^2 f ^" dv and the direction-cosines (49) of the normal to 2 are X.A aw is, r = ^, l aw Z1==^, ^ that the normals to 2 are parallel to the corresponding tangents to the curves v = const, on S. Hence we have the the following theorem : When the linear element of a surface is in face 2 whose coordinates are given by (48) form (41), the sur has the same spherical 362 DEFORMATION OF SURFACES its representation of normals as the tangents to the curves v = const. on S. If p and 2q denote the distance from the origin to the tangent plane to S and the square of the distance to the point of contact, they have the values (45). Moreover, if the change of parameters defined by these equations be expressed in the inverse form /cn (50) , M = d&lt;f&gt; dp the principal radii of s v= d(f&gt; * dq 2 satisfy the condition and (52) the coordinates of S are given by quadratures of the form dx with the same representation as 2, and whose determines by equa functions p v /? 2 p, q satisfy (51) for the same tions of the form (52) a surface applicable to S.* Moreover, ever// surface , (f&gt;, 149. Surfaces applicable to a surface of revolution. When the linear element of a surface applicable to a surface of revolution is written (53) d?=du* + p*(u z-axis of the l )dv*, is and the v moving is trihedral tangent to the curve const., the function r equal to zero, as follows (8), from (4). In order to obtain the conditions of variables we effect the transformation u = v^ ds* 2 v = u^ so that the linear element becomes (54) = p du*+dv 2 . Now (55) r =p f , element in 7^=0, and consequently in order the reduced form we must take u to have the linear = u, v=p (v). * For a direct proof of this theorem the reader is referred to a memoir by Goursat, Sur un theoreme de M. Weingarten, et sur la the orie des surfaces applicables, Toulouse Annales, Vol. V (1891) also Darboux, Vol. IV, p. 316, and Bianchi, Vol. II, p. 198. ; SURFACES OF REVOLUTION From surface these results and (32 ) 363 we find that the coordinates of the 2 are given by p cu v f p du^ p dv l p dv l i== _lJl + !!l.^, p cu^ p d Vl to and the direction-cosines of the normals .A., 2 ~ Zs* are 1 fo Y = -- &gt; JL v 1 1 fy -- 5 1 8* = -- p dv l p cv l 2 p dv 1 Also, (56) we have P =^xX^v v : 2? =^ =^ + ~ Hence we have the theorem To a curve which i* the deform of a meridian of a surface of revo planes origin, lution there corresponds on the surface 2 a curve such that the tangent to 2 at points of the curve are at a constant distance from the a deform of a parallel there corresponds a curve such that the projection of the radius vector upon the tangent plane at a to and point is constant. For the present case 77 =f = 1 ; consequently we have, from (38), Sf This equation (57) is satisfied by 2 &lt;/&gt;(&gt;, #)=/(2 q jt? ), where / is any function whatever. In terms of this function we have, from (38), where the accents indicate differentiation with respect to the argument, 2qp 2 . By means of (55) the linear element (54) can be transformed into the function a)(v) being defined by 364 Since 77 DEFORMATION OF SURFACES = fl = 0, we have and we know that r = v. Now equations (58) become and these are consistent because of the relation 2^ p~ = \/v 2 which results from (56). Hence we have the theorem : , When is &lt; (p, q) is a function of 2 q to applicable p\ the corresponding surface S a surface of revolution, the tangents to the deforms of the form (57) and put 2 the parallels being parallel to the corresponding normals to 2. If we give &lt;j&gt; ^ = 2f, 2 2 the linear element of Sis ds (59) = (^q 2 2 p ) d^ + ^fr dp , as follows from (40) or (58). 150. Minimal lines on the sphere parametric. In 147 we re marked that the parametric curves on the sphere may be any what ever. An interesting case is that in which they are the imaginary In generatrices. 35 we saw that the parameters of these lines, say a and (60) /3, can be so chosen that rp X,= + a/3 - a/3-1 a/3 Consequently (61) da 2 =dX? + (32) 4 dad /3 From are we find that the coordinates of 2, the envelope of the plane Xx + Yy+Zz p = (62) z = From (63) these we obtain 2 MINIMAL LINES ON THE SPHEKE By means of p and its found, and 365 of (34) the expressions for p l in terms p 2 and derivatives with respect to a and yS can be readily thus the fundamental equation (37) put in a new form. + p^ not with the general case that we shall now concern but with a particular form of the function ourselves, q). This function has been considered by Weingarten * it is However, it is &lt;j&gt;(p, ; (64) In this case so that equation (37) reduces to (65) /&gt;i + ft = -(2;&gt; + be written (P) which, in consequence of (34), (66) may dad j3 (l + a/3) 2 When the values from (62) are substituted in (52), we obtain (67) z = *& - Cu dZ + J 1 l \ occ where (68) From (42) and (64) we have u qp 2 2 a&gt; f ( p), v=p. in this case, Hence (69) the linear element (43) of ds 2 *S is, =du +2v dudv + 2 [u + v + w (v)] dv 2 2 . *Acta Mathematica, Vol. XX (1896), p. 195. 366 DEFORMATION OF SURFACES (68) it is However, from (70) seen that Ul -, =u + v 2 so that (69) (71) may be written ds 2 = dul + 2 [ MI + a *&gt; . (t&gt;)] theorem Gathering together these results, we have the 2%e determination of reduces to the : all the surfaces ivith the linear element (71) integration of the equation for o(_p) arbitrary integral of this equation However, the integral is known in certain cases. The is not known. consider We several of these. 151. Surfaces of Goursat. oloids. Surfaces applicable to certain parab When we take (73) v (p)=im(l-m)p\ m being any constant, equation (72) becomes m(l-m)p dad ft (l + aj3f can be found by the method general integral of this equation of Laplace,* in finite form or in terms of definite integrals, accord The ing as m The (75) integral or not. linear element of the surface is S is ds 1 = du? + [2 u^ H- m (1 - m) ^] dv\ are such that And (76) the surfaces 2 p 1 + pt is, +2p = m(m-I)p, of the principal radii is proportional to the dis tance of the tangent plane from a fixed point. These surfaces were first studied by Goursat, f and are called, consequently, the that the sum surfaces of G- our sat. *Darboux, Vol. II, p. 66. t American Journal, Vol. X (1888), p. 187. SURFACES OF GOURSAT 367 Darboux has remarked* that equation (71) is similar to the linear element of ruled surfaces (VII, 53). In fact, if the equations of a ruled surface are written in the form (77) x , = x +lu ; l, y w, n are functions of v alone, which now is not necessarily the arc of the directrix, the linear element of the surface will have the form (71), provided that where # /, (78) 2J 2 = 1, 2a^ = 0, 2X = 2 w (^), 2a# = l, 2 2/ 2 =0. In consequence of the equations 2ft it =0, follows that a ruled surface of this kind admits an isotropic = 0, that is, if plane director. If this plane be x + iy we have where V is a function of v. By means form *dv of these values and equa tions (78), we can put (77) in the (79) = %Vu^+% Cv v dv - C~ dv, /y We shall find that dv - yt among these surfaces there is an imaginary which are applicable certain surfaces to which Weingarten called attention. To this end we consider the function paraboloid to _2p^ (80) (_p)=: ^ficp 2 tee v " where K denotes a constant. Now A equation (66) becomes _2 Vic _ 1 * Vol. IV, p. 333. 368 DEFOKMATION OF SURFACES In consequence of the identity the preceding equation is equivalent to dad/3 If log(l + a/3)V^ = JL we put this equation takes the Liouville form -20 &gt; dad IB of which the general integral is 1 AS and /3 where A and B are functions of a respectively, accents indicate differentiation with respect to these. and the Hence the general integral of (81) c is %= VAB (l + a/3) is l 2^ + AS) and the linear element of S (82) If ds 2 = du^2\u - V^K - 2 to (80), now, in addition we take V the equations (79) take such a form that (83) (x+iy)x = icz. Hence the surfaces with the linear element (82) are applicable to the imaginary paraboloid (83). The generator x + iy = Q of this paraboloid in the plane at infinity is tangent to the imaginary circle at the point (x:y:z = l:i:Q), which is a different point from that in which the plane at infinity touches the surface, that is, the point of intersection of the two generators. DEFOBMATION OF PARABOLOIDS Another interesting case value (84) If 2. is 369 afforded when (71) m in (73) has the Then u&gt; = (v) ds 2 v 2 , and equation ( becomes = duf + 2 Ml we take V=v/^/2^c, we obtain from equations (79) from which we (85) find, by the elimination (z of it/) u l and . v, + i 2 = K (x The generator x + iy = in the plane at infinity on the paraboloid : 1 i: 0), circle at the point (x: y: z (85) is tangent to the imaginary but the paraboloid (85) is just as in the case of the paraboloid (83), same point. tangent to the plane at infinity at the GENERAL EXAMPLES 1 moving trihedral can be associated with a surface in an infinity of ways so that as the vertex of the trihedral describes a curve u = const, the z-axis generates a ruled surface whose line of striction is this curve. . A 2. The tangents to the curves v = const, on a surface at the points where these curves are met by an integral curve of the equation form a ruled surface for which the 3. latter curve is the line of striction. If the ruled surface formed by an its line locus of the points of contact for deformations of S. infinity of tangents to a surface S has the of striction, this relation is unaltered by D, D face with the linear element 4 . Show that if , D" (53), are the second fundamental coefficients of a sur the equation of the lines of curvature of the associated surface S is reducible to Ddii! +D ; dvi V dui + D"d dui pp dv\ p 5 . Show that the surface S associated by the method of Weingarten with a sur with parallelism of tangent to the surface S complementary to S with respect to the deforms of the planes meridians and that the lines of curvature on S and S correspond. face S applicable ; to a surface of revolution corresponds 370 6. DEFORMATION OF SURFACES Show that when has the form (57), the equation (51) is reducible to hence the determination of all the surfaces applicable to surfaces of revolution is equivalent to the determination of those surfaces S which are such that if MI and is the projection 2 are the centers of principal curvature of 2 at a point I/, and of the origin on the normal at M, the product NMi 2 is a function of ON. M N NM 7. Given any surface S applicable to a surface of revolution. Draw through a fixed point O segments parallel to the tangents to the deforms of the meridians and of lengths proportional to the radii of the corresponding parallels, and through the extremities of these segments draw lines parallel to the normals to S. Show that these lines form a normal congruence whose orthogonal surfaces 2 have the same spherical representation faces of the equation of Ex. 8. 6. of their lines of curvature as S and are integral sur Let fi complementary be a surface applicable to a surface of revolution and S the surface to S with respect to the deforms of the meridians let also S and ; S be Show surfaces associated with that S and S respectively after the manner of Ex. 7. corresponding normals to S and S are perpendicular to one another, and that the common perpendicular to these normals passes through the origin and is divided by it into two segments which are functions of one another. 9. Show that a surface determined 2q by the equation PIPZ + K + (PI + pz)p + = 0, where a constant, possesses the property that the sphere described on the seg ment of each normal between the centers of principal curvature with this segment K is K in great for diameter cuts the sphere with center at the origin and of radius circles, orthogonally, or passes through the origin, according as K is positive, nega tive, or zero. These surfaces are called the surfaces of Bianchi. V 10. Show that for the surfaces of Bianchi the function 0(p, q) . is of the form = V2 q - p 2 + 1 /c, and that the linear element of revolution is of the associated surface S applicable to a surface Show ds2 also that according as /c = 0, &gt; 0, or &lt; the linear element of S is reducible to the respective forms , = dw 2 + e2M dv 2 ds 2 = tanh 4 u du 2 + sech 2 w du 2 , ds 2 = coth4 u du 2 -f csch 2 u dv 2 . On account of this result and Ex. 10, p. 318, the surfaces of Bianchi are said to be 0. of the parabolic, elliptic, or hyperbolic type, according as K 0, or 0, = &gt; &lt; 11. Let S be a pseudospherical surface with its linear element in the form Find (VIII, 32), and Si the Bianchi transform whose linear element is (VIII, 33). the coordinates x, y, z of the surface S associated with Si by the method of Weingarten, and show that by means of Ex. 8, p. 291, the expression for x is reducible to x = | aea (cos 6X1 + sin 0JT2 ) 4- fX, sc-axis of the where X\, JT2 , X are the direction-cosines with respect to the S and of the tangents to the lines of curvature of normal to the latter. GENERAL EXAMPLES 371 12. Show that the surfaces S and S of Ex. 11 have the same spherical represen tation of their lines of curvature, that S is a surface of Bianchi of the parabolic type, and that consequently there is an infinity of these surfaces of the parabolic type which have the same spherical representation of their lines of curvature as a given pseudospherical surface S. 13. Show that if Si and S 2 are two surfaces of Bianchi of the parabolic type which have the same spherical representation of their lines of curvature, the locus of a point which divides in constant ratio the line joining corresponding points of Si and S 2 is a surface of Bianchi with the same representation of its lines of cur vature, and that it is of the elliptic or hyperbolic type according as the point divides the segment internally or externally. 14. When S is a pseudospherical surface with its linear element in the (VIII, 32), the coordinates x~i, yi, z\ of the surface of Weingarten are reducible to Xi S determined by the form method = A (ae J- a cos 6 + y sin 6) X\ z 1? -f (ae a sin 77 cos 0} JF2 , ; JT2 , 2 , Z 2 are the JTi, FI, direction-cosines of the tangents to the lines of curvature of S. Show also that S has the same spherical representation of its lines of curvature as the surface Si with and analogous expressions for yi and where Zx F the linear element (VIII, 33). 15. Derive by means of from the equations X2 xXi + yYi + zZ t =p, and (49), the equations (44), (48), + 2 + 2 = 2 g, where 16. x, y, 2 are the coordinates of S. Show that the equations for S similar to (IV, 27) are reducible to dudv cv* \cu* z. cucv (cf. and similar expressions in y and Derive therefrom D"dv Ex. 15) the equations D du dpdq du + + r(Ddu + \dq 2 lYdv) - 0, dp* cpdq where D, 17. D , D" are the second fundamental coefficients of 8. Show Ex. that the lines of curvature on S correspond to a conjugate system on S (cf. 16). 1 8. Show that for the surface S we have dx dx ~ dX\ plp&lt; dXi dp . dXi dq * 2p 19. Let eq dq~ yi S be the surface defined by (67) and Si the surface whose coordinates are Xi = x u\X\, = y WiFi, z\ z u\L\. Show that Si is an involute of /S, that the curves p = const, are geodesies on lines of curvature on Si, and that the radii of principal curvature of Si are S and 372 20. DEFORMATION OF SURFACES Show that when m /3 in (73) is or 1, the function p is the trary functions of a and respectively, that the linear element of sum of two S is arbi ds*= dw 12 that + 2i*idw 2 , S is an evolute of a minimal surface (cf. Ex. 19), and that the mean evolute of S is a point. 21. Show that when m in (73) is 2, the general integral of equation (74) is where /i and /2 are arbitrary functions surface 2 is minimal (cf. 151). 22. of a and respectively. Show also that the Show that the mean evolute of a surface of Goursat is a surface of Goursat homothetic to the given one. 23. Show that when u&gt;(p) p= = ^op 2 then a log(l + a/3) +/i(nr) +/2 (/3), , where /i and /2 are arbitrary functions, that the linear element of S ds 2 is = is du? + 2 (ui + 2 aw) du , and that the mean evolute of 2 24. tion a sphere. Show that the surfaces S of Ex. 23 are applicable to the surfaces of revolu S whose equations are v . v a where a 25. is a = J I c vu* /~17 a2 ia, is an arbitrary constant. Show also that when a that = a paraboloid. Show when the surfaces tive ; S also that the surfaces are spherical or pseudospherical according as are applicable to the surface &lt;S m z is positive or nega x + iy = v, x-iy = 1)2 M2 2 TT~~ 2m mu = w which is a paraboloid tangent to the plane at infinity at a point of the circle at infinity. CHAPTER XI INFINITESIMAL DEFORMATION OF SURFACES 152. General problem. of isometric surfaces The preceding chapters deal with pairs which are such that in order that one may be applied to the other a finite deformation is necessary. In the present chapter we shall be concerned with the infinitesimal deformations which constitute the intermediate steps in such a finite deformation. , S and Let x^y,z\ x y\ z respectively be the coordinates of a surface a surface S\ the latter being obtained from the former by a If very small deformation. (1) we put f x e ^x + ex^, y =y + cyv z ^z + ez^ where tions of denotes a small constant and x v y^ z l are determined func u and v, these functions are proportional to the direction- cosines of the line through corresponding points of these equations we have S and S f . From dx * + * dy + dz n = da? + dy + dz + 2 2 2e(dx dx l + dy dy l + dz dzj If the functions satisfy the condition (2) dx dx l -f dy dy^ + dz dz l = r 0, corresponding small lengths on S and S are equal to within terms 2 of the second order in e. When e is taken so small that e may be neglected, the surface S defined by (1) is said to arise from S by an infinitesimal deformation of the latter. In such a deformation each point of S undergoes a displacement along the line through it whose direction-cosines are proportional to x v y v z r These lines are called the generatrices of the deformation. It is evident that the problem of infinitesimal deformation is equivalent to the solution of equation 373 (2). Since x v y^ z l are 374 INFINITESIMAL DEFORMATION v, functions of u and surface they may be taken for the coordinates of a the fact that the tangent to r Equation (2) expresses curve on S is perpendicular to the tangent to the correspond any We say that in this case ing curve on Sl at the homologous point. S and S ments. l linear correspond with orthogonality of corresponding And so we have: ele The problem of equivalent the infinitesimal deformation of a surface S is to the determination of the surfaces corresponding to it with orthogonality of linear elements. 153. Characteristic function. of these surfaces We proceed to the determination (2) Sv and to this end replace equation by the equivalent system Weingarten (4) * replaced the last of these equations by the two ^ ex Sssw-** fix. ^-v X^w=~* 7/ dx dx l thus denning a function , which Bianchi has called the character = V EG F*. istic function ; as usual If the first of equations (3) be differentiated with respect to v, H and the second with respect to M, we have * dv du dv dudv ** fa -uv fr fa du dv With the aid of these identities, of the formulas (V, 3), and of the Gauss equations (V, 7), the equations obtained by the differentia tion of equations (4) with respect to . reducible to u and v respectively are H v * Crelle, Vol. H C (1887), pp. 296-310. CHARACTERISTIC FUNCTION Excluding the case where S &lt;r-\ 375 is a developable surface, * we solve these dx. equations for &gt;,^ *-l du 2*X-zr anc ^ dv TT-\ dx. obtain cu (5) u dv S. dv KH where and K denotes and the total curvature of If we solve equations (3), (4), v, (5) for the derivatives of x^ y^ zl with respect to u we obtain v cu CU (6) KH dv dv dv KH function and similar expressions in y^ and z r Hence, when the characteristic is known, the surface S can be obtained by quadratures. l Our problem reduces If equations (5) therefore to the determination of &lt;. be differentiated with respect to v and u respec and the resulting equations be subtracted from one another, tively, we have +4 V 1 to du V 2j i fa. fa. When the derivatives of Jf, F, Z 8), in the right-hand member are replaced by the expressions (V, the above equation reduces to dv u d du dv KH Bianchi du KH (7) is H reducible to calls this the characteristic equation. In consequence of (IV, 73, 74) equation ft 376 INFINITESIMAL DEFORMATION &lt;", where c^, are the coefficients of the linear /S, element of the spherical representation of (9) namely 2 da* = &du* + /K ^ dudv reducible to and By means of (V, 27) equation (8) is where the Christoffel symbols are formed with respect Since X, Y, to (9). Z tions of (10), and latter equation may be written are solutions of equations (V, 22), they are solu consequently also of equation (7). Therefore the dX dv du KH du v _ But this we have is the condition of integrability of equations : (6). Hence the theorem Each solution of the S^ and consequently an characteristic equation determines a surface infinitesimal deformation of S. 154. Asymptotic lines parametric. When the asymptotic lines on 8 are parametric, equation (10) (VI, 15), to is reducible, in consequence of dudv 2 dv du 2 du dv where If we put &lt;f, V- ep = 6, ASYMPTOTIC LINES PARAMETIC e 377 is being +1 or 1 according as the curvature of S positive or negative, equation (11) (12) becomes ** Since X, Y, vl Z are = XV solutions of (11), the functions e/j, v2 = FV ep, are solutions of (12). Now (13) equations (6) may be put in the form e dx, e dv cB cv du du The reader should compare formulas ( 79), which give the expressions these equations with the Lelieuvre for the derivatives of i/ i/ the coordinates of S in terms of it 1? 2 , i&gt; 3 . From these results follows that any three solutions of an ffQ equation of the form = MQ, where isany function of u and w, determine a surface S upon which the parametric curves are the asymptotic lines, and every other solution linearly independent of these three gives by ratures an infinitesimal deformation of S. M quad EXAMPLES 1. A tion (2) be applicable 2. ft, necessary and sufficient condition that two surfaces satisfying the condi is that they be minimal surfaces adjoint to one another. If x, y, z and x l9 T/I, zi satisfy the condition (2), so also do , 17, f and &, ^j, the latter being given by 77 = aix + =2+ = a 3 x -f f , biy &22/ 68 y , + + + ciz C2 Z c3z + + + di, xi 2/1 d2 , d8 , Zi = a^ + a2 Tn + a s ft + ei, = &ll + &2^?l + &3ft + C = Ci^ + c 2 + c 3 ft + c 8 2 , &gt;?i , where a 1? a 2 3. , ei, e 2 e3 are constants. necessary condition that the locus of the point (xi, ?/i, z$$ be a curve is that S be a developable surface. In this case any orthogonal trajectory of the tangent planes to S satisfies the condition. 4. 5. A Investigate the cases If Si = and to = c, where c is a constant different from zero. of linear elements, so also does the locus of a point dividing in constant ratio the line joining corresponding points on Si and S{. and S{ correspond S with orthogonality 378 INFINITESIMAL DEFORMATION The expressions in the parentheses of 155. Associate surfaces. equation efficients, (10) differ D , Z&gt; Z&gt;J, ", only in sign from the second fundamental co of the surface /7 enveloped by the plane (14) Hence equation (15) (10) may D"D be written Q + DDJ - 2 D D[ = 0. the condition that to the asymptotic lines upon either of the surfaces S, S there corresponds a conjugate system on is ( 56). Bianchi applies the term associate to two sur whose tangent planes at corresponding points are parallel, and for which the asymptotic lines on either correspond to a conjugate system on the other. Since the converse of the pre This the other faces ceding results are readily shown to be true, of Bianchi f : we have the theorem When from a two surfaces are associate the expression for the distance fixed point in space to the tangent plane to one is the char acteristic function for an infinitesimal deformation of the other. the problems of infinitesimal deformation and of the determination of surfaces associate to a given one are equivalent. consider the latter problem. Hence We Since the tangent planes to are parallel, S and SQ dxn -5 dv 2 at corresponding points we have dzn -2 dx = X -- dx fji , = 0- dx -- r dx du du dv du X, ft, du and similar equations in y Q and of u and v to be determined. J , where by &lt;r, r are functions If these equations be multiplied ~y ~Y likewise by dv dv 2V- and added, we dv &M obtain$U -- and added, and dU \D&lt; t * Cf. J 67. Lezioni, Vol. II, p. 9. The negative signs before p. and r are taken so that subsequent results may have a suitable form. ASSOCIATE SURFACES where Z&gt; 379 When (17) are the second fundamental quantities for $. these values are substituted in (15), we find " , Z&gt; , D X-r=0. Consequently the above equations reduce to du If du of the dv dv du dv we make use Gauss equations (V, is 7), the condition of integrability of equations (18) reducible to du dv where A and B 2, are determinate functions. Since similar equations Calculating the following equations hold in y and both A and B must be identically zero. we have &lt;r the expressions for these functions, to be satisfied by X, /A, and : JL d\ . f22i . rm fl21 I1J . rii (19) da_d\ du dv J22\ v fill I 2 1J 1J To (20) these equations we must add 2 \D - &lt;rZ&gt; = pl&gt;" 0, obtained from the last of (16). The determination of the asso ciate surfaces of a given surface referred to any parametric system requires the integration of this system of equations. Moreover, shall now every set of solutions leads to an associate surface. consider several cases in which the parametric curves are of a We particular kind. face Suppose that S is a sur which the parametric curves form a conjugate system. upon We inquire under what conditions there exists an associate sur face upon which also the corresponding curves form a conjugate 156. Particular parametric curves. system. 380 INFINITESIMAL DEFORMATION this hypothesis On we have, from (16), /* = a- = 6, so that equations (19) reduce to &lt;&gt; which are consistent only when that is, when the point equation of S, namely j^ dudv fi21&lt;tf \l J cu \ZJfo ri2|&lt;tf has equal invariants (cf. 165). the function X Conversely, when condition (22) is satisfied, the equations makes given by the quadratures (21) of an associate surface are compatible, and thus the coordinates have the theorem of Cosserat*: obtained by quadratures. Hence we a surface S is the same problem infinitesimal deformation of as the determination of the conjugate systems with equal point invari The ants on S. reciprocal and the both surfaces, these curves on parametric curves are conjugate for also have equal point invariants. Since the relation between S and S is lines asymptotic lines, the corresponding In this case, as is seen from (16), on SQ form a conjugate system. X is zero and equations (18) reduce to If S be referred to its . /24 &gt; dxn 2 ) = u dx dxn dv dx a- . ; du ^dv 3u moreover, equations (19) become n Toulouse Annales, Vol. VII (1893), N. 60. KULED SUEFACES 381 The solution of this system is the same problem as the integra tion of a partial differential equation of the second order, as is seen by the elimination of either unknown. When a solution of the former is obtained, the corresponding value of the other unknown is given directly by one of equations (25). We make an application of these results to a ruled surface, which we suppose to be referred to its asymptotic lines. If the curves v const, are the generators, they are geodesies, and conse quently (VI, 50) p 1 12 can be found by a quadrature. When this value is sub stituted in the second of equations (25), we have a linear equa tion in and consequently also can be obtained by quadratures. Now /* &lt;r &lt;r, Hence we have the theorem : When the curved asymptotic lines on a ruled surface are known, its associate surfaces can be found by quadratures. its If S were referred to (24). asymptotic lines, we should have equations similar to as follows: These equations may be interpreted The tangent is parallel to on one of two associate surfaces the direction conjugate to the corresponding curve on to an asymptotic line the other surface. EXAMPLES 1. If two associate surfaces are applicable to one another, they are minimal surfaces. 2. Every surface of translation admits an associate surface of translation such that the generatrices of the two surfaces constitute the 3. 4. common conjugate system. The surfaces associate to a sphere are minimal. When the equations of the right helicoid are x u cos v, y u sin u, z cro, 2 2 the characteristic function of any infinitesimal deformation is = ( V) (u + )~ functions of u and v respectively. Find the surfaces are arbitrary and where U+ , U V Si and So, and 5. show that the latter are molding surfaces. If S and S are associate surfaces of a surface S, the locus of a point dividing in constant ratio the joins of corresponding points of So associate of S. and S6 is an 382 INFINITESIMAL DEFOKMATION S1? S . 157. Relations between three surfaces S, Having thus discussed the various ways in which the problem of infinitesimal deformation may be attacked, we proceed to the consideration of other properties which are possessed by a set of three surfaces$, Stf SQ . We recall the differential equation dxdxl + dydy v + dzdz l = 0, and remark that (26) if it may yQ dz, , be replaced by the three dXi=zQ dy dy^ x dz Q z Q dx, dz^y^dx x^dy, the functions # ?/ , z are such a form that the conditions of integrability of equations (26) are satisfied. These conditions are du dv du dv dx_d_z_o dv du dv du d_x_dz, = fe?5, dv du dv du du dv dx dy, du dv dv du du dv If these equations du dv dy dx dv du ( be multiplied by i " - - respectively and F, ^, added, and likewise by ? and by JT, we obtain, 0y " by (IV, (27) 2), (28) From the first two of these equations it follows that the locus of the point with coordinates XQJ T/O z corresponds to S with paral lelism of tangent planes. , from In order to interpret the last of these equations we recall 61 that a d(Y, Z} ft d(u, v) Y= a d(Z, ft d(u, X) v) a d(X, Y) /if d (u, v) KELATIONS BETWEEN where a is S, S 19 AND S 383 or negative. 1 according as the curvature of the surface is positive If we substitute these values in the left-hand mem bers of the following equations, and add and subtract dU dv dx dX dX from these equations respectively, the resulting and --members --dv du dv expressions are reducible to the form of the right-hand du/ \ du dv dv \ du dv (28) can be By means of these and similar identities, equation transformed into ,_ ^. __ D,^ is , * Y o Since this equation quantities # when a surface , a; ZQ is in equivalent to (15) because of (27), the Hence (26) are the coordinates of S . ing surface S and Sr This result enables us to find another property of If X^ Y^ Zt denote the direction-cosines of the normal to S^ they are given by 1 l I l Sl known, the coordinates of the correspond are readily found. d(u, v) HI F*, d(u, v) H^ d(u, v) where 7/ = t ^^ E^ F^ G being l the coefficients of the , , linear element of values of the derivatives of x v y v z x r as given by (26), be substituted in these expressions, we have, If the in consequence of (14), (30) X^-^ normal to Y the theorem : As an immediate consequence we have A S t is parallel to the radius vector of S at the corre sponding point. 384 INFINITESIMAL DEFORMATION of (30) we find readily the expressions for the second coefficients J9 t D[, If we notice that J of $L , By means fundamental D . and substitute the values from 1 (6) and (30) in du du ^ du dv ^4 dv du we obtain (31) From (32) these expressions follow Combining S, or, this result lines to with (15), we have : The asymptotic Sv SQ correspond upon any one of a group of three surfaces a conjugate system on the other two; in other words: lines The system of f which to the is conjugate for any two of three surfaces *S , Sv S If the corresponds asymptotic lines on the other. negative, its curvature of S be asymptotic lines are 1 real, is and consequently the common conjugate system on S and S real. If these lines be parametric, the second of equations (32) reduces to As an odd number of the four quantities in this equation must be negative, either SQ or S1 has positive curvature and the other negative. Similar results follow if we begin with the assumption that Sl or SQ has negative curvature. If the curvature of l S be positive, the conjugate system common to it and S is real (cf. 56) ; consequently the asymptotic lines RELATIONS BETWEEN on SQ are real, S, S AND S 385 and the curvature the curvature of is of the latter we saw that of that when SQ : is is negative. But negative, and of S positive, S 1 also negative. Hence S, Given a set of three surfaces has positive curvature. S^ SQ ; one and only one of them Suppose that S is to asymptotic lines on SQ referred to the conjugate system corresponding The point equation of S is . We If shall prove that this is the point equation of S^ also. we differentiate the equation with respect to v, and make use of the fact that tions of (33), we have, in consequence of (26), ~~ y and z are solu dudv \dv du~~l)vduj I1 / #M t 2 J~dv is zero in consequence of equa and hence xl is a solution of (33). Since the parametric curves on S are its asymptotic lines, the and consequently of S must satisfy spherical representation of But the expression in parenthesis tions similar to (24), the condition ^ f!2V d fl2V Hence we have the theorem The problem of invariants of Cosserat: infinitesimal deformation of a surface is the same as the determination of the conjugate systems with equal tangential upon the surface. 158. Surfaces resulting pass to tesimal deformation of 8. from an infinitesimal deformation. We the consideration of the surface S arising from an infini Its coordinates are given by where a small constant whose powers higher than the first are for neglected. Since the fundamental quantities of the first order$ namely G are equal to the corresponding ones for by is , J" , F f , , , 386 INFINITESIMAL DEFORMATION means of (26) the expressions for the direction-cosines of the normal to S are reducible to X 1 , FZ , and similar expressions for Y and Z . The means derivatives of of (29) to X / with respect to u and v are reducible by dX^_d_X dY_ OJL 77 dZ\ ea / D ,^_ D 3X _ T) .1 dX dv dX i fly \ I vZ\ I I d - I ( uX 7) i vJL dv dv dv / /i- \ du dv where a When 1 according as the curvature of $is positive or negative. is these results are combined with (26) and (34), we obtain ^^_V ^4 du du e du du /if V du (D ^\ **l \9* 9* The +du du cu) -D dv v *\du du du du it last expression is identically zero, as one sees by writing for out in full. a- aTl From this , and similar expressions , X , and dv du V dv ** , v/ dv VA fix 1 flX 9 ay ^ the values for the second fundamental in the form coefficients of S can be given =(30) 2) u T** cu T =D + jf We know of that ff is equal to UK according as the curvature 157, one and only 60). Also, by positive or negative (cf. surfaces S, Sv S has positive curvature. Recalling one of three 1 according as the curvature of that a in the above formulas is S is positive or negative, we can, in consequence of (31), write equations (36) in the form S is Z&gt; .*" = .ZX J&gt;, where the upper sign holds when Sl has positive curvature. ISOTHERMIC SUBFACES From these equations it is 387 seen that zero. & and D r can be zero sim ultaneously only when D[ is Hence we have: infini The unique conjugate system which remains conjugate in an tesimal deformation of a surface is the one corresponding to a conju thing, to the asymptotic lines gate system on S^, or, what is the same on SQ . In particular, in order that the curves of this conjugate system be the lines of curvature, it is spherical representation be orthogonal, a minimal surface (cf. 55). From this necessary and sufficient that the and consequently that be it representation of the lines of curvature of versely, if follows that the spherical S is isothermal. Con unique minimal sur same representation of its asymptotic lines, and this surface can be found by quadratures. Hence the required infinites a surface is of this kind, there is a face with the imal deformation of the given surface can be effected by quadra tures (26), and so we have the theorem of Weingarten * : sufficient condition that a surface admit an infini tesimal deformation which preserves its lines of curvature is that the A necessary and spherical representation of the latter be isothermal; when such a surface is expressed in terms of parameters referring to its lines of curvature, the deformation can be effected by quadratures. 159. Isothermic surfaces. By means of the results of 158 we obtain an important theorem concerning surfaces whose lines of curvature form an isothermal system. They are called isothermic surfaces (cf. Exs. 1, 3, p. 159). it From equations (23) follows that if the common conjugate system on two associate surfaces is orthogonal for one it is the same for the other. In this case equation (22) reduces to of which the general integral is E where U Hence the 41). G=r Akademie zu Berlin, 1886. functions of u and v respectively. lines of curvature on S form an isothermal system (cf. * U and V are Sitzungsberichte der Konig. 388 If the INFINITESIMAL DEFORMATION parameters be isothermic and the linear element written ds 2 =r(du 2 +dv 2 ), it follows from (21) that (37) X (23) = i, ~ and equations become ~~ du r du dv r dv From these results : we derive the following theorem of Bour * and to its lines Christoffel If the linear element of an isothermic surface referred of curvature be ds * _r / du * _|_ dl?\ It is asso a second isothermic surface can be found by quadratures. ciate to the given one, and its linear element is 1 ds? r (du + dv ). From equations (16) and (17) it follows that the equation of the common conjugate system (IV, 43) on two associate surfaces$, S is reducible to fi (38) du 2 + 2 X dudv + a dv = 0. 2 tion that The preceding results tell us that a necessary and sufficient condi S be an isothermic surface is that there be a set of solu is tions of equations (19) such that (38) of curvature on S. Hence there must 2 the equation of the lines be a function p such that &lt;r p p (ED* FD), X == p (ED 1 GD), =?&lt;p (FD" GD f ) satisfy equations (19).f equations of the form Upon S-s- substitution we are brought to two = a;, = p\ u and v. du dv where a and ft are determinate functions of In order that S be isothermic, these functions must ~~ satisfy the condition dv du When it is satisfied, p and consequently p, X, a are given by quad / ratures. * Journal de f Cf. I Ecole Poly technique, Cahier 39 (18G2), p. 118. Bianchi, Vol. II, p. 30. ISOTHEKMIC SURFACES Consider furthermore the form (39) 389 H(p du* + 2 X dudv + o- dv~). lines of curvature are para (37) it is seen that when the Hence its curvature is to 2 dudv. metric, this expression reduces From zero (cf. From ratures. V, 135 12), it follows that this form and consequently the curvature of (39) is zero. is reducible to du 1 dv l by quad the theorem of Weingarten : Hence we have The lines of curvature upon an isothermic surface can be found by quadratures. We of Ribaucour. conclude this discussion of isothermic surfaces with the proof of a theorem He introduced the term limit surfaces of a group of applicable sur faces to designate the members of the group whose or minimum. According to Ribaucour, mean curvature is a maximum The limit surfaces of a group of applicable surfaces are isothermic. In proving vature. (36) the Its it we consider a member S of the group referred to its lines of cur mean curvature is given by D/E + D"/G. In consequence of equations mean curvature of a near-by surface is, to within terms of higher order, A or necessary and sufficient condition that the mean curvature of S be a maximum minimum is consequently /j) j&gt;"\ Excluding the case of the sphere for which the expression in parenthesis we have that DO is zero. Hence the common conjugate system of S and posed of lines of curvature on the former, and therefore S is isothermic. &lt;S is zero, is com GENERAL EXAMPLES the coordinates of two surfaces corresponding with surfaces orthogonality of linear elements, the coordinates of a pair of applicable are given by =+ = y + ty\, =x+ n fc 1. If x, y, and xi, y\, z\ are tei, txi, m 2 &lt;zi, 2 =x -r} y tyi, f2 = z tei, where 2. any constant. two surfaces are applicable, the locus of the mid-point of the corresponding points admits of an infinitesimal deformation in which t is If line joining this line is the generatrix. 3. tion (a, 6, Whatever be the surface S, the characteristic equation (7) admits the solu = aX + bY + cZ, where a, 6, c are constants. Show that S is the point c) and that equations (26) become infinitesi - bz + d, ex -\- e, ay + /, z\ = bx yi = az xi = cy where d, e, /are constants; that consequently Si is a plane, and that the mal deformation is in reality an infinitesimal displacement. 390 4. INFINITESIMAL DEFORMATION Determine the form of the results of Exs. 1, 2, where has the value of Ex. 3. 5. Show that the first fundamental coefficients EI, FI, GI of a surface Si are of the form E= 1 E&lt;f&gt;* , , = -F0 2 _.- dv Let S denote the locus of the point which bisects the segment of the normal S between the centers of principal curvature of the latter. In order that the lines on 2 corresponding to the lines of curvature on S shall form a conju 6. to a surface it is necessary and sufficient that S correspond to a minimal surface with orthogonality of linear elements, and that the latter surface and S correspond with parallelism of tangent planes. gate system, 7. Show that when face S satisfies the the spherical representation of the asymptotic lines of a sur condition a (92 \\y cu ( 2 } cv ( 1 = and equations (25) admit two pairs of solutions which are such that /x = On the two associate surfaces S SQ thus found by quadratures the parametric systems are isothermal-conjugate, and S Q and S Q are associates of one another. &lt;r /j. &lt;r. , 8. Show two surfaces associate 9. that the equation of Ex. 7 is a necessary and sufficient condition that to S be associate to one another. Show that when the sphere is referred to its minimal lines, the condition of Ex. 7 is satisfied, and investigate this case. 10. On any surface associate to a pseudospherical surface the curves correspond ing to the asymptotic lines of the latter are geodesies. A surface with a conjugate system of geodesies is called a surface of Voss (cf. 170). 11. Determine whether minimal surfaces and the surfaces associate to pseudo- spherical surfaces are the only surfaces of Voss. 12. When the equations of a central quadric are in the form (VII, 35), the asso ciate surfaces are given by 2/o = 2 V6 Fj Uu du + f Vv dv\ u and , z =i v respectively ; where 13. and are arbitrary functions of are surfaces of translation. U V hence the associates When the equations of a paraboloid are in the form x=Va(u + 1&gt;), y=Vb(u-v), z = 2uv, ; the associate surfaces are surfaces of translation whose generators are plane curves their equations are x = Va(U + V), y =Vb(V-U), z = 2fuU du where U and V are arbitrary functions of u and v respectively. GENERAL EXAMPLES 14. 391 Show its lines of curvature, that a quadric admits of an infinitesimal deformation which preserves and determine the corresponding associate surface. &lt;S between S arid Si is reciprocal, there is a surface 3 which bears to S a relation similar to that of SQ to Si. Show that the asymptotic lines on S and S 3 correspond, and that these surfaces are polar 2 2 + z 2 + 1 = 0. reciprocal with respect to the imaginary sphere z + 15. Since the relation associate to Si ?/ reciprocal, there is a surface S% cor responding to S with orthogonality of linear elements which bears to S a relation is 16. Since the relation between S and So similar to that of Si to So. Show that the asymptotic lines on Si and that the coordinates of the latter are such that Sz correspond, xi-xz = and that the 17. yzo - zy , yi-y z = zx - zz , z\ is z* = xy - yx , line joining corresponding points on Si and S 2 tangent to both surfaces. Show that of linear elements S& are related to 18. S 5 denotes the surface corresponding to S 3 with orthogonality which is determined by Si, associate to SB, the surfaces S and one another in a manner similar to Si and Sz of Ex. 16. if Show that the surface &lt;S the polar reciprocal of 19. If S 4 which is the associate to Sz determined by So, with respect to the imaginary sphere x 2 + y 2 + z 2 + 1 = , is 0. we continue the process introduced in the foregoing examples, we obtain two sequences of surfaces S, Si, So, $3, Sj, S7 , Sg,$8, Sn, Sio, - , S, S2 , 84, Se, Show that Sn and S 10 are the same surface, likewise ; quently there is a closed system of twelve surfaces faces of Darboux. 20. lines S i2 and S 9 and that conse they are called the twelve sur , A necessary and sufficient condition that a surface referred to be isothermic is that j) jj D" its minimal F iso where 21. of curvature on an thermic surface be represented on the sphere by an isothermal system is that U and V are functions of u and v respectively. A necessary and sufficient condition that the lines P* Pi_U ~ F where and are functions of u and v respectively, the latter being parameters referring to the lines of curvature. Show that the parameters of the asymptotic lines on such a surface can be so chosen that = G. U V E 22. Show that an isothermic surface is transformed by an inversion into an isothermic surface. 23. If Si and S 2 are the sheets of the envelope of a family of spheres of two parameters, which are not orthogonal to a fixed sphere, and the points of contact of any sphere are said to correspond, in order that the correspondence be conformal, it is necessary that the lines of curvature on Si and surfaces be isothermic (cf. Ex. 15, Chap. XIII). S 2 correspond and that these CHAPTER XII RECTILINEAR CONGRUENCES 160. Definition of a congruence. Spherical representation. A two- parameter system of straight lines in space is called a rectilinear congruence. The normals to a surface constitute such a system ; likewise the generatrices of an infinitesimal deformation of a sur face (cf. 152). Later we shall find that in general the lines of a congruence are not normal to a surface. Hence congruences of normals form a special class ; they are called normal congruences. of light. They were the first studied, particularly in investigations of the effects of reflection and refraction upon rays The first purely mathematical treatment of general rectilinear congruences was given by Kummer in his memoir, Allgemeine Theorie der gradlinigen Strahlensysteme.* We begin our treatment of the subject with the derivation of certain of methods similar to his own. Rummer s results by the definition of a congruence it follows that its lines meet a given plane in such a way that through a point of the plane one line, or at most a finite number, pass. Similar results hold if a surface be taken instead of a plane this surface is ; From called the surface of reference. And so we . r rnay define a con gruence analytically by means of the coordinates of the latter surface in terms of two parameters u, v, and by the directioncosines of the lines in terms of these parameters. Thus, a con gruence is defined by a set of equations such as *f$$u where the functions i v} &gt; y = fz( u v -&gt; z )-&gt; jz( u v ) -&gt; &gt; / and &lt; under consideration, and the functions are analytic in the domain of are such that (/&gt; u and v * Crelle, Vol. LVII (1860), pp. 189-230. 302 NORMAL CONGRUENCES 393 a representation of the congruence upon the unit sphere by drawing radii parallel to the lines of the congruence, and call it the spherical representation of the congruence. When We put We make the linear element of the spherical representation (3) is da 2 = 2 &lt;f(^ + 2 &dudv dX dx If we put = ^A we have (5) dx dX ,, ex dX dx the second quadratic form ] dxdX= e du + (/ +/ 2 ) rfwdv + g dv\ If which fundamental in the theory of congruences. 161. Normal congruences. Ruled surfaces of a congruence. is there be a surface of (6) S normal to the congruence, the coordinates S are given by y x t =x + tX, =y + tY, z =z + tZ, r . where Since (7) measures the distance from the surface of reference to S is normal to the congruence, we must have which is equivalent to du du dv 3v If these equations be differentiated tively, (9) with respect to v and u respec and the resulting equations be subtracted, we obtain /=/ . Conversely, when this condition is satisfied, the function t given by the quadratures (8) satisfies equation (7). Since t involves an additive constant, equations (6) define a family of parallel surfaces normal to the congruence. Hence : A that necessary and sufficient condition for a normal congruence be equal. f and is f the congruence which pass through a curve on surface of reference S form a ruled surface. Such a curve, and The lines of the 394 RECTILINEAR CONGRUENCES is consequently a ruled surface of the congruence, relation (10) determined by a between u and v. Hence a differential equation of the form Mdu+Ndv = consider defines a family of ruled surfaces of the congruence. a line l(u, v) of the congruence and the ruled surface 2 of this family upon which I is a generator ; we say that 2 passes through I. 103, 104. apply to 2 the results of We We If dsQ denotes the linear element of the curve it C in follows from (VII, 54), 2 the quantities a and b for 2 have the values the surface of reference, which 2 cuts (3), and (5) that \* da 2 ~ ^ dX From common v dx (VII, 58) we have I that the direction-cosi ies X, //., v of the 4- du, perpendicular to is and to the line (10), + dv, where dv/du given by I of parameters u have the values (12) - \ da da which, by means of (V, 31), are reducible to dX - ^dX\ & (13) , \= dv )du du / / + ^dX - 3 dv , *&gt;dX\-, { &lt;?- dv \ du/ &* da . and similar expressions for /A and v. From (12) it follows that .dX X ^~ + da , dY ^^~" da hz/ t dZ n ^~ =0 da - Since dX/da, dY/da, dZ/da are the direction-cosines of the tangent to the spherical representation of the generators of 2, we have the theorem : Given a ruled surface 2 of a congruence ; let C be the curve on the point of C correspond the unit sphere which represents 2, and M ing to a generator to L of S; the limiting position of the common per to the pendicular tangent to L and M. a near-by generator of 2 is perpendicular C at PRINCIPAL SURFACES 395 162. Limit points. Principal surfaces. By means of (VII, 62) and (12) we find that the expression for the shortest distance 8 between I and V is, to within terms of higher order, dx dsf da- dy dz X Y dS z dZ dX dY When member (14) the values (13) for X, /*, v are substituted in the right-hand of this equation, the result is reducible to &lt;odu /{do- e du du + gdv + g dv , If jY denotes the point where this line of shortest distance meets the locus of jVis the line of striction of 2. Hence the distance of N from it the surface r, be denoted by . measured along Z, we have, from (11), , jC-J.2 is given by (VII, 65) ; if n/ 10 V r 2 _ edu*+(f+f) dudv + g dv i For the present* we exclude the case where the coefficients of the two quadratic forms are proportional. Hence r varies with the value of dv/du, that is, with the ruled surface 2 through I. If we limit our consideration to real surfaces 2, the denominator is always positive, and consequently the quantity r has a finite mum and minimum. In order to find the surfaces , 2 for maxi which r has these limiting values, (16) we replace dv/du by and obtain If we equate , to zero the derivative of the right-hand member with respect to we get a quadratic in t. Since &*&gt; 0, we may apply to this equation reasoning similar to that used in connection with equation (IV, 21), * Cf. Ex. 1, 171. 396 RECTILINEAR CONGRUENCES and thus prove that it has two real roots. The corresponding values of r follow from (16) when these values of t are substituted in the latter. Because of (17) the resulting equation may be written r where t When we indicates a root of (17) and r the corresponding value of write the preceding equations in the form r. r \ + e] + [r +\ -o, and eliminate t, we obtain the following quadratic in r: If r^ and r 2 denote the roots of this equation, we have (19) The which points on limit points. lie corresponding to these values of r are called its They are the boundaries of the segment of I upon I near-by line of the congruence. the feet of each perpendicular common to it and to a The ruled surfaces of the con-, I gruences which pass through and are determined by equation (17) are called the principal surfaces for the line. There are two of them, and their tangent planes at the limit points are determined by I and by the perpendiculars of shortest distance at the limit points. They are called the principal planes. In order to find other properties of the principal surfaces, we imagine that the parametric curves upon the sphere represent these surfaces. If equation (17) be written du (20) iu + = l 0, PRINCIPAL SURFACES it is 397 surfaces v seen that a necessary and sufficient condition that the ruled = const., u = const, be the principal surfaces, is From thes"e it follows that since the coefficients of the two funda mental quadratic forms are not proportional, we must have (21) ^=0, the first /+/ =0. From of these equations and the preceding theorem follows the result: The principal surfaces of a congruence are represented on the sphere by an orthogonal system, and the two principal planes for each line are perpendicular to one another. For this particular parametric system equation (13) reduces to &lt;o ^9X, du ^dX. dv & (22) so that the direction-cosines is X x, JJL^ v l of the the limit point on I corresponding to v 1 = const, perpendicular whose foot have the values l ay dz Hence the angle and those with GO between the is lines with these direction-cosines . (22) given by cos &&gt; = 7 du The values of r 1 and r 2 are now e ri=--r r ^~S ^ 2 a so that with the aid of (23) equation (15) can be put in the form (24) r is =r l cos 2 &&gt; -f r 2 sin co. This Hamilton s equation. We remark that it is independent of the choice of parameters. 398 KECTILINEAR CONGEUENCES 163. Developable surfaces of a congruence. Focal surfaces. In order that a ruled surface be developable, it is necessary and suffi cient that the perpendicular distance between very near generators be of the second or higher order. From (14) it follows that the ruled surfaces of a congruence satisfying the condition (25) e du -}-fdv, f du + g dv are developable. fying this equation are Unlike equation (20), the values of dv/du satis not necessarily real. We have then the theorem : Of all the ruled surfaces of a congruence through a line of it two are developable, but they are not necessarily real. The normals to a real surface afford an example of a congruence with real developables for, the normals along a line of curvature form a developable surface ( 51). Since /and/ are equal in this ; case, equations (20) and (25) are equivalent. And, conversely, they are equivalent only in this case. Hence : When a congruence is normal, and only then, the principal surfaces are developable. When them. a ruled surface is developable its generators are tangent to a curve at the points where the lines of shortest distance is meet Hence each line of a congruence tangent to two curves in space, real or imaginary according to the character of the roots of equation (25). The points of contact are called the focal points for the line. By means e of (25) we find that the values of r for these points are given by du -\-fdv _ f du+g dv If these equations be written in the form (&lt;&!&gt; (p 4- e)du + (&p +f)dv = 0, +/ du 4- (gp +g}dv = 0, ) and (26) if du, dv be eliminated, we have DEVELOPABLE SURFACES If p l 399 and p 2 denote the roots of this equation, it follows that (27) A= ^-// (19) From and (27) it is seen that (28) These results may be interpreted The mid-points of limit points as follows : the two segments bounded respectively by the and by the focal points coincide. its This point is called the middle point of the line and middle surface of the congruence. locus the The distance between the focal points between the limit points. is never greater than that the congruence is normal. They coincide when Equation (24) may 2 be written in the forms cos &) = ri A* T -1 r2 sin 2 &) = r i &) -r r i Hence if a) 1 and &) 2 denote the values of corresponding to the developable surfaces, we have ^, A* From these and the cos 2 first ft) of (28) 2 it follows that sin 2 ft) 1 = sin &) 2, 1 = cos 2 ft) 2, so that (29) cos2&) 1 +cos2ft) 2 =0, and consequently (30) jor w 1 +o) 2 ~ft) 2 =|ww, = (31) ft) 1 400 RECTILINEAR CONGRUENCES where n denotes any integer. If the latter equation be true, the developable surfaces are represented on the sphere by an orthog onal system, as follows from the theorem at the close of 161. But by tem on the sphere is/=/ that , 34 the condition that equation (25) define an orthogonal sys is, the congruence must be normal. Since in this case the principal surfaces are the developables, equa tion (30) as well as (31) is satisfied. Hence equation (30) is the general solution of (29). The planes through I which make the angles o^, = are called the focal planes for the principal plane w are the tangent planes to the &&gt; 2 with the ; line they two developable surfaces through the line. Incidentally we have proved the theorem: that the congruence A necessary and sufficient condition that the two focal planes for is each line of a congruence be perpendicular be normal. And from equation (30) it follows that The focal planes are symmetrically placed with respect to the prin cipal planes in such a way that the angles formed by the two 2iairs of planes have the same bisecting planes. If 6 denote the angle between the focal planes, then ^ sin 6 and (32) cos 2 a) l cos 2 o) 1 cos 2 o&gt; 2 =- l The loci of the focal points of a surfaces. Each line of the congruence are called its focal congruence touches both surfaces, being it. tangent to the edges of regression of the two developables through By reasoning of center ( similar to that employed in the discussion of surfaces : 74) we prove the theorem A congruence faces. regarded as two families of developable sur Each focal surface is touched by the developables of one family may be along their edges of regression and enveloped by those of the other family along the curves conjugate to these edges. The preceding theorem shows that of a line I one is tangent to the focal surface the two focal planes through SL and the other is the ASSOCIATE NORMAL CONGRUENCES 401 osculating plane of the edge of regression on /S\ to which I is tan gent similar results hold for Sz When the congruence is nor mal these planes are perpendicular, and consequently these edges ; . of regression are geodesies on true ( 76), we have: S and Sz l . Since the converse is necessary and sufficient condition that the tangents to a family of curves on a surface form a normal congruence is that the curves be geodesies. A EXAMPLES 1. If JT, Y", Z are the direction-cosines of the normal to a minimal surface at the point (cc, T/, z), the line whose direction-cosines are F, through the point (x, y, 0) generates a normal congruence. 2. X, Z and which passes tact of a Prove that the tangent planes to two confocal quadrics at the points of con common tangent are perpendicular, and consequently that the common tangents to two confocal quadrics form a normal congruence. 3. Find the congruence of common tangents x 2 to the paraboloids + y 2 = 2az, x2 + L L y* =- 2 az, and determine the 4. focal surfaces. line If two ruled surfaces through a lines, their lines of striction orthogonal meet are represented on the sphere by at points equally distant from the middle point. 5. same angle, In order that the focal planes for each line of a congruence meet under the it is necessary and sufficient that the osculating planes of the edges of under regression of the developables meet the tangent planes to the focal surfaces constant angle. 6. A ence be its necessary and sufficient condition that a surface of reference of a congru middle surface is g - (/ + )&lt;^+ e& = 0. / 164. Associate normal congruences. If we put dx dx equations (34) (8) may be replaced by t =c Now I 7 du + y^dv, where c is a constant. equation (9) is equivalent to 402 RECTILINEAR CONGRUENCES may be written In consequence of this condition equation (34) (36) t = c-(u^ orthogonal be taken as parametric curves vl If the where u vl l is a function of u and v thus denned. trajectories of the curves = u^ const, const., it follows from (36) and from equations in u l and (34) that analogous to (33) and From The this result follows the lines of theorem : a normal congruence cut orthogonally the curves on t the surface of reference at whose points is constant. If denotes the angle which a line of the congruence makes with the normal to the surface of reference at the point of inter section, we have sin (37) *= is where the linear element of the surface If S be taken for the surface of reference of a second congruence whose direction-cosines Xv Yv Z l satisfy the conditions u i) where normal and 4&gt;i( ig anv function whatever of u^ has the value this congruence is any function, there is a family of these normal congru ences which we call the associates of the given congruence and of Since 1 is one another. Through any point of the surface of reference there lie passes a line of each congruence, and all of these lines const, through the point. plane normal to the curve u l in the : Hence The two lines of two associate congruences through the same point of the surface of reference lie in a plane normal to the surface. DERIVED CONGRUENCES Combining with equation gruence, (38) (37) a similar 403 one for an associate con we have E* = sin^ : #K) &) =/(W) &lt;*&gt; Hence we have the theorem The congruences make with ratio of the sines of the angles which the lines of two associate is the normal to their t surface of reference constant. con stant along the curves at whose points is When in particular f(u^ in (38) and equation is a constant, the former theorem (38) constitute the laws of reflection and refraction of rays of light, according as the constant is equal to or different from minus one. And so we have the theorem of Malus and Dupin : If a bundle of rays of light forming a normal congruence be reflected or refracted any number of times by the surfaces of successive homo geneous media, the rays continue to constitute a normal congruence. By means of (37) equation (36) can be put in the t form = c l \l E sin 6 du r : From a this result follows the theorem of Beltrami * be If a surface of reference of a normal congruence deformed in such that the directions of the lines of the congruence with respect to the surface be unaltered, the congruence continues to be normal. way 165. Derived congruences. It is evident that the tangents to the curves of any one-parameter family upon a surface S constitute a congruence. If these curves be taken for the parametric lines v and their conjugates for u = const., the developables in one family have the curves v = const, for edges of regression, and = const., u the developables of the other family envelop S along the curves const. may take S for the surface of reference. If Sl be We the other focal surface, the lines of the congruence are tangent to the curves u = const, on Sr The tangents to the curves v = const, S1 form a second congruence of which Sl is one focal surface, and the second surface 2 is uniquely determined. Moreover, the on * Giornale di matematiche, Vol. II (1864), p. 281. 404 lines of the . RECTILINEAR CONGRUENCES second congruence are tangent to the curves u = const. on Sz In turn we may construct a third congruence of tangents This process may be continued const, on Sz to the curves v indefinitely unless one of these focal surfaces reduces to a curve, . or is infinitely distant. In like manner the curves we get u = const, on on S_ l form a congruence by drawing tangents to S, which is one focal surface, and the other, S_ v is completely determined. The tangents to the curves u const, still another, and so on. In this way we obtain a suite of surfaces terminated only when a surface reduces to a curve, or points are infinitely distant. Upon each of these surfaces the parametric curves form a conjugate system. The congruences thus which is its obtained have been called derived congruences by Darboux.* It is clear that the problem of finding all the derived congruences of a given one reduces to the integration of the equation of its developables (25); for, when its the developables are focal surfaces. known we have the conjugate system on In order to derive the analytical expressions for these results, we recall ( 80) that the coordinates x, y, z of of an equation of the form S are solutions (39) du dv b are du cv v. where a and nates of l determinate functions of u and 2t, If the coordi S be denoted by x^ y^ dx they are given by *-x+\-* fc-jr By + x,-. v dz. + Xi S , , dz . where \^J~E measures the distance between the focal points. But as the lines of the congruence are tangent to the curves u = const. on Sv we must have dx. (40) 1 dv = Pl dx M 1 dy. -^I du dv * Vol. = Pl dy u 1 * i =u dz l , du dv du II, pp. 16-22. DERIVED CONGRUENCES where is /-^ 405 is a determinate function of is u and v. When the above value for x l substituted in the first of these equations, the result reducible, by means L^LI of (39), to _ flX dv n\te + (1 __ fog l/ to dv V du = 0. Since the same equation is true for y and theses must be zero, that is, 1 a1 z, the quantities in paren a Hence the surface S , l is defined by 1 ex \dy 1 dz , and equations /42\ i (40) become __ __ ^y Vav b b]du dv \dv b b/du dv \dv b is bjdu defined by Proceeding in a similar manner, we find that _i the equations /4 ox - 1 a* - 1 to . . 1 V*; - * and that and similar expressions in y_^ and 2_i. From (41) and (43) it is seen that the surface Sl or S.i a and 5 are both infinity, according as b or a is zero. When is at zero, S is a surface of translation generators of a surface of each of which the other focal surface ( 81). Hence the tangents to the translation form two congruences for is at infinity. In order that S^ be a curve, x^ y^ z l must be functions of u alone. From (42) it follows that the condition for this is d1 ~dv _a ~6* is 5 In like manner the condition that _i be a curve l=i du a a 406 RECTILINEAR CONGRUENCES &, The functions h and denned by , if h __da du * dv are called the invariants of the differential equation (39). the above results may be stated : Hence A _i be necessary and sufficient condition that the focal surface Sl or a curve is that the invariant k or h respectively of the point equation of S be zero. 166. Fundamental equations of condition. have seen ( 160) that with every congruence there are associated two quadratic dif ferential forms. Now we shall investigate under what conditions two quadratic forms determine a congruence. assume that we We We a corresponding congruence. The tangents to the parametric curves on the surface of reference at a point are determined by the angles which they make with the is have two such forms and that there tangents to the parametric curves of the spherical representation of the congruence at the corresponding point, and with the normal to the unit sphere. Hence we have the relations ,__._+ I Zti 7)ti (44) and similar equations of in y and 2, where #, /3, 7; a r (S v y l r are functions respec- u and v. If we multiply also these equations by * &gt; tively, and add; dX dY dZ by dv dv dv du Y, dU 3U and by A", Z\ we obtain from which we derive a /\ = e& f \ j c/ ^2 * (O& ~~ cy j p=- & (n iy ~~~ c-c/ * 7 c/ =^.A 01^ &gt; (45) foo/ Ctf CA M* fl (Q^/ CV^ *&gt;2 c/ 1 yi^^rr 1 FUNDAMENTAL EQUATIONS In order that equations (44) be consistent, we must have 407 du \dv 2v \du is which, in consequence of equations (V, 22), reducible to the form ]t?X+S -X du dv d where J?, S, be satisfied T are determinate functions. Since by Y and Z also, we must have R a, y3, this equation 0, must S= 0, T= 0. When the values of a^ fi v from (45), are substituted in these equations, we have (47) Conversely, is +1, it may when we have a quadratic form whose curvature be taken as the linear element of the spherical rep , resentation of a congruence, which is determined by any set of functions e, f, / #, 7, 7^ satisfying equations (4T). For, when these equations are satisfied, so also is (46), and consequently the coordinates of the surface of reference are given by the quadratures (44). Incidentally we remark that when the congruence is normal, and the surface of reference is one of the orthogonal surfaces, the last of equations (47) is satisfied identically, and the first two reduce to the Codazzi equations (V, 27). apply these results to the determination of the congruences We with an assigned spherical representation of their principal surfaces, and those with a given representation of their developables. 167. Spherical representation of principal surfaces and of devel opables. A necessary and sufficient condition that the principal surfaces of a congruence cut the surface of reference in the para metric lines is given by (21). 408 If RECTILINEAR CONGRUENCES we require that the surface of reference be the middle surface of the congruence, and if r denote half the distance between the limit points, (48) we have, from (15), e r&lt;o, g = r& first When these values are substituted in (47), the two become 12, (49) f? a/ / \ 7 = ^ 1 d i , (r &lt; and the ,50) 2 last is reducible to * av i al g^ g? dv \ dwy cu aiog^gTjg iog^,. du dv dudv + YL a F W d { f l\4-2f * + sLNtfsfe;J 4N?s\^Jr^ d \ d i f Moreover, equations (44) become ao: ax / ax aa: , / ax ax in where 7 and y l are given by (49) ; and similar equations y and z. reduces, therefore, to the determination of pairs of functions r and / which satisfy (50). Evidently either of these Our problem functions may be chosen arbitrarily and the other is found by the solution of a partial differential equation of the second order. Hence any orthogonal system on the unit sphere serves for the representation of the principal surfaces of a family of congruences, whose equations involve three arbitrary functions. In order that the parametric curves on the sphere represent the developables of a congruence, it is necessary and sufficient that from (25). If the surface of reference be the middle sur and p denotes half the distance between the focal points, it face, follows from (15) that e as is seen p " c I, * Cf. Bianchi, Vol. p. 314. DEVELOPABLES PARAMETRIC Combining these equations with the above, we have (52) 409 e=-p& f = -f = p & for ff = p& (47) When these values are substituted in the first and the resulting equations are solved two of equations 7 and 7^ we find and the last of equations (47) reduces to d ri2V a n2 = of this equation determines a congruence with the given representation of its developables,* and the middle surface is Each solution given by the quadratures (54) and similar expressions in y and 2. When the values (52) are substituted in (18) the latter becomes Consequently equation (32) reduces to a sin ^ 2P = -^ = Referring to equation (III, 16), we have: the focal planes of a congruence is equal to the the lines on the sphere representing the corresponding angle between The angle between developables. This result 168. is obtained readily from geometrical considerations. for the focal surfaces. Fundamental quantities We shall make use of these results in deriving the expressions for the funda , mental quantities of the focal surfaces Sl and 2 which are defined by * This result pp. 342-344. is due to Guichard, Annales de I Ecole Normale, Ser. 3, Vol. VI (1889), 410 RECTILINEAR CONGRUENCES these and (54) From we get The from coefficients of the linear elements of ^ and 2, as derived these formulas, are (56) and (57) . The tions direction-cosines of the normals to JT2 , X^ Yv Z^ r^ 2, 2 respectively are ^ and S2 denoted by found from the above equa and (V, 31) to have the values v 1 . Si) ZJ- ._ = 3V /~^ ^V ^/^s /./V^ ^ _ /}?&gt; 2/^, i\ \ d(u, v) ^ and similar expressions for Y and Z If these equations be differ of (V, 22), entiated, and the resulting equations be reduced by means they can be put in the form . t { K == "a^" \\ fi2Vax ^^ "VL^^lJ"^ x I a*t_^/22Vwr \ dv-^lif dv Til2/ FOCAL SURFACES From these expressions and (55) 411 we obtain ^ du (58) ~ du V^ \du ^2 o, D[ = Y dv du y du cv ^ ^1 Mi = _ ^ ?i Mi = ~2 ^r = "^7 * A"= and the foregoing formulas we derive the following expressions for the total curvature of Sl and of Sz From : 22V {1J (60) EXAMPLES 1. If upon a surface of reference S of a normal congruence the curves orthog onal to the lines of the congruence are defined by 0(u, u) = const., and 6 denotes the angle between a line of the congruence and the normal to the surface at the 2 = AiF(0) where the differential parameter point of meeting, then sin with respect to the linear element of S. Show that 6 is constant along a line is formed = const. only 2. when the latter is a geodesic parallel. a surface, namely , , When in the point equation of c2 , du cv n - + a c0 + 6 30 = 0, du cv a or 3. 6 is zero, the coordinates of the surface can be found by quadratures. tetrahedral surface (Ex. face Si or 5-i is a curve. 4. Find the derived congruences of the tangents to the parametric curves on a the sur 2, p. 267), and determine under what conditions Find the equation of the type given by (41). (39) which admits as solutions the quantities *i, yi, zi 5. When a congruence consists of the tangents to the lines of curvature in one system on a surface, the focal distances are equal to the radii of geodesic curvature of the lines of curvature in the other system. 412 6. RECTILINEAR CONGRUENCES Let S be a surface referred to its lines of curvature, let i and s 2 denote the = const, and u = const, respectively, ri and r2 their radii of curvature, and RI and JR 2 their radii of geodesic curvature for the second ; arcs of the curves v first focal sheet Si of the congruence of tangents to the curves v = const, the linear element is reducible to 2 hence the curves 7. Si = const, are geodesies. is Show that 2t of Ex. 6 r\ so developable when n =/(si), and determine the most general form of 8. that 2i shall be developable. Determine the condition which p must satisfy in order that the asymptotic on either focal surface of a congruence shall correspond to a conjugate system on the other, and show that in this case lines where 9. denotes the angle between the focal planes. In order that the focal surfaces degenerate into curves, it is necessary and sufficient that the spherical representation satisfy the conditions 12 { \ du 10. ={ cv \ 12 \ ~ = ( \ 12 1 \ 1 ) 2 } Show that the surfaces orthogonal to a normal congruence of the type of Ex. 9 are cyclides of Dupin. 11. A necessary and sufficient condition that the second sheet of the congruence developable is of tangents to a family of curves on a surface S be that the curves be plane. 169. Isotropic congruences. isotropic congruence is one whose focal surfaces are developables with minimal edges of regression. An In 31 we saw that H= is a necessary and sufficient condition that a surface be of this kind. Referring to (56) and (57), we see that we must have From (54) it is seen that if p were zero the middle surface would be a point, and from (55) that if the expressions in parentheses were zero the surfaces Sl and 2 would be curves. Consequently (61) &lt; = g= 0. Conversely, if this condition be satisfied, S and l S.2 are isotropic developables. isotropic congruence opables are represented on the sphere by minimal lines. Hence an is one whose devel ISOTROPIC CONGRUENCES In consequence of (61) 413 we have, from (52), and since (62) f+f also is is zero, it follows that dxdX+ dydY+ dzdZ= 0. zero, so that all the lines of striction lie Therefore r on the Since (61) is a consequence of (62), we have the following theorem of Ribaucour,* which is sometimes taken for the definition of isotropic congruences middle surface. : All the lines of striction of an isotropic congruence dle surface ; and, conversely, lie on the mid when is all the lines ; middle surface, the congruence face corresponds linear elements. to the isotropic of striction lie on the moreover, the middle sur spherical representation with orthogonality of Ribaucour has established also the following theorem f : TJie middle envelope of an isotropic congruence is a minimal surface. Since the minimal lines on the sphere are parametric, in order to prove this theorem it is only necessary to show that on the envelope of the middle planes, denotes the corresponding lines form a conjugate system. If the distance of the middle plane from the origin, the condition middle envelope, that is, the W necessary and sufficient that the parametric lines be conjugate is that satisfy the equation W (63) r + &lt;^0 = 0. By definition and with the aid of (V, 22) we find ft du dv cu dv o2 Since equation (53) reduces to satisfies - + /&gt;&lt;^=0, the function W&gt; (63). * Etude des Elasso ides ou Surfaces a Courbure Moyenne Nulle, Memoires Couronnts t L.c., p. 31. par r Academic de Belgique, Vol. XLIV (1881), p. 63. 414 RECTILINEAR CONGRUENCES Guichard* proposed and solved 170. Congruences of Guichard. the problem : To determine the congruences whose focal surfaces are met by developables in the lines of curvature. the With Bianchi we call them congruences of Gruichard. remark that a necessary and sufficient condition that a con gruence be of this kind is that Fl and F2 of 168 be zero. From (56) and (57) it is seen that this is equivalent to We Comparing this result with 78, we have the theorem: of a con gruence meet the focal surfaces in their lines of curvature is that the congruence be represented on the sphere by curves representing also necessary the asymptotic lines on a pseudospherical surface. A and sufficient condition that the developables In this case the parameters can be so chosen thatf &lt;F=^=1, c? = COSQ), where co is a solution of = sin dudv In this case equation (53) (65) is ft). - = p cos &lt;0. In particular, this equation is satisfied by X, F, in (54), we have replace p by X Z (V, 22). If we consequently, for the congruence determined by this value of the middle surface is a plane. /&gt;, From (55) it follows that the lines of the to the lines of curvature v const, on *Sy congruence are tangent = Consequently they are (64). *L.c., p. 346. f This is the only real solution of CONGRUENCES OF GUICHARD call it 415 parallel to the normals to one of the sheets of the evolute of (cf. Sl 74) ; 2 . X Hence the conjugate system on 2 t is corre represented on the sphere by the same lines as the developables of the congruence. Referring to (VI, 38), we see that condition (64) is equivalent to sponding to the lines of curvature on S^ where the Christoffel symbols are formed with respect element of to the linear are the conditions that the parametric Surfaces with a conjugate curves (cf. 85). X of geodesies were studied by Voss, * and on this account system are called surfaces of Voss. Since the converse of the above results true, 2 r But these on 2 be geodesies is we have the following theorem of Guichard : and sufficient condition that the tangents to the lines curvature in one family of a surface form a congruence of of Guichard is that one sheet of the evolute of the surface be a sur face of Voss, and that the tangents constituting the congruence be those which are parallel to the normals to the latter. If A necessary W to the surface of (cf. denotes the distance from the origin to the tangent plane is a solution of equation (65) Voss 2 X then l Hence W^ + Kp is a solution of this equation, provided K 84). l , W be a constant. since the tangent plane to 2 X passes through the corresponding point of Sv the above result shows that a plane normal to the lines of the congruence, and which divides in con But stant ratio the segment between the focal points, envelopes a sur face of Voss. In particular, we have the corollary : The middle envelope of a congruence ofGruichardis a surface of Voss. 171. Pseudospherical congruences. The lines joining correspond ing points on a pseudospherical surface S and on one of its Backhand transforms S1 (cf. 120) constitute an interesting congruence. We between corresponding points is constant, and that the tangent planes to the two surfaces at these points meet under constant angle. From (32) it follows that the distance recall that the distance between the limit points also is constant. (1888), pp. 95-102. *Miinchener JSerichte, Vol. XVIII 416 Conversely, KECTILINEAB CONGRUENCES is gruence the angle between the focal planes of a con constant, and consequently also the angle 6 between when the parametric lines on the sphere representing the developables, we have, from (V, 4), 111112 i Furthermore, if the distance between the focal points 2 have p = a, and by (60) gjn Q is constant, we K =K = ^ * "4^" Hence the two Congruences called focal surfaces have the same constant curvature. first studied by Bianchi.* He them pseudospherical congruences. In order that the two focal surfaces of the congruence be Back- of this kind were lund transforms of one another, curvature correspond. the equation of these lines is it is necessary that their lines of It is readily found that for both surfaces reducible by means of (66) to 12V f 12V { is\2j dw ~[^ + f\is + H2V 2 n n^v ! 2 2 ) J surface Moreover, the differential equation of the asymptotic lines on each 2 is dv? 0. Hence we have the theorems: ^/di) On the focal surfaces of a pseudospherical congruence the lines of curvature correspond, and likewise the asymptotic lines. The focal surfaces of a pseudospherical congruence are Backlund transforms of one another. EXAMPLES any whatsoever, and likewise the surface of reference, a condition necessary and sufficient that a congruence be 1. When is the parameters of a congruence are isotropic ~~ f + f = e g 2^ ^ necessary and sufficient condition that a congruence be isotropic is that two points on each line at an equal constant distance from the middle surface shall describe applicable surfaces. 2. A the locus of 3. Show that equation (65) admits is and 3u dv as solutions. Prove that in each case one of the focal surfaces a sphere. pp. 161-172; also Lezioni, Vol. I, *Annali, Ser. 2, Vol. XV (1887), pp. 323, 324. JF-CONGRUENCES 4. is 41T Determine all the congruences of Guichard for which one of the focal surfaces a sphere. 5. When a surface is referred to its lines of curvature, a necessary and suffi cient condition that the tangents to the curves v of Guichard is a/1 = const, shall form a congruence 3u\^ Determine the surfaces which are such that the tangents curvature in each system form a congruence of Guichard. 6. to the lines of 172. TF-congruences. We have just seen that the asymptotic lines on the focal surfaces of a pseudospherical congruence correspond the same is true in the case of the congruences of normals to a ; 124). For this reason this property are called W-congruences. erties of these congruences. JF-surface (cf. all congruences possessing shall derive other We prop The condition that asymptotic lines correspond, namely takes the following form in consequence of (58) and (59): 22V Hence from (60) it follows that a necessary for a JF-congruence is and sufficient condition In order to obtain an idea of the analytical problem involved in the determination of TF-congruences, we suppose that we have two surfaces S referred to their asymptotic lines, and inquire under , what conditions the lines joining corresponding points on the surfaces are tangent to them. We assume that the coordinates of the surfaces are defined* by means of the Lelieuvre formulas (cf. 79), thus: dx dx du du (68) dx_ du dx dv dv du du du ~dv dv *Cf. Guichard, Comptes Rendus, Vol. CX (1890), pp. 126-127. 418 RECTILINEAR CONGRUENCES y, z, and similar equations in vv i&gt; y, and z. The functions v v 2 , v s respectively are solutions of equations of the form (69) dudv and they are such that (70) v? + l + vl = a, vl + v + vl = a, wliere a and a are defined by J5T (71) =, ,, a- K = -~. a 2 Since v^ v^ v s and v r vz v s are proportional to the directioncosines of the normals to S and S, the condition that the lines joining corresponding points be tangent to the surfaces S and S is v^x -z)+ v z (y -y}+ v^(z - z) = v z z 0. Hence x x y where value, w we denotes a factor of proportionality. In order to find notice that from these equations follow the relations 2 its (2 /&gt;) = ^(x - x)*= 2 7?i 2(^ =w 3 a 2) iy&gt; where ^ denotes the angle between the focal planes. If this value of 2p and the values of and from (71) be substituted in (67), = 1. We take w=l, thus fixing the signs of it is found that m* K K i&gt; i/j, 2, i&gt; 3, and the above equations become x (72) x = V& vfa y y = v^t *&, z - * = W- v v r 2 If the first of these equations be differentiated with respect to w, the result is reducible by (68) to JF-CONGRUENCES 419 Proceeding in like manner with the others, and also differentiating with respect to v, we are brought to /- 7 /- (73) . = 1,2,8) where Z and & are factors of proportionality to be determined. If the first of these equations and in the reduction we make be differentiated with respect to v, use of the second and of (69), we find In like manner, if tiated with respect to u, the second of the above equations be differen we obtain Since these equations are true for i=l, parentheses must be zero. , This gives 2, 3, the quantities in x In accordance with the last =-^ + cl 3k dv du we put = and the others become a , IQOT * i- Hence equations (69) may be written Bdudv i dudv l dufo\0.if \ from which it follows that l is a solution of the first of equa tions (69) and l/0 l of the second. Moreover, equations (73) may now be written in the form 0, v&lt; du dv 420 RECTILINEAR CONGRUENCES if Q l be a known solution of the first of equations (69), we obtain by quadratures three functions v which lead by the quadra tures (68) to a surface S. The latter is referred to its asymptotic lines and the joins of corresponding points on S and are tangent Hence f, to the latter. And so we have : If a surface S be referred to its asymptotic lines, and the equations of the surface be in the Lelieuvre form, each solution of the corre sponding equation ffQ = \0 S and S are dudv determines a surface S, found by quadratures, such that the focal surfaces of a W-congruence. Comparing (74) with (XI, 13), we i see that if we put ^1=^1. yi= i=^8 , the locus of the point (x^ y^ zj corresponds to S with orthogo to the nality of linear elements. Hence v v v 2 v s are proportional of an infinitesimal deformation direction-cosines of the generatrices of , so that we have : focal surface of a W-congruence admits of an infinitesimal deformation whose generatrices are parallel to the normals to the Each other focal surface. Since the steps in the preceding argument are reversible, have the theorem : we trices of The tangents to a surface which are perpendicular to the genera an infinitesimal deformation of the latter constitute a Wto the congruence of the most general kind ; and the normals surface are parallel to the other generatrices of the deformation. In his study of surfaces corre with orthogonality of linear elements Ribaucour consid sponding ered the congruence formed by the lines through points on one surface parallel to the normals to a surface corresponding with the 173. Congruences of Ribaucour. calls such a congruence a con and the second surface the director surface. gruence of Ribaucour, In order to ascertain the properties of such a congruence, we former in this manner. Bianchi * recall the results of 153. Let S be taken for the surface of l *Vol. II, p. 17. CONGRUENCES OF RIBAUCOUR reference, 421 If the latter and draw its lines parallel to the lines, it normals to S. be referred to asymptotic follows from (XI, 6) that civ du HK du dv 9 =y ~^ dx l dX Since these values satisfy the conditions the ruled surfaces since also u = const., v = const, equal to zero, p^+ p 2 But the parametric curves on 8 form a conjugate system when the asymptotic lines on S are parametric. Hence we l is S is are the developables. And the middle surface of the t congruence. have the theorem : The developable surfaces of a congruence of Ribaucour cut the middle surface in a conjugate system. Guichard the first * ences of Ribaucour. proved that this property is characteristic of congru In order to obtain this result, we differentiate make use of equations (54) with respect to v, and in the reduction of the fact that and p satisfy equations (V, 22 ) and X (53) respectively. This gives /isyy*. dv is log p \ fi2\as. \\ }/ du iu and similar equations in y and z it follows that a and sufficient condition that the parametric curves form necessary this v From v a conjugate system is ^ f!2V 1 d T12V J du\ S to\ 2 When this condition is satisfied by a system of curves on the sphere, they represent the asymptotic lines on a unique surface S, whose coordinates are given by the quadratures (VI, 14) *Annales L Ecole Nonnale, Ser. 3, Vol. VI (1889), pp. 344, 345. 422 RECTILINEAR CONGRUENCES for and similar expressions with (54), y and z. Combining these equations dx ~ we find that _ dx _ = du i ^\ 3x l dx du dv Z ** ^ dx l = ^\ dx v dx = dv du ^~dv~dv of linear elements, Hence S and S correspond with orthogonality and the normals gruence. Hence to the : former are parallel to the lines of the con A necessary and sufficient condition that the developables of a congruence cut the middle surface in a conjugate system is that their representation be that also of the asymptotic lines of a surface, in which case the latter and the middle surface correspond with orthogonality of linear elements. EXAMPLES 1. When metric curves are asymptotic lines. is the coordinates of the unit sphere are in the form (III, 35), the para Find the IF-congruences for which the sphere one of the focal sheets. 2. Let vi =fi(u) + 0i (w), where /; and - &lt;/&gt; t and and i = 1, 2, 3, be three solutions of the first are functions of u and u respectively, of equations (09), in which case X = 0, dle surface be unity. Show that for the corresponding ^-congruence the mid a surface of translation with the generatrices u = const., v = const., that the functions / and 0,- are proportional to the direction-cosines of the binorlet 61 in (74) is - t inals to these generatrices, and that the intersections of the osculating planes of these generatrices are the lines of the congruence. 3. Show that isotropic congruences and congruences of Guichard are congru ences of Ribaucour. 4. A mal is necessary and sufficient condition that a congruence of Ribaucour be nor that the spherical representation of its developables be isothermic. to quadrics 5. The normals and to the cyclidesof Dupin constitute congruences is of Ribaucour. 6. When Show is the middle surface of a congruence is plane, the congruence of the Ribaucour type. 7. helicoid, 8. that the congruence of Ribaucour, whose director surface is a skew a normal congruence, and that the normal surfaces are molding surfaces. Show be normal is that a necessary and sufficient condition that a congruence of Ribaucour that the director surface be minimal. GENERAL EXAMPLES Through each line of a congruence there pass two ruled surfaces of the con gruence whose lines of striction lie on the middle surface their equation is 1 . ; edu* + (f + f )dudv they are called the mean ruled surfaces of the congruence. GENERAL EXAMPLES 2. 423 sphere by an orthogonal system of real ruled surfaces of a congruence are represented on the lines, and that their central planes ( 105) bisect the angles between the focal planes. Let u = const. v = const, be the mean 167. ruled surfaces and develop a theory analogous to that in that the , Show mean 3. If the two focal surfaces of a congruence intersect, the intersection is the envelope of the edges of regression of the two families of developable surfaces of the congruence. 4. If a congruence consists of the lines joining points on two twisted curves, the focal planes for a line of the congruence are determined by the line and the tangent to each curve at the point where the curve is met by the line. 5. In order that the lines which join the centers of geodesic curvature of the curves of an orthogonal system on a surface shall form a normal congruence, it is necessary and sufficient that the corresponding radii of geodesic curvature be func tions of one another, or that the curves in one family have constant geodesic curvature. 6. Let S be a surface whose lines of curvature in one system are circles; let C denote the vertex of the cone circumscribing S along a circle, and L the corre sponding generator of the envelope of the planes of the circles a necessary and sufficient condition that the lines through the points C and the corresponding lines L ; form a normal congruence is that the distance from C to the points of the correspond ing circle shall be the same for every circle if this distance be denoted by a, the ; radius of the sphere is given by _ p /2 / jn \ a 2\ to the arc of the curve of where the accent indicates differentiation with respect centers of the spheres. 7. Let -S be a surface referred to its lines of curvature, Ci and C2 the centers of principal normal curvature at a point, GI and G 2 the centers of geodesic curva ture of the lines of curvature at this point; a necessary and sufficient condition of Pg u that the line joining C2 and G\ form a normal congruence or that one of these radii be a constant. , is that p 2 be a function 8. Let S be a surface of the kind defined in Ex. 6; the cone formed by the normals to the surface at points of a circle A is tangent to the second sheet of the evolute of the vertex -S in a conic T (cf. 132). Show that the lines through points of T and C of the cone which circumscribes 8 along the plane of F. A generate a normal con gruence, and that 9. C lies in Given an isothermal orthogonal system on the sphere for which the linear is Z element _ * + 2 cto ) ; on each tangent to a curve v = const, lay off the segment of length X measured from the point of contact, and through the extremity of the segment draw a line parallel to the radius of the sphere at the point of contact . Show that this congruence is iso tropic. its a congruence is iso tropic and 10. 35), equation (53) reduces to (III, When direction-cosines are of the form 8uBv (1-f-ww) Show that the general integral p is = 2 O0(v) - vf(u)](l + uv) v respectively. where / and are arbitrary functions of u of the middle surface. and Find the equations 424 11. RECTILINEAR CONGRUENCES Show that the intersections of the planes (1 - M 2 )x v 2 )z -f i (1 + w2 ) y (1- - i(l + v z )y + + 2 uz + 4/(w) = 0, 2vz +40(u)= ; constitute an isotropic congruence, for which these are the focal planes that the locus of the mid-points of the lines joining points on the edges of regression of the developables enveloped by these planes is the minimal surface which is the middle envelope of the congruence, by rinding the coordinates of the point in which the tangent plane to this surface meets the intersection of the above planes. 12. Show that the middle surface of an isotropic congruence is the most general surface which corresponds to a sphere with orthogonality of linear elements, and that the corresponding associate surface in the infinitesimal deformation of the sphere 13. is the minimal surface adjoint to the middle envelope. to the Find the surface associate middle surface of an isotropic congruence when is the surface corresponding to the latter with orthogonality of linear elements a sphere, and show that it is the polar reciprocal, "with respect to the imaginary 2 2 2 sphere x -f y -f z of the congruence. +1= 0, of the minimal surface adjoint to the middle envelope 14. The lines of intersection of the osculating planes of the generatrices of a surface of translation constitute a IT-congruence of which the given surface is the middle surface if the generatrices be curves of constant torsion, equal but of ; opposite sign, the congruence is normal to a TF-surface of the type (VIII, 72). 15. If the points of a surface S be projected orthogonally upon any plane A, and if, after the latter has been rotated about any line normal to it through a parallel to the corresponding nor right angle, lines be drawn through points of mals to -S, these lines form a congruence of Ribaucour. 16. A necessary and sufficient condition that the tangents to the curves v const. on a surface, whose point equation is (VI, 26), shall form a congruence of Ribaucour is A aa_S6 du dv dudv 17. Show that the tangents to each system of parametric* curves on a surface form congruences of Ribaucour when the point equation is where Ui and V\ are functions of u and v respectively, and the accents indicate differentiation. 18. Show that if the parametric curves on a surface S form a conjugate system, and the tangents to the curves of each family form a congruence of Ribaucour, the same is true of the surfaces Si and S_i, which together with S constitute the focal surfaces of the two congruences. 19. Show that the parameter of distribution is p of the ruled surface of a con gruence, determined by a value of dv/du, given by -f P= 1 e du + /du, f du -f g dv GENERAL EXAMPLES 20. 425 Show by that the mean ruled surfaces (cf. Ex. 1) of a congruence are char the property that for these surfaces the parameter of distribution has the maximum and minimum values. acterized 21. If S and SQ are two associate surfaces, and through each point of one a line be drawn parallel to the corresponding radius vector of the other, the developables of the congruence thus formed correspond to the common conjugate system of S and SQ, 22. In order that two surfaces S and SQ corresponding with parallelism of tangent planes be associate surfaces, it is necessary and sufficient that for the and MQ of these sur congruence formed by the joins of corresponding points faces the developables cut S and SQ in their common conjugate system, and that M the focal points M and MQ form a harmonic range. 23. In order that a surface S be iso thermic, it is necessary and sufficient that there exist a congruence of Ribaucour of which S is the middle surface, such that the developables cut S in its lines of curvature. CHAPTER XIII CYCLIC SYSTEMS 174. General equations of cyclic systems. The term congruence not restricted to two-parameter systems of straight lines, but is * applied to two-parameter systems of any kind of curves. Darboux is has made a study of these general congruences and Ribaucourf has considered congruences of plane curves. Of particular interest is the case where these curves are circles. Ribaucour has given the name cyclic systems to parameter family of orthogonal surfaces. to a study of cyclic systems. congruences of circles which admit of a oneThis chapter is devoted begin with the general case where the planes of the circles associate with the latter envelop a nondevelopable surface S. We We a moving trihedral ( 68), and for the present assume that the parametric curves on the surface are any whatever. As the circles lie in the tangent planes to S, the coordinates of a point on one of them with respect to the corresponding trihedral are of the form (1) a a, b + Rcos0, b+Rsm0, 0, where are the coordinates of the center, latter to a R the radius, and the angle which the given point makes with the moving In 69 we found the following expressions for the projections of a displacement of a point with respect to the moving axes t : (") (dx+%du + ^dv + (qdu + q^v) z du 4- rj^dv + (rdu + r dv] x \ dy + 77 v + r dv) y, (pdu + p^dv) z, (rdu \dz * Vol. +(p du +p 1 dv) y (qdu+ q v dv) x, Math. II, pp. 1-10; also Eisenhart, Congruences of Curves, Transactions of the Amer. Soc., Vol. IV (1903), pp. 470-488. t Memoire sur la theorie generale des surfaces courbes, Journal des Mathtmatiques, Ser. 4, Vol. VII (1891), 117 et. seq. 426 GENERAL EQUATIONS where the translations f f 1? , 427 p,q,r-&gt; T;, ^ and the rotations p^ qv , r^ satisfy the conditions dp_di_ (3) __ d|__Mi = dr dr. ^Wl (1) PH Ph-M^flin (2) When the values are substituted the latter are reducible to J du + J^v cos QdR (dd + rdu+ r^dv) R sin ^, # ^w + ^jrfu + sin 6dR + (dd+rdu + r^dv] R cos 6, (y du + q^dv) (a+R cos (p du -f ^^v) (b + R sin 0) -|- where we have put, for the sake of brevity, ^ du (5) The conditions that du \dv] are reducible, by (\ = dv (^ means \du/ du \dv \du of (3), to (6) _ dv du The point (7) direction-cosines of the tangent to the given circle at the (1) are sin0, cos0, 0. Hence the condition that the be orthogonal to the circle multiplied respectively by is locus of the point, as u and v vary, that the sum of the expressions (4) the quantities (7) be zero. This gives 428 CYCLIC SYSTEMS In order that the system of circles be normal to a family of sur faces this equation must admit of a solution involving a parameter. Since it is of the form (9) the condition that such an integral exist is that the equation be satisfied identically. * For equation (8) this condition is reducible to In order that this equation be satisfied identically, the expressions in the brackets must be zero. If they are not zero, it is possible that the two solutions of this equation will satisfy (8), and thus determine two surfaces orthogonal to the congruence of Hence we have the theorem of Ribaucour: circles. If the circles of a congruence are normal to more than two surfaces, they form a cyclic system. The equations consequently of condition that the system be cyclic are dR . dR . The total curvature of S is given by (cf. 70) * Equations, Murray, Differential Equations, p. 257. London, 1888. p. 137. New York, 1897; also Forsyth, Differential THEOREMS OF RIBAUCOUR 429 From this and (5) it is seen that equations (12) involve only functions relating to the linear element of S and to the circle. Hence we have the theorem of Ribaucour: If the envelope of the planes of the circles of a cyclic system be deformed in any manner without disturbing continues to the size or position of the circles relative to the point of contact, the congruence of circles form a if cyclic system. Furthermore, we put t = tan Q z* &gt; equation (8) assumes the Riccati form, dt + (af +a t + 2 a 3 ) du + (b/ + bjt + b3 ) dv = 0, : where the # s and 5 s are functions of u and v. Recalling a funda mental property of such equations ( 14), we have Any four orthogonal surfaces of a cyclic system meet the circles in is constant. four points whose cross-ratio Since by hypothesis be replaced by S is nondevelopable, equations (12) may du (13) ^ dv AB, - - trf) JBT = 0. By (5) the first two of these equations are reducible to du (14) a The condition (15) of integrability of these equations is ^{ + g,-^f -g, -r(f^-J )-r (,-^. cu du cv dv 1 l 1 1 Instead of considering this equation, by the equation 2 2 we 2 , introduce a function &lt;j&gt; (16) 24&gt;=,K -a -& 430 CYCLIC SYSTEMS must &lt;/&gt; and determine the condition which a and b the satisfy. We is take for expressions obtained by solving (14); that (17) Now the equation (15) vanishes identically, and the only other condition to be satisfied is the last of (13); this, by the substi tution of these values of a, 6, R, becomes a partial differential equation in (18) &lt; of the form du -r ^_r du dv 7, _(__r_] -\-J^ \dudv / +L ducv &lt;&gt;, + M ~ + N= 0, du where X, JHf, JV denote functions of f , rt, and their deriva tives of the first order. Conversely, each solution of this equation lie gives a cyclic system whose circles in the tangent planes to S. EXAMPLES 1. Let S be a surface of revolution defined by is (III, 99), and let Tbe the trihedral whose x-axis function \f/ (u) tangent to the curve v = const. Determine the condition which the must satisfy in order that the quantities a, b in (1) may have the values a = _*w_. 6 = l, also the expression for R. necessary and sufficient condition that all the circles of a cyclic system whose planes envelop a nondevelopable surface shall have the same radius, is that 2. and determine A the planes of the circles touch their envelope that S be pseudospherical. 3. S at the centers of the circles, and Let S be a surface referred to an orthogonal system of lines, and let T be With reference to the is tangent to the curve v = const. trihedral the equations of a curve in the tangent plane are of the form x = p cos 0, z = 0, y = p sin 0, the trihedral whose z-axis where in general p is a function of 0, w, and is v. Show that the condition that there 0, be a surface orthogonal to these curves that there exist a relation between u, and v which satisfies the equation U sin When this condition is satisfied stant, there is -f prji cos 30 by a function which involves an arbitrary con an infinity of normal surfaces. In this case the curves are said to by the planes in form a normal congruence. 4. When the surface enveloped is of the curves of a normal con gruence of plane curves deformed such a way that the curves remain invari ably fixed to the surface, the congruence continues to be normal. CYCLIC CONGRUENCES 175. Cyclic congruences. 431 tem constitute a rectilinear of the circles of a cyclic sys * congruence which Bianchi has called The axes a cyclic congruence. In order to derive the properties of this con gruence and further results concerning cyclic systems, we assume that the parametric curves on S correspond to the developables of the congruence. The coordinates of the focal points of a line of the congruence with reference to the corresponding trihedral are of the form a, ft, p^ a, ft, /&gt; 2 . On we coordinates of the focal the hypothesis that the former are the point for the developable v = const, (2), through the cu line, have, from rft -+ ? +?/i = 0, cu - + T)pp +ra = l Q. of similar equations. Proceeding in like manner with the other point, we obtain a pair All of these equations may be written in the abbreviated form (19) A+ qPl =0, Ji- PPl =Q, ^+ ?lft =0, 1 J?,-^p2 =0, in consequence of (5). last of equations (13), (20) When it is these values are substituted in the found that Sf=-P lines joining a point If pf circle to the focal points are Hence the on the perpendicular. we put thus indicating by 2 p the distance between the focal points, and by 8 the distance between the center of the circle and the mid-point of the line of the congruence, we find that We (21) replace this equation by the two 8 p cos a-. &lt;r, R p sin cr, thus defining a function /5 1 Now we have 1), =/E)(coso-+l), /? 2 =/o(cos&lt;r so that equations (19) may be written A= (22) ^* i o. o- = _ q ip (cos 1), B = prf (cos ol 1). *Vol. II, p. 161. 432 CYCLIC SYSTEMS of (5) equation (15) can be put in the By means form When the values (22) are substituted in this equation, it becomes Since by (3) the expression in the first parenthesis is true of the second, and so we have is zero, the same these are the conditions (V, 67) that the parametric curves on S form a conjugate system. Hence we have the theorem of But Ribaucour : On the envelope of the planes of the circles of a cyclic system the curves corresponding to the developables of the associated cyclic con gruence form a conjugate system. 176. Spherical representation of cyclic congruences. When the expressions (22) are substituted in (6), we obtain do da. dp dp, Since pq l p l q =t= unless Sis developable, the preceding equations be replaced by may 12 [ /3 (coso--l)]=2 a -hi)] /0 { 2 }VH-^), 1J \ ( d _ cv [/3(cos = 2psL /12V + a( l\P$$ i where the Christoff el symbols are formed with respect to (24) (pdu+p 2 l dv) -\-(qdu-{-q l dvY, S. the linear element of the spherical representation of SPHERICAL REPRESENTATION 433 of equations When (13), in like manner we J? 2 substitute in the 2 first two taking =p 2 sin cr =p 2 (1 cos 2 o-), we obtain . cos dp a-) du p cos o-- d --cos a 1 4- cos a du P cos 1 * , =pb p o r qa, (1 n+ , \ p cos o) -J7 cos o- a cos From these equations and (23) we find The condition of integrability of equations (25) is reducible to obtained from this equation be substi tuted in (25), we find two conditions upon the curves on the sphere in order that they may represent the developables of a cyclic con If the expression for cos a- gruence. isfied, A when particular case is that in which (27) the two conditions are is identically sat 121 ri O"\ (28) ll 1/121 _ O"^ " ^ n J 0v I2J 2 n o^ f 121 ll if / f f1 121 o"\ / J 12 / It is now our purpose to show that any system of curves on the sphere satisfies either set of conditions, all the congruences whose developables are thus represented on the sphere are cyclic. We assume that the sphere a solution p of is referred to such a system and that we have 434 CYCLIC SYSTEMS the By method of 167, or that hereinafter explained, we find the middle surface of the congruence. Then we take the point on each line at the distance p cos a- from the mid-point as the center of the circle of radius p sin a and for which the line is the axis. These cir cles form a cyclic system, as we shall show. In the first place we determine the middle surface with reference 2-axis coincides to a trihedral of fixed vertex, whose If with the radius of the sphere parallel to the line of the and #-axes are any whatever. # , # , z congruence and whose xdenote the coordinates of the mid-point of a line with reference to the corresponding tri hedral, the coordinates of the focal points are From ables (2) it is seen that if v = const, and u = const, these points correspond to the developrespectively, we must have Since pq l p l q =t= 0, the conditions of integrability of these equa tions can be put in the form (30) It is readily found that the condition of integrability of these equa tions reducible to (29). It will be to our advantage to have also the coordinates of the is point of contact of the plane of the circle with its envelope S. If x, y, Z Q p cos a denote these coordinates with reference to the above trihedral, it follows from (2) that (z - p cos v) + py-qx = 0, o (z - p cos a) + p^y - q,x = 0. CYCLIC CONGRUENCES 435 If these equations be subtracted from the respective ones of (30), the results are reducible, by means of (25), to (cos a -1) - + 2 p cos 1 a- gj + p(y - y) - q(x - x) = 0, Q (cos cr + 1) + 2 p cos a- V^- y)- ftfo- x) = 0, For, the quantities x x, y Q y are the coordinates of the center of the circle with reference to the tri (26). which are the same as hedral parallel to the preceding one and with the corresponding point on S for vertex. If, then, we have a solution b cr of (25) and p of (29), the corre (22), since the sponding values of a and given by (26) satisfy latter are the conditions that the parametric curves on the sphere the developables of the congruence. However, we have represent seen that when the values (22) are substituted in (12), we obtain equations reducible to (25) and (26). Hence the circles constructed as indicated above form a cyclic system. Since equations (25) admit only one solution (27) unless the con dition (28) is satisfied, we have the theorem: With each tem unless case there is cyclic congruence there is associated a it is at the unique cyclic sys same time a congruence of Ribaucour, in which an infinity of associated cyclic systems. Recalling the results of 141, we have the theorem of Bianchi * : When the total curvature of a surface referred to its asymptotic lines is of the form - ~ [* it is the surface generatrix of a congruence of Ribaucour which is cyclic in an infinity of ways, and these are the only cyclic congru ences with an infinity of associated cyclic systems. In this case the general solution of equations (25) (31) is cos is &lt;,=&gt;-* &lt;#&gt; + *, +^ where a an arbitrary constant. * Vol. II, p. 165. 436 CYCLIC SYSTEMS 177. Surfaces orthogonal to a cyclic system. In this section we consider the surfaces Sl orthogonal to the circles of a cyclic sys tem. Since the direction-cosines of the normals to the surfaces with reference to the moving trihedral in 174 are sin 6, cos 0, 0, the spherical representation of these surfaces is given by the point whose coordinates are these with respect to a trihedral of fixed vertex parallel to the above trihedral. From (2) we find that the expressions for the projections of a displacement of this point are + rdu + ^ dv), sin 6 (dd + r du + r dv), (p du + p^dv) cos 6 -f (q du + q dv) sin 0. cos 0(d0 1 v Moreover, by means of (32) (8), (21), (22), we obtain the identity ( sin a- (dd + r du + r^dv) = (1 4- cos a) p cos 6 -f q sin 6) du + (1 Hence the (33) cos a) ( p l cos 6 + &lt;?! sin 0) dv. linear element of the spherical representation of ^ is da*= T -L - COS (p cosO + q sin 0fdu* O~ 1\ + COSOcurvature, if r 1 Since the parametric curves on the sphere form an orthogonal system, the parametric curves on the surface are the lines of that this condition they form an orthogonal system. In order to show is satisfied, we first reduce the expressions (4) v for the projections of a displacement of a point of (21), (22), (25), (26), and (32), to on Sv by means cos v sin Cdu &lt;r ( ,1 cos a 1 (34) , Cdu cos &lt;r sin sin &lt;T i Ddv \ + cos 07 Ddv \ 1 + cos cr, ) Cdu + Ddv, where we have put = pi(b + R sin 0)q (a+E cos 0). 1 NORMAL CYCLIC CONGRUENCES Hence the (36) 437 linear element of S 1 is ds* = 2 it is du . 1 COS + (T 2 1 D V , + COS cr from which seen that the parametric curves on Sl form an orthogonal system, and consequently are the lines of curvature. Furthermore, it is seen from (34) that the tangents to the curves v = const., u = const, make i/l 1 tan" : with the plane of the 1 tan" circle the respec tive angles .-. (37) COScrX ), ./ ! l-fCOSCT\ sin \ sin cr / \ a / But it cumference of a follows from (21) that the lines joining a point on the cir circle to the focal points of its axis make the angles (37) with the radius to the point. Hence we have : to a cyclic system of the congruence of axes of the circles, correspond developables and the tangents to the two lines of curvature through a point of the surface meet the corresponding axis in its focal points. The lines of curvature to the on a surface orthogonal 178. Normal cyclic congruences. Since the developables of a cyclic congruence correspond to a conjugate system on the enve lope S of the planes of the circles, this system consists of the lines of curvature this case (cf. when 83). If, of the trihedral tangent to the lines of curvature, edges the congruence is normal, and only in under these conditions, we take two of the we have and equations (25) become d -, 03, " d , . cr d By a suitable choice of parameters tt we have so that (39) if we put &&gt; = cr/2, the linear element of the sphere 2 2 . is d(r*= sinW% + cosWv ( Comparing of its lines this result with 119), we have the theorem: The normals to a surface 2 with the same spherical representation curvature as a pseudospherical surface constitute the of only kind of normal cyclic congruences. 438 Since the surface CYCLIC SYSTEMS of the planes of the circles have the same representation of their lines of curvature, the tangents to the latter at corresponding points on the two surfaces are parallel. Hence with reference to a trihedral for 2 parallel to the trihedral for 2 and the envelope the coordinates of a point on the circle are R cos 0, R sin 6, p, where ft remains to be determined and 6 is given by (32), which can be put in the form ,*(\^ S cd H d(D (40) du n = cos co sm 0, . dO 1 da) = . sin &&gt; cos 6. a dv dv du If we express, by means of (2), the condition that all displace ments of this point be orthogonal to the line whose direction- cosines are sin 0, cos 0, 0, the resulting equation is reducible, by means of (40), to sin 9 (R cos o&gt; fi sin &&gt; o&gt; f) //, du &&gt; cos (R sin -f- cos 77^ dv 0. Hence the (41) quantities in parentheses are zero, from which i; we o&gt;. obtain R= cos &) -h T] I sin w, ^ = f sin o&gt; + rj l cos When, in particular, 2 is 2 -I/a we have (VIII, 22) , a pseudospherical surface of curvature f so that R= a and /x and the envelope of (cf. = a cos = 0. Hence &), rj l = a sin &&gt;, the circles are of constant radius is their planes the locus of their centers is sat 174). in this follows from (13) that is constant. Moreover, case p^ and p 2 as defined in 175, are the principal radii of the surface, which by (20) is pseudospherical. When these values Ex. 2, Conversely, when the latter condition &lt; isfied, it R , are substituted in (36) and (33), ment of each orthogonal surface it is is found that the linear ele ds* = a? (cos 2 6 du* -f sin 2 6 dv z ), and (42) of its spherical representation d&lt;r*=sm*0du*+ cos 2 &lt;W. Hence these orthogonal surfaces are the transforms of ( 2 by means of the Bianchi transformation 119). PLANES OF CIRCLES TANGENT TO A CUKVE The expression (42) is 439 the linear element of the spherical rep resentation of the surfaces orthogonal to the circles associated with be pseudospherical or not, whose spherical representation given by (39). Since these orthogonal surfaces have this representation of their lines of curvature, they are of the any surface 2, whether is it same kind as 2. We have thus for all surfaces with the same rep resentation of their lines of curvature as pseudospherical surfaces, a transformation into similar surfaces of which the Bianchi trans formation is a particular case transformation.* ; we call it a generalized Bianchi 179. Cyclic systems for circles is which the envelope of the planes of the a curve. We consider now the particular cases which have for been excluded from the preceding discussion, and begin with that which the envelope S of the planes of the circles is a curve C. take the moving trihedral such that its zy-plane, as before, that of the circle, and take the z-axis tangent to C. If s denotes the arc of the latter, we have is We ds = f; du + and by (43) ^dv, ?? = rj 1 = 0, (3) r ^-r^ = Q, (16) it rfx-ftf = 0. are functions of &lt;/&gt; From (14), (15), and follows that a and s, so that these equations (44) may .K 2 be replaced by 2 2 =a +& on the sphere represent the developables of the congruence, the conditions (19) must hold. But from (5), _Q (15), and (43) we obtain If the parametric curves ^. _^ If the values from (19) be substituted in this equation, we have, from (43), ^-^=0. focal surfaces coincide. If Hence the we put P in (19) = Pi=P* we obtain and substitute in the last of (12), 2 1 (^+^ )(^ -^)==0. *Cf. American Journal, Vol. XXVI (1905), pp. 127-132. 440 CYCLIC SYSTEMS of The vanishing p pq l p l q is the condition that there be a single infinity of planes, which case we exclude for the present. Hence the developables of the cyclic congruence are = iR ; that is, imaginary. Instead of retaining as parametric curves those representing the take the arc of C developables, we make the following choice. We for the parameter u consequently f =1, also, we have, from (3), ; ^=0. Since 77 =^ = x .hence we dq 2MB-* parameter v so that may it choose the p= 0, ^=1. From (3) follows, furthermore, that dr i= of T &gt; B-V = -* which the general integral is q=U where U^ and l cos v +U 2 sin v, r = U l sin v -f U z cos v, U 2 are arbitrary functions of u. From (5) we have A= j&gt;"(u)+\-rl, ^=0, is A = 7T #y U so that the third of equations (12) (*"+!)- reducible by (44) to cv d U sin v cos v Hence if we 6, (45) gives take for a any function of u denoted by and R follows directly from (44). r &lt;f&gt; (u), equation 180. Cyclic systems for which the planes of the circles pass through a point. If the planes of the circles of a cyclic system a point 0, we take it for the origin and for the pass through vertex of a moving trihedral whose z-axis is parallel to the axis of the circle under consideration. In this case equations (14) may (46) be replaced by 2 A&gt; = 2 tf +6 2 -*, in where c denotes a constant. But this is the condition that all the it circles are cut orthogonal to a sphere with center at 0, or PLANES OF CIKCLES THROUGH A POINT diametrically opposite points, or pass through 0, according as positive, negative, or zero. Hence we have the theorem : 441 c is If the planes of the circles of a cyclic system pass through a point, the circles are orthogonal to a sphere with its center at the point, or meet the sphere in opposite points, or pass through the center. From geometrical considerations is we see that the converse of this theorem true. When by (21) (47) c in (46) is zero all the circles pass through 0. Then we have a = p sin &lt;r cos 6, b = p sin a sin 6, and equations (26) become (cos or 1) ^ = . 2 cos a- \ f , + sin a- (p sin 6 q cos (cos &lt;r +1) - S-L. 2cos&lt;rj f -\-smo-(p l sm0 These equations are obtained likewise when we substitute the values (47) in equations (22) and reduce by means of (25) and (32). Because of (22) the function p given by (26) arid therefore p is given by (48) is a solution. But a solution of (29), the solution 6 of of Bianchi * : (32) involves a parameter. Hence we have the theorem Among all the cyclic congruences with the same spherical repre sentation of their developables there are an infinity for which the circles of the associated cyclic system pass through a point. If we take the line through the z-axis of the trihedral, equation (11) tion = TT, and the center of the circle for must admit of the solu and consequently must be of the form In order that zero and the system cyclic. combine this result with the preceding theorem to obtain the following: L and M must be this equation admit of a solution other than TT, both We A to two-parameter family of circles through a point and orthogonal any surface constitute a cyclic system, and the most general spher ical representation of the developables of a cyclic congruence is afforded by the representation of the axes of such a system of * Vol. II, p. 169. circles.^ t Bianchi, Vol. II, p. 170. 442 CYCLIC SYSTEMS consider finally the case where the planes of the circles depend upon a single parameter. If we take for moving axes the tangent, principal normal, and binormal of the edge of regression of these planes and its arc for the parameter w, we have We and comparing (V, 50) with p (2), we see that -&lt;7 T = 0, r =p of the where p and r are the edge of regression. b da A = --hi -, radii of first and second curvature Now du p A= 8a dv , ; = -B _#,hdu p r&gt; 2?! = *b dv The equations (12) reduce to two. One of the functions a, b may be chosen arbitrarily then the other and R can be obtained by the solution of partial differential equations of the first order. EXAMPLES 1. Show is helicoid cyclic, that a congruence of Ribaucour whose surface generator and determine the cyclic systems. is the right 2. congruence of Guichard is a cyclic congruence, and the envelope of the planes of the circles of each associated cyclic system is a surface of Voss. A 3. The surface generator of a cyclic congruence of Ribaucour surface of the planes of the circles of each associated cyclic system. 4. is an associate If S is tion as a pseudospherical surface, a surface whose lines of curvature have the same spherical representa and Si is a transform of S resulting from a gen eralized Bianchi transformation ( 178), the tangents to the lines of curvature of 81 pass through the centers of principal curvature of S. 5. When the focal segment of each line of a cyclic congruence is divided in constant ratio by the center of the circle, the envelope of the planes of the circles is a surface of Voss. 6. The circles of the cyclic system whose axes are normal to the surface S, defined in Ex. 11, p. 370, pass through a point, and the surfaces orthogonal to the circles are surfaces of Bianchi of the parabolic type. 7. If the spheres with the focal segments of the lines of a congruence for diameters pass through a point, the congruence is cyclic, and the circles pass through the point. 8. Show that the converse of Ex. 7 is true. GENERAL EXAMPLES GENERAL EXAMPLES 1. 443 Determine the normal congruences of Ribaucour which are If the cyclic. 2. Voss whose envelope of the planes of the circles of a cyclic system is a surface of conjugate geodesic system corresponds to the developables of the asso ciated cyclic congruence, any family of planes cutting the focal segments in con stant ratio and perpendicular to them envelop a surface of Voss. 3. necessary and sufficient condition that a congruence be cyclic is that the developables have the same spherical representation as the conjugate lines of a sur face which remain conjugate in a deformation of the surface. If the developables A of the congruence are real, the 4. deforms of the surface are imaginary. The planes Ribaucour touch their respective envelopes straight line. of the cyclic systems associated with a cyclic congruence of in such a way that the points of con tact of all the planes corresponding to the same line of the congruence lie on a 5. If the spheres described on the focal segments of a congruence as diameters cut a fixed sphere orthogonally or in great circles, the congruence is cyclic and the circles cut the fixed sphere orthogonally or in diametrically opposite points. 6. If one draws the circles which are normal to a surface S and which cut a fixed sphere S Q in diametrically opposite points or orthogonally, the spheres described on the focal segments of the congruence of axes as diameters cut SQ in great circles or orthogonally. 7. Determine the cyclic systems of equal circles whose planes envelop a devel opable surface. 8. Let Si be the surface defined in Ex. 14, p. 371, and let S be the sphere with center at the origin and radius r. Draw the circles which are normal to Si and which cut S orthogonally or in diametrically opposite points. Show that the of the axes of these circles is a normal congruence, and that the cyclic congruence coordinates of the normal surfaces are of the form ( -a?e a - 2 (r? [1 L2 + 2 -) K) e cos 6 -f 77 sin 8 T \Xi j + fJL ja2 e~ ( (T; + K )(P I sind - rj cose] X 2 + tX, or in ) r2 or + r2 according as the circles cut K is equal to diametrically opposite points, and where t is given by where , S orthogonally ( 2e 2a( [1 \a a a 2 (i? -I- a K) e ^) [ cos 6 -) + "1 -r\ sin 6 sin du _ [1 9. ( -a?e ~\ 2 j (j 4- K) e" sin 6 j t\ cos cos u dv. Show same spherical representation of their surface S referred to in Ex. 14, p. 371. that the surfaces of Ex. 8 are surfaces of Bianchi which have the lines of curvature as the pseudospherical Ex. 8 are surfaces 10. Show that the surfaces orthogonal to the cyclic system of Bianchi of the parabolic type. of 444 11. CYCLIC SYSTEMS Let S be a surface referred to an orthogonal system, and is let T be the trihe . dral whose x-axis tangent to the curve u = = const. 0, The equations z x = p(l + cos0), y = /&gt;sin0 define a circle normal to S. Show that the necessary and sufficient conditions that the circles so defined form a cyclic system are cu when an orthogonal necessary and sufficient condition that a cyclic system remain cyclic surface S is deformed is that S be applicable to a surface of revolution and that 12. A where 13. c is a constant and the linear element of S is ds2 = du 2 + 2 (w) dv* (cf . Ex. 11). Determine under what conditions the lines of intersection of the planes of the circles of a cyclic system and the tangent planes to an orthogonal surface form a normal congruence. M Let Si and S 2 be two surfaces orthogonal to a cyclic system, and let MI and be the points of intersection of one of the circles with Si and S 2 Show that the normals to Si and S 2 at the points MI and 2 meet in a point equidistant 14. 2 . M M points, and show that Si and S 2 constitute the sheets of the envelope of a two-parameter family of spheres such that the lines of curvature on Si and S 2 from these correspond. 15. Let -S variable radius spheres. be the surface of centers of a two-parameter family of spheres of JR, and let Si and S 2 denote the two sheets of the envelope of these Show that the points of contact MI and 2 of a sphere with these sheets are symmetric with respect to the tangent plane to S at the corresponding point M. Let S be referred to a moving trihedral whose plane y = is the plane 2 and M MiMM , let if the parametric curves be tangent to the x- and y-axes respectively. Show that ff denotes the angle which the radius MMi makes with the x-axis of the trihedral, the lines of curvature on Si are given by sin &lt;r (sin &lt;rp r cos &lt;r) du 2 + in[qi \ I sin &lt;r ) dv 2 dv/ H fllfl -) (cos &lt;rri + p sin &lt;r) \dudv = 0. 16. Find the condition that the lines of curvature on S! and S 2 of Ex. 15 corre spond, and show that in this case these curves correspond to a conjugate system on S. 17. Show that the circles orthogonal to two surfaces form a cyclic system, pro vided that the lines of curvature on the two surfaces correspond. 18. Let &lt;S lines of curvature being parametric, be a pseudospherical surface with the linear element (VIII, 22), the and let A be a surface with the same spher ; ical representation of its lines of curvature as S furthermore, let AI denote the envelope of the plane which makes the constant angle a with the tangent plane at a point of A and meets this plane in a line I/, which forms with the tangent to the curve u = const, at an angle defined by equations (VIII, 35). If MI M M GENERAL EXAMPLES 445 denotes the point of contact of this plane, we drop from MI a perpendicular on L, and NMi, meeting the latter in N. Show that if X and p denote the lengths MN they are given by X = ( V2? cos o&gt; + V6? sin o&gt;) sin &lt;r, ^ = ( Vj sin w + Vt? cos w) sin o-, where 19. E and (? are the first fundamental coefficients of A. then that Show that when the surface A in Ex. 18 is the pseudospherical surface S, and AI is the Backhand transform Si of S by means of the functions (0, when A is other than S the lines of curvature on the four surfaces S, .4, Si, &lt;r), AI correspond, and the last two have the same spherical representation. 0-, is given all values satisfying equations (VIII, 35) for a given 20. Show that as the locus of the point Jfi, defined in Ex. 18, is a circle whose axis is normal to at M. the surface A 21. Show p. that when A in Ex. 18 is a surface of Bianchi of the parabolic type a-. (Ex. 11, 370) the surfaces AI are of the same kind, whatever be CHAPTER XIV TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 181. Triple system of surfaces associated with a cyclic system. Let S be one of the surfaces orthogonal to a cyclic system, and 1 let its lines of curvature be parametric. The locus 2 t of the circles which meet S orthogonally. Hence, by Joachimsthal s theorem ( 59), the line of intersection is a line of curvature for 2 r In like manner, the locus 2 2 of the circles l a point M l in the line of curvature v = const, through is a surface which cuts S u = const, through cuts and the curve of intersection is a line of curva S^ orthogonally, ture on S 2 also. Since the developables of the associated cyclic which meet S: in the line of curvature M congruence correspond to the lines of curvature on all of the orthogonal surfaces, each of the latter is met by 2 X and 2 2 in a line of curvature of both surfaces. At each point of the circle through M the of curvature v = const, tangent to the circle is perpendicular to the line on 2 t through the point and to u = const, , Hence the circle is a line of curvature for both 2 X and 2 2 and these surfaces cut one another orthogonally along the circle. Since there is a surface 2 X for each curve v = const, on Sl and a surface 2 2 for each u = const., the circles of a cyclic system and . on 2 a the orthogonal surfaces may be looked upon as a system of three families of surfaces such that through each point in space there passes a surface of each family. Moreover, each of these three sur faces meets the other two orthogonally, and each curve of intersec ( 96) that the confocal quadrics form such a system of surfaces, and another example is afforded by a family of parallel surfaces and tion is a line of curvature on both surfaces. We have seen the developables of the congruence of normals to these surfaces. When three families of surfaces are so constituted that through each point of space there passes a surface of each family and each of the three surfaces meets the other 446 two orthogonally, they are GENERAL EQUATIONS 447 said to form a triply orthogonal system. In the preceding examples the curve of intersection of any two surfaces is a line of curvature for both. Dupin showed that this is a property of all triply orthog onal systems. We shall prove this theorem in the next section. of 182. General equations. Theorem Dupin. The simplest exam ple of an orthogonal system is afforded by the planes parallel to the coordinate planes. The equations of the system are 3 = 1*!, y =M a , z = i* 8, where u# u^ u 3 are parameters. Evidently the values of these parameters corresponding to the planes through a point are the rectangular coordinates of the point. In like manner, the surfaces of each family of any triply orthogonal system may be determined by a parameter, and the values of the three parameters for the three surfaces through a point constitute the curvilinear coordi nates of the point. Between the latter and the rectangular coor dinates there obtain equations of the form (1) x =/ (w 1 1, i* 8, i*,), y =/ K, 2 i* a , i*,), z =/,(!*!, i* a , i* 8 ), where the functions example of this is domain considered. An / afforded by formulas (VII, 8), which define space are analytic in the it is referred to a system of confocal quadrics. In order that the system be orthogonal necessary and cient that these functions satisfy the three conditions suffi v dx Any given dx ^aST By ment _ v dx *to t dx y dx is dx _ when u t t to.- Zto.dut defined by (1) is one of the surfaces u this constant value. = const, the linear element of space at a point we mean the linear ele at the point of any curve through it. This is which, in consequence of (3) may be written in the 2 = Hl du* + H* du* + HI dui, ds (2), parametric form As thus assume defined, the functions H# HH 2, 3 are real and we shall that they are positive. 448 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES (3) From we have at once the linear element of faces of the system. any of the sur For instance, the linear element of a surface , 2 U= COnst. is rrl J. Now we shall find that the second quadratic i forms of these surfaces are expressible in terms of the functions and their derivatives. If X^ F., Z denote the direction-cosines of the normals to the H surfaces u i = const, we have (5) , du. We (6) choose the axes such that = 4-1. the second fundamental coefficients of a sur In consequence of face u { (5) const, are defined by 1 _ ~ where 1 dx d*x _, 4 , = ^ dx y d 2 x_ ^u t _ ** H y du 1 i dx d*x t du? t, /c, I take the values 1, 2, refers to the summation of terms in 3 in cyclic order, and the sign 2 In order #, ?/, 2, as formerly. differentiate equations (2) to evaluate these expressions we with respect to u^ u v u 2 respectively. This gives dx . _JL__ = f o, dx dx ^ du. 0. If of the three, each of these equations be subtracted from one half of the sum we have ^ du z du = o, 0. dx d*x = du 3 du l 0, V = 0; consequently D-= THEOREM OF DUPIN. EQUATIONS OF LAME If the first 449 and third u z and u s respectively, respect to u v we have dx d*z of (2) be differentiated with respect to and the second and third of (4) with dx tfx ~~ 2 2 a# ~ 2 du^ dx tfx y J5T t dx tfx 3H 3 S Hence we have Z&gt; Proceeding in like manner, we find the expressions for the other s, which we write as follows : 3 in cyclic order. From the sec ond of these equations and the fact that the parametric system on each surface is orthogonal, follows the theorem of Dupin where i, K, I take the values 1, 2, : The surfaces of a lines triply orthogonal system meet one another in of curvature of each. Lame". 183. Equations of conditions to be satisfied by Hv H^ H By means of these results z, we find the in order that (3) may be the linear element of space referred to a triply orthogonal system of surfaces. For each surface the Codazzi and Gauss equations must be satisfied. When the above values are substituted in these equations, we and sufficient that the functions H satisfy find the following six equations : which it is necessary PH /ox { 1 dH^H, 1 ~ gJgjgjr^ H t, /^, where I take the values Lame", 1, 2, 3 in cyclic order. These are first the equations of being named for the geometer who deduced them.* * Lemons sur les coordonntes curvilignes et leurs diverses applications, pp. 73-79. Paris, 1859. 450 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES of the surfaces there For each form (V, 16). equations When is a system of equations of the the values from (7) are substituted in these we have &lt; t du, Recalling the results of 65, we have that each set of solutions of equations (8), (9) determine a triply orthogonal system, unique to within a motion in space. In order to obtain the coordinates of space referred to this system, we must find nine functions JQ, r;., Zf which satisfy (10) and 1, 2^=0. + H^X du + H X du Z 2 Z Z (=*=*) Then the coordinates of space are given by quadratures of the form x = I H^XI du l s . denotes the principal radius of a surface u = const, in the direction of the curve of parameter U K we have, from (7), If p. K f , rm l 1Pi. eter Let p denote the radius of first curvature of a curve of param u r In accordance with 49 we let w 1 and w[ ?r/2 denote the angles which the tangents to the curves of parameter u 3 and u 2 respectively through the given point make, in the positive sense, with the positive direction of the principal normal of the curve of parameter /i ur Hence, by (IV, 16), we have o\ Pi Pn Pi Pzi these equations and similar ones for curves of parameter u and u s we deduce the relations From , (13) 1 = 1 + 1, Pi Pl&gt; tan5 = 6t, ( Pfi P ONE FAMILY OP SURFACES OF REVOLUTION where 2, /c, I 451 take the values 1, 2, 3 in cyclic order. it Moreover, since the parametric curves are lines of curvature, that the torsion of a curve of parameter u is i follows from (59) (14) l-l^i. r { Hi du, 184. Triple systems containing one family of surfaces of revolution. of plane curves and their orthogonal trajectories the plane be revolved about a line of the plane as an axis, the two families of surfaces of revolution thus generated, and the planes Given a family ; if through the axis, form a triply orthogonal system. We inquire whether there are any other triple systems containing a family of surfaces of revolution. Suppose that the surfaces u s = const, of a triple system are sur faces of revolution, and that the curves u 2 = const, upon them are the meridians. Since the latter are geodesies, we must have From (8) it follows that either dH. i du s = n 0, or dffs 8 n = 0. du 2 follows from (11) that l//o 31 = 0. Consequently, the surfaces of revolution w 3 = const, are developables, that is, either circular cylinders or circular cones. Furthermore, from (15) and In the first case it (11), we have l//o 21 =0, so that the surfaces u2 = const, also are developables, and in addition we have, from (13), that l//^ = 0, that lines and consequently is, the curves of parameter u^ are straight the surfaces u^= const, are parallel. The latter are planes when the surfaces u s = const, are cylinders, and surfaces with circular lines of curvature when u s = const, are circular cones. Conversely, ( 132, it follows 187) and from with parallel generators, or that any system of circular cylinders locus any family of circular cones whose axes are tangent to the of the vertex, leads to a triple system of the kind sought. consider now the second case, namely from the theorem of Darboux We 452 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES (11) From we it surfaces w a = const, are planes. meridians, = 0; consequently the l//o 28 Since these are the planes of the follows that the axes of the surfaces coincide, and find that l//o 21 =0, and consequently the case cited at the beginning of this section only one for nondevelopable surfaces. is the it In 119 185. Triple systems of Bianchi and of Weingarten. was found that all the Bianchi transforms of a given pseudo- of the same total spherical surface are pseudospherical surfaces curvature, and that they are the orthogonal surfaces of a cyclic system of circles circles of constant radius. Hence the totality of these and surfaces constitutes a were first triply orthogonal system, such that the surfaces in one family are pseudospherical. of this sort As systems 119), they considered by Ribaucour (cf. are called the triple si/stems of Ribaucour. proceed to the consideration of all triple systems such that the surfaces of one We family are pseudospherical. * These systems were first studied by Bianchi, and consequently Darboux f has called them the systems of Bianchi. From 119 it ture of a pseudospherical surface of curvature chosen that the linear element takes the form (1 6) follows that the parameters of the lines of curva a l/a can be so d o&gt; a = cos a o) du* + sin a o&gt; dv\ where is a solution of the equation d a o&gt; d a o&gt; _ sin o&gt; cos a) In this case the principal radii are given by 1 tan &) 1 cot a) (18) Pl a p, a In general the total curvature of the pseudospherical surfaces of a system of Bianchi varies with the surfaces. If the surfaces w = const, are the pseudospherical surfaces, we may write the 3 curvature in the form Annali, 8er. Vol. t II, 7&gt;vow 1/f^, where (188,",), ?/8 is a function of ?/ alone. 2, Vol. XIII pp. 177-234; Vol. et les XIV (1880), pp. 115-130; Lezioni, chap, xxvii. wr les ni/stemes orthogonaux coonlonntes curvilignes, pp. 308-323. Paris, 18U8. TRIFLE SYSTEMS OF BIANCHI In accordance with (11) and (18) _1 P* , 453 we put tan&lt;w 1 dff, (19) 1 g// cot &lt;a s) tf - ?) ff = If these values of - and be substituted in equations (8) for* r/r, (K,, equal to 1 and 2 respectively, 1 we , obtain 1 BJf, l = 3to tan dlfa = o&gt; = do) cot a) From (20) these equations we &lt; have, by integration, cos co, ffl = 13 H= z &lt;/&gt;., 3 sin CD, where $13 and$ 23 are functions independent of w a and u respectively. \Vo shall show that both of them are independent of u 3 v . When we have the values of respectively jff H l and !! from to to (20) are substituted in (19), = fr = fr it /. C ot cw ( tan 3 log &lt;. I (21) / // tan o&gt; [ cot o&gt; \ A d da) --- - log z ,5w, ^3 \- \ ! / From these equations follows that Hence, unless l;l and ^&gt;., ;l the ratio of a function of u are independent of w 3 , tan is equal to and w 3 and of a function of w 2 and 3 v o&gt; ?&lt; . consider the latter case and study for the moment a partic ular surface i/ 3 = c. By the change of parameters (^..(MP We cjau^ the linear element of the surface reduces to (16), and (22) becomes tan co = v respectively. where f and V is are functions of u and obtain V a , When this substituted in (17), we w^Ox /C\^,,, iv i n , *S v I-- \ * -w-r-ft . (u"T 454 TKIPLY ORTHOGONAL SYSTEMS OF SURFACES with respect to u If this equation be differentiated successively and v, we find /U"\ f 1 r \u) unless ~uu~ + /V"\ 1 = \r) ~vv this it follows that V or V is equal to zero. From where K denotes a constant. Integrating, we have U"=2icU*+aU, V"=-2tcV s + l a and y3 being constants, and another integration gives U *=KU*+a(r*+&lt;y, F 2 =-*F + /3F 4 2 +S. find When these expressions are substituted in (23), we This condition can be Hence alone. U satisfied only when the curvature is zero. be zero, that is, &) must be a function of u or v In this case the surface is a surface of revolution. In accord 1 or V must ance with 184 a triple system of Bianchi arises from an infinity of pseudospherical surfaces of revolution with the same axis. (f&gt; When exception is made of this case, the functions . 13 and &lt;/&gt; 23 in (20) are independent of u s Hence the parameters of the sys tems may be chosen so that we have /2^x H IT H f7 When ~ these values are substituted in the six equations (8), (9), they reduce to the four equations 2 &) 2 ft) sin to cos &) = duf 0M* cot U, III ft) - -f- tan to = 0, 1 2 g&lt; (25) 0/1 cu v \cos _d_ / ft) g 2 ft) \ 1 d 3 /sinftA ft) du 1 duj d 2 U 1 du 3 \ d U =() G) 3 / ft)\ sin 1 du 2 du 2 du 3 1 &) &) \ /cos do ft) d^co _ du 2 \sin du 2 duj U 3 du 3 \ U Q 3 / cos cu v du l du z TRIPLE SYSTEMS OP WEINGAKTEN 455 Darboux has inquired into the generality of the solution of this system of equations, and he has found that the general solution involves five arbitrary functions of a single variable. shall not give a proof of this fact, but refer the reader to the investi We gation of Darboux.* turn to the consideration of the particular case where the total curvature of all the pseudospherical surfaces is the same, 1 without any loss of generality. which may be taken to be We As we triple systems of this sort were first discussed by Weingarten, follow Bianchi in calling them systems of Weingarten. kind are the triple systems of Ribaucour. Of this For (26) this case we have U3 = 1, 2 so that the linear element of space is ds = cos 2 &) 2 dul + sin a) du* Since the second of equations (25) the forms c may \ cos ft) be written in either of a / \ n ft) ff w \ cw c z a) , du 2 du s / )= du 2 du^ du s du 2 \cos if ft) du^ du 3 / 2 sin / -j ft) du l du 2 du s o2 We pUt / -j \2 \2 /o ycos it ft) du 1 du^l V sin &) du z cu z j \^ 3 follows from the last two of (25) and from (27) that eter is a function of u s alone. Hence u z an operation which will not &lt;J&gt; But by changing the param affect the , form of (26), we can give (28 ) &lt;& a constant value, say c. \cos -- Bu du ft) -+ ---= l Consequently we have c. 3 J \sm co du 2 du 3 / \^ 3 / Bianchi has shown f that equation (28) and the first of (25) are = Consequently the equivalent to the system (25), when Z78 l. of the determination of triple systems of Weingarten is problem the problem of finding common solutions of these two equations. *L.c., pp. 313, 314; Bianchi, Vol. II, pp. 531, 532. t Vol. II, p. 550. 456 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES EXAMPLES Show that the equations 1. x = r cos u cos v, y = r cos u sin u, z =: r sin w define space referred to a triply orthogonal system. 2. A necessary and sufficient condition that the surfaces u s = const, of a triply What are the s be a function of u 3 alone. orthogonal system be parallel is that other surfaces u\ = const., u% = const.? H 3. Two near-by surfaces u s = const, intercept equal segments on those orthog onal trajectories of the surfaces w 3 const, which pass through a curve s = const. on the former; on this account the curves #3 = const, on the surfaces u 3 = const. H are called curves of equidistance. 4. Let the surfaces w 3 = const, of a triple system be different positions of the in the direction of its axis. same pseudosphere, obtained by translating the surface Determine the character of the other surfaces of the system. 5. Derive the following results for a triple system of Weingarten : C+ V/8w\2 U where the surface u 3 differential parameter is formed with respect to the linear element of a = const., and p g is the radius of geodesic curvature of a curve = w3 const const. on that the curves of equidistance on the surfaces u s are geodesic parallels of constant geodesic curvature. this surface. 6. Show = Show that when c in (28) is is curves of parameter u 3 to (12) equal to zero, the first curvature l/p 8 of the constant and equal to unity; that equations similar a2 , become 2 = - sin o&gt; cos w 3 last _ 8&lt;a u&gt; = dot cos u sin u&gt; 3 ; that if we put cd w 3 the , two of equations dO ^H (25), where U= & ; 1, may be written gw -- -- = sm 6 cos w, 1 aw -- = cos sin u and that &d -- = 8u ^ Sin 6 COS 0, / / cos6 is --- --1 8*6 \ 2 J -f / / 1 8*6 \ ] 2 = /de\* / ) - . When c = in (28) the system said to be of constant curvature. 7. A necessary and sufficient condition that the curves of parameter u s of a system of Weingarten be circles is that w 3 be independent of u 3 In this case (cf. Ex. 6) the surfaces u s = const, are the Bianchi transforms of the pseudospherical surface with the linear element . ds* = cos*0du* THEOREM OF RIBAUCOUR Theorem Ribaucour * 186. : 457 is of Ribaucour. The following theorem due to Griven a family of surfaces of a triply orthogonal system and their orthogonal trajectories; the osculating circles to the latter at their points of meeting with any surface of the family form a cyclic system. In proving this theorem we fied first derive the conditions to be satis by a system of circles orthogonal to a surface S so that they may form a cyclic system. Let the lines of curvature on S be parametric and refer the surface to the moving trihedral whose x- and ?/-axes are tangent to the curves v (29) If &lt; = const., u = const. We have (V, 63) ^=n=p = qi =0. P denotes the angle which the plane of the circle through a the angle which point makes with the corresponding zz-plane, the radius to a point of the circle makes with its projection in the z^-plane, and the radius of the circle, the coordinates of with reference to the moving axes are R P x = R(\ -f cos 0) cos $, sin 6 cos y =^(1+ cos 0) sine/), sin 6 sin$, z=lism0. circle at 6. Moreover, the direction-cosines of the tangent to the &lt;, P are cos If we express the condition that every displacement of P must be at right angles to this line, we have, from (29) and (V, 51), dB - [sin B( [_ - f sin 6 . \R du J\ dR \jri + lL^i)+ q cos 0(1 + cos 0)1 du R/ J [_ cv + A__ K 77, sin&lt;f&gt;\ _p sm 0(i + -."I cos B)\dv J is 7 = 0. / The condition that this equation admit an integral reducible to cosjAI [sin L 4, cose/) E /J R Hence, as remarked before ( 174), if there are three surfaces orthog is cyclic. onal to a system of circles, the system * Comptes Rendus, Vol. LXX (1870), pp. 330-333. 458 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES that (f&gt;\ The condition d_ /i ?1 it be cyclic d_ is sin cu \ (30) R cos (/ / dv\ R , sin &lt;f&gt; d /sin (f) \ d icos&lt;f) \_ /&gt; Since the principal radii of (31) S are given by i= -|. i= J. S : the second of equations (30) reduces to the first when or a plane. Hence we have incidentally the theorem is a sphere A any two-parameter system of circles orthogonal to a sphere and to other surface constitute a cyclic system. return to the proof of the theorem of Ribaucour and apply the foregoing results to the system of osculating circles of the We curves of parameter u 3 of an orthogonal system at their points of intersection with a surface u= const. o From equations similar to (12) cos &lt;j) we have, by (11), 1 &lt;f&gt; 1 d//., sin dH z and the equations analogous 1 to (31) are 1 /&gt;, q 1 211^ Z _ 2 p l _ 1 3H Z pn ^ H^H du 3 //2 7/2 // 3 du s these values are substituted in equations (30) the first vanishes identically, likewise the second, in consequence of equa tions (8). Hence the theorem of Ribaucour is proved.* naturally arises whether any family of surfaces whatever forms part of a triply orthogonal system. This question will be answered with the aid 187. of When Theorems Darboux. The question of the following theorem of Darboux, f which we establish by his methods : necessary and sufficient condition that two families of surfaces orthogonal to one another admit of a third family orthogonal to both is that the first two meet one another in lines of curvature. * For a geometrical proof the reader is A referred to Darboux, I.e., p. 77. t L.c., pp. 6-8. THEOREM OF DARBOUX Let the two families of surfaces be defined by (32) 459 a(x, y, b are the z) = a, @(x, y, z) = b, is where a and parameters. The condition of orthogonality dx ~dx dy^y ~dz~dz~ In order that a third family of surfaces exist orthogonal to the surfaces of the other families, there satisfying the equations . must be a function 7(2, y, z) , _ dz _ ~dx dx dx dydy dz~ dx ~d^ ~dz~dz~ If dx, dy, dz ment of a point denote the projections on the axes of a displace on one of the surfaces 7 = const., we must have dx da dx dy da dy dz da dz = 0. Idx ~dy ~dz that This equation is of the form (XIII, 9). The condition (XIII, 10) it admit of an integral involving a parameter is da dx dz* dz dxdz dx dz* dz dxdz ~^ydxdy~~dx^y* dydxdy ^x ty*\~ where S indicates the sum of the three terms obtained by permut ing x, y, z in this expression. If we add to this equation the identity d (a, /3) 01* \da + { ^ -^-\ v f~f u f-f ^-\ ^ i/ LV ~i i f* =~~ \ 5 the resulting equation may be written in the form d/3\ da T~ dx da (34) dp T~ dx dj3 Ja \ ^~ dx ,/ p ~ ^l& da = 0, S( I ill r\ I da dz d/3 )_ gr/3 *1 I _ r\ . dz 460 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES we have introduced the symbol where, for the sake of brevity, defined by 8(0, 4&gt;), equation (33) be differentiated with respect to be written If x, the result may Consequently equation (34) da dx r\ is reducible to df3 dx ccc cp dy O /O (35) =o, da dz dz which is therefore the condition upon a and /S in order that the desired function 7 exist. a = const, displacement along a curve orthogonal to the surfaces A is given by ^ = ^_^. da da da dx dy dz /3 Such a curve it satisfies lies upon a surface = const, and since, by (35), the condition = 0, it is a line of curvature on the surface (cf. Ex. 3, p. 247). fi Hence the curves of intersection of the surfaces a = const., = const., are lines of being the orthogonal trajectories of the above curves, = const. And by Joachimsthal s theo curvature on the surfaces ft rem ( 59) they are lines of curvature on the surfaces a = const, also. Having thus established the theorem answer the question at the of Darboux, we are in a position to beginning of this section. TRANSFORMATION OF COMBESCURE ; 461 Given a family of surfaces a const. the lines of curvature in one family form a congruence of curves which must admit a family of orthogonal surfaces, if the surfaces a = const, are to form part of an orthogonal system. If this condition is satisfied, then, accord ing to the theorem of Darboux, there is a third family of surfaces which together with the other two form an orthogonal system. If Xv Yv Z 1 lines of curvature in denote the direction-cosines of the tangents to the one family on the surfaces a const., the ana be a family of surfaces orthogonal to that the equation lytical condition that there these curves is admit an integral involving a parameter. The condition for this is In order to find X^ Y^ Z we remark 1 that since they are the direc tion-cosines of the tangents to a line of curvature we must have and similar equations in Y, Z, where the function X is a factor of proportionality to be determined arid Jf, Y, Z are the directioncosines of the normal to the surface a const. Hence, if the = surfaces are defined by a = const., the functions Xv Y^ Z^ of a, are expressible in terms of the first and second derivatives and so equation (36) is of the third order in these derivatives. fore we have the theorem of Darboux*: There The determination of all triply orthogonal systems requires the integration of a partial differential equation of the third order. Darboux has given the name family of Lame to a family of surfaces which forms part of a triply orthogonal system. 188. Transformation of Combescure. We close our study of triply orthogonal surfaces with an exposition of the transformation of Combescure,^ by means of which from a given orthogonal system others can be obtained such that the normals to the surfaces of one system are parallel to the normals to the corresponding sur faces of the other system at corresponding points. * L.c., p. 12. f Annales de I Ecole Normale Superieure, Vol. IV (1867), pp. 102-122. 462 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES make use of a set of functions /3iK , We introduced by Darin space of boux * in his development of a similar transformation n dimensions. By definition In terms of these functions equations the form (8), (9) are expressible in 37 &lt; &gt; $-* t (10) + and formulas (38) become A.jr.-arr,, Equations (37), (38) are the the expression ^^ ^^ ^^ + + From their necessary and sufficient conditions that we have another set of functions H[, 772 //8 satisfying the six conditions it is be an exact differential. form seen that if , 39 &lt; &gt; *--/3tK where the functions have the same values as for the given system, the expression XJI[ dUl -f JT2 //2 du 2 + Xfi du a and similar ones tures in F, Z, are exact differentials, , and so by quadra desired property. we obtain an orthogonal system possessing the In order to ascertain the analytical character of this problem, we eliminate H[ and H^ from equations (39) and obtain the three equations n _ .._ , -^ du, cu^ . The general integral of a system of equations of this kind involves three arbitrary functions each of a single parameter u When one t *L.c., p. 161. GENERAL EXAMPLES has an integral, the corresponding values of by (39). Hence we have the theorem : 463 H^ H^ are given directly With every triply orthogonal system there is associated an infinity of others, depending upon three arbitrary functions, such that the normals to the surfaces of any two systems at corresponding points are parallel.* 1. EXAMPLES In every system of Weingarten for which c in (28) is zero, the system of cir cles osculating the curves of parameter u s at points of a surface w 3 = const, form a system of Ribaucour 2. ( 185). orthogonal trajectories of a family of Lame" are twisted curves of the same constant first curvature, the surfaces of the family are pseudospherical If the surfaces of equal curvature. 3. Every triply orthogonal system which is derived from a cyclic system by a transformation of Combescure possesses one family of plane orthogonal trajectories. 4. If the system of circles osculating these trajectories family may be obtained from the given system 5. Determine the triply orthogonal systems the transformation of Combescure to a system orthogonal trajectories of a family of Lame* are plane curves, the cyclic at the points of any surface of the by a transformation of Combescure. which result from the application of of Ribaucour ( 185). GENERAL EXAMPLES 1. If an inversion by reciprocal radii ( 80) be effected upon a triply orthogonal system, the resulting system will be of the same kind. 2. Determine the character of the surfaces of the system obtained by an inversion from the system of Ex. 1, 185, and show that all the curves of intersection are circles. 3. Establish the existence of a triply orthogonal system of spheres. 4. necessary and sufficient condition that the asymptotic lines correspond on the surfaces u% = const, of a triply orthogonal system is that there exist a relation A of the form 0j, 2 &lt;t&gt;s 03 are functions independent of w 3 is satisfied, . = ? where 5. When us the condition of Ex. 4 those orthogonal trajectories of the surfaces u s face = const, = const, asymptotic lines 6. which pass through points of an asymptotic line on a sur constitute a surface S which meets the surfaces u s = const, in of the latter and geodesies on &lt;S. that the asymptotic lines correspond on the pseudospherical surfaces of a triple system of Bianchi. 7. Show that there exist triply orthogonal systems for which the surfaces in one Show family, say u$ const., are spherical, and that the parameters can be chosen so that sinh 6, HI = Find the equations of 8. cosh 8, Hz = H 3 = US CUz is . Lame" for this case. Every one-parameter family of spheres or planes *Cf. Bianchi, Vol. II, p. a family of Lame . 494. 464 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 9. In order to obtain the most general triply orthogonal system for which the surfaces in one family are planes, one need construct an orthogonal system of curves in a plane and allow the latter to roll over a developable surface, in which case the curves generate the other surfaces. When the developable determination of the system reduces to quadratures. 10. of is given, the Show that the most general triply orthogonal system for which one family Lame" consists of spheres passing through a point can be found by quadratures. . 11. Show Show that a family of parallel surfaces is a family of Lame 3 that the triply orthogonal systems for which the curves of parameter are circles passing through a point can be found without quadrature. 12. 13. By means of Ex. 6, 185, show that for a system of Weingarten of constant curvature the principal normals to the curves of parameter w 3 at the points of meet ing with a surface u 3 = const, form a normal pseudospherical congruence, and that the surfaces complementary to the surfaces w 3 = const, and their orthogonal tra jectories constitute a system of 14. By means of Ex. 13 Weingarten show that for a of constant curvature. triple system arising from a system of of Weingarten of constant curvature by a transformation Combescure the osculat ing planes of the curves w 3 = const., at points of a surface u s = const., envelop a surface S of the same kind as this surface M 3 = const. ; and these surfaces S and their orthogonal trajectories constitute a system of the same kind as the one result ing from the Combescure transformation of the given system of Weingarten. 15. Show system of Bianchi be plane d(*t that a necessary condition that the curves of parameter u\ of a triple is that w satisfy also the conditions = 023 sm w, . Set = 0i 3 sin w, (cf. di/2 dui where 23 Show that and 0i 8 are independent of HI and w 2 respectively if 0is and 02 3 satisfy the conditions Ex. 5, p. 317). where a and given by b are constants and U& is an arbitrary function of w 8 the function w, g0 23 a0i 8 , COS W = determines a triply orthogonal system of Bianchi of the kind sought. 16. (25) When Z73 = 1 and w is independent of u 2 the , first and fourth of equations may be replaced by gw = sin w. dui Show that for a value of satisfying this condition / ( the expressions HI = cos w rfadu s I \J sin w - \ + _(- 0i ) - r^cosu, 3 du / J : / sin w and the other equations -f 0i, (25) H z = sin w ( \J ( ( C^J^ sin w I 0A- Jr0 3 cZM8 + / 2 , HB ~ \J ffoduz sin w + 0i \ du / 5u 3 GENERAL .EXAMPLES 3 are functions of MI, u 2j 2 0i, differentiation, define a triply orthogonal 465 where , w 3 respectively, and the accent indicates system for which the surfaces M 3 = const, are molding surfaces. 17. Under what conditions do the functions d2 sin w Z72 and Z7s are functions of M 2 and M 3 respectively, determine a triply orthog onal system arising from a triple system of Bianchi by a transformation of Combescure ? Show that in this case the surfaces w 2 = const, are spheres of radius Z72 and where , that the curves of parameter M 2 in the system of Bianchi are plane or spherical. 18. Prove that the equations y Z B(UI b) C) m i(u 2 i(lt 2 b) C) m *(u s 2(w 3 6) c) m s, "3, = - C(Ui wl w are constants, define space referred to a triple system of surfaces, such that each surface is cut by the surfaces of the other two families in a, 6, c, t where A, B, C, m a conjugate system. 19. Given a surface 8 and a sphere S; the circles orthogonal to both constitute a cyclic system hence the locus of a point upon these circles which is in constant cross-ratio with the points of intersection with S and S is a surface Si orthogonal to the circles Si may be looked upon as derived from S by a contact transformation which preserves lines of curvature such a transformation preserves planes and ; ; ; spheres. 20. When S all of Ex. 19 surface which is is a cyclide of Dupin, so are the surfaces Si, and also the the locus of the circles which meet S in any line of curvature ; hence of these surfaces form a Z7, triple system of cyclides of Dupin. 21. Given three functions defined by Ui = imuf + 2 mm + p^ - (i = 1, 2, 3) where m t -, Wj, pi are constants satisfying the conditions Sm = 0, - t Sn = t 0, Sp - t = ; and given also the function N= where ,-, tti(M 2 - u 3 )VUi -f a 2 (M 8 - MI) VU^ + &lt;* 3 (MI - + pZniiUi + 0, 7 (PiMaWa + PZ^UI 7 are constants determine under what condition the functions U S - Ui W 2 - US MI -Ma TT &l - ; ;= N^lfi determine a triply orthogonal system. and that they are cyclides of Dupin. 22. Determine N^U 7= ^33= 2 -p= N-VU 3 Show that all of the surfaces are isothermic, whether there exist triply orthogonal systems of minimal surfaces. INDEX The numbers refer to pages. References to an author and his contributions are made in the form of the first Bianchi paragraph, whereas when a proper name is part of a title the reference is given the form as in the second Bianchi paragraph. Acceleration, 15, 60 spherical lines of curvature), ; ; 315 ; Angle between curves, 74, 200 Angle of geodesic contingence, 212 Applicable surfaces, definition, 100 to the plane, 101, 156 invariance of invariance geodesic curvature, 135 ; ; ; (associate surfaces), 378 (cyclic con gruences of Ribaucour), 435 (cyclic of total curvature, 156 solution of the problem of determining whether ; systems), 441 Bianchi, transformation of, 280-283, 290, 318, 320, 370, 456 surfaces of, 370, 371, 442, 443, 445 generalized ; ; two given surfaces are applicable, 321-326 pairs of, derived from a given pair, 349. See Deformation of ; surfaces Area, element of, 75, 145 Area of a portion of a surface, 145, 250 minimum, 222 Associate surfaces, definition, 378 de termination, 378-381; of a ruled surface, 381; of the sphere, 381; ap plicable, 381; of the right helicoid, 381 of an isothermic surface, 388 of of pseudospherical surfaces, 390 characteristic .quadrics, 390, 391; property, 425 Asymptotic directions, definition, \2S *Asymptotic lines, definition, 128 para metric, 129, 189-194 orthogonal, 129 spherical represen straight, 140, 234 tation, 144, 191-193; preserved by protective transformation, 202 pre served in a deformation, 342-347 ; ; ; ; transformation of, 439 triply orthog onal systems of, 452-454, 464, 465 Binormal to a curve, definition, 12 spherical indicatrix, 50 Bmormals which are the principal nor mals to another curve, 51 ; ; Bonnet (formula of geodesic curvature), (surfaces of constant curvature), 179; (lines of curvature of Liouville type), 232 (ruled surf aces), 248 (sur faces of constant mean curvature), ; ; 136 ; 298 IJour (helicoids), 147; (associate isother mic surfaces), 388 ; Canal surfaces, definition, 68 of center, 186 Catenoid, definition, 150 ; ; surfaces ; ; ; ; ; Backhand, transformation of, 284-290 Beltrami (differential parameters), 88, 90; (geodesic curvature), 183; (ruled W-surfaces), 299 (applicable ruled surfaces), 345 (normal congruences), ; ; adjoint sur 267 surfaces applicable to, 318 Cauchy, problem of, 265, 335 Central point, 243 Central plane, 244 Cesaro (moving trihedral), 8S Characteristic equation, 375 Characteristic function,, 374, 377 face of, ; Characteristic lines, 13Q, 131 ric, ; paramet 203 ; Characteristics, of a fanUly of surfaces, 59-61 of the tangent pfones to a sur--/ face, 126 Christoffel (associate isothermic sur faces), 388 Christoffel symbols, definition, 152, 153 ; relations between, for a surface and its 403 Bertrand curves, definition, 39 proper parametric equations, 51 ties, 39-41 on a ruled surface, 250 ; deformation, 348 Bianchi (theorem of permutability), 286-288 (surfaces with circular lines of curvature), 311; (surfaces with ; ; ; spherical representation, 162, 193, ; ; 201 Circle, of curvature, 14 osculating, 14 * surfaces are listed under the latter. References to asymptotic lines, geodesies, lines of curvature, etc., on particular kinds of 467 468 INDEX ; Circles, orthogonal system of, in the plane, 80, 97 on the sphere, 301 Circular lines of curvature, 149, 310, Correspondence with orthogonality of linear elements, 374-377, 390 Corresponding conjugate systems, 130 Cosserat (infinitesimal deformation), 380, 385 Cross-ratio, of four solutions of a Riccati equation, 26 of points of intersection of four-curved asymptotic lines on a ruled surface, 249 of the points in which four surfaces orthogonal to a cyclic system meet the circles, 429 Cubic, twisted, 4, 8, 11, 12, 15, 269 Curvature, first, of a curve, 9; radius of, 9; center of, 14; circle of, 14; constant, 22, 38, 51 Curvature, Gaussian, 123 geodesic (see ; ; ; 316, 423, 446 Circular point on a surface, 124 Codazzi, equations of, 155-157, 161, 168, 170, 189, 200 Combescure transformation, of curves, of triple systems, 401-465 surface, 184, 185, 283, 290, 370, 464 Conforinal representation, of two sur faces, 98-100, 391; of a surface and its spherical representation, 143 of a surface upon itself, 101-103 of a plane upon itself, 104, 112 of a sphere upon the plane, 109 of a sphere upon itself, 110, 111; of a pseudospherical surface upon the plane, 317 Conformal-con jugate representation of ; 50 Complementary ; ; ; ; Geodesic) Curvature, mean, of a surface, 123, 126, 145 surfaces of constant (see Sur ; face) two surfaces, 224 Congruence of curves, 426 normal, 430 Congruence of straight lines (rectilinear), ; definition, 392 normal, 393, 398, 401, 402, 403, 412, 422, 423, 437; associate ; normal, 401-403, 411; ruled surfaces, 393, 398, 401 limit points, 396 prin cipal surfaces, 396-398, 408 principal ; ; ; Curvature, normal, of a surface, radius of, 118, 120, 130, 131, 150; principal radii of, 119, 120, 291, 450 center of, 118, 150; principal centers of, 122 Curvature, second, of a curve, 16 con stant, 50. See Torsion Curvature, total, of a surface, 123, 126, ; ; planes, 396, 397; developable*, 398, 409, 414, 421, 432, 437; focal points, 398, 399, 425; middle point, 399; middle surface, 399, 401, 408, 413, 421-424 middle envelope, 413, 415 focal planes, 400, 401, 409, 416 focal ; 211 145, 155, 156, 160, 172, 186, 194, 208, radius of, 189 surfaces of con ; ; ; ; stant (see Surface) Curve, definition, 2; of constant first curvature, 22, 38, 51; of constant form of a, 18 torsion, 50 Cyclic congruences. See Congruences ; surfaces, 400, 406, 409-411, 412, 414, 416, 420 derived, 403-405, 411, 412 isotropic, 412, 413, 416; of Guichard, 414,415,417,422,442 pseudospherical, 184, 415, 416, 464 W-, 417-420, 422, 424 of Ribaucour, 420-422, 424, 425, 435, 442, 443 mean ruled surfaces, 422, 423, 425 cyclic, 431-445 spher ical representation of cyclic, 432-433 cyclic of Ribaucour, 435, 442, 443 developables of cyclic, 437, 441 ; ; ; ; ; ; ; ; Cyclic system, 426-445 definition, 426 of equal circles, 430, 443 surfaces orthogonal to, 436, 437, 444, 457; planes envelop a curve, 439, 440 planes through a point, 440, 441 planes depend on one parameter, 442 triple system associated with a, 446 associated with a, triple system, 457; ; ; ; ; ; ; ; ; 458 Cyclides of Dupin, 188, 312-314, 412, 422, 465 ; ; normal cyclic, 437 Conjugate directions, 126, 173 radii in, 131 ; normal ; D, Z7, Conjugate system, definition, 127, 223 parametric, 195, 203, 223, 224 spher ical representation of, 200 of plane curves, 224 preserved by projective 202 preserved in a transformation, deformation, 338-342, 348, 349 Conjugate systems in correspondence,- 130 Conoid, right, 56, 58, 59, 68, 82, 98, 112, ; ; ; A Darboux 170 191 195 ; for the definition, 115 ing trihedral, 174 definition, 386 Jb i 7)", ; &"&gt; mov (moving trihedral), 168, 169, ; (asymptotic lines parametric), (conjugate lines parametric), (lines of curvature preserved by an inversion), 196 (asymptotic lines ; ; ; and conjugate systems preserved by projective transformation), 202 (geo desic parallels), 216, 217 (genera tion of new surfaces of Weingarten), 298 (generation of surfaces with plane lines of curvature in both sys 304 ; (general problem of tems), ; ; 120, 195, 347 Coordinates, curvilinear, on a surface, 55 curvilinear, in space, 447 sym metric, 91-93 tangential, 163, 194, ; ; ; ; 201; elliptic, 227 INDEX deformation), 332 (surfaces appli cable to paraboloids), 367 (triply ; 469 ; ; orthogonal systems), 458-461 Darboux, twelve surfaces of, 391; de rived congruences of, 404, 405 Deformation of surfaces (see Applicable of surfaces of revolution Surfaces of revolution) of mini mal surfaces, 264, 269, 327-330 of surfaces of constant curvature, 321323 general problem, 331-333 which changes a curve on the surface into a given curve in space, 333-336 which preserves asymptotic lines, 336, 342, 343 which preserves lines of curva which preserves ture, 336-338, 341 conjugate systems, 338-342, 349, 350, 443 of ruled surfaces, 343-348, 350, surfaces) (see ; ; Element, of are.a, 75, 145 linear (see Linear element) normal sec Ellipsoid, equations, 228 tion, 234 polar geodesic system, 236-238; umbilical geodesies, 236, 267; surface corresponding with par allelism of tangent plane, 269. See Quadrics ; ; ; ; ; ; ; Elliptic coordinates, 227 Elliptic point of a surface, 125, 200 Elliptic type, of pseudospherical sur of surfaces of Bianchi, faces, 274 370, 371 Enneper (torsion of asymptotic lines), 140 (equations of a minimal surface) ; ; , ; 256 Enneper, minimal surface ; 367; method of Weingarten, 353-369 of the of paraboloids, 348, 368, 369 envelope of the planes of a cyclic ; ; system, 429, 430 Developable surface, definition, 61 ; particular kinds, 69 equation, 64 polar, 64, rectifying, 62, 64, 112, 209 applicable to the plane, 65, 112, 209 101, 156, 219, 321, 322 formed by nor mals to a surface at points of a line of curvature, 122 principal radii, 149 ; ; ; ; of, 269; sur faces of constant curvature of 317, 320 Envelope, definition, 59, 60 of a oneparameter family of planes, 61-63, 64, 69, 442 of a one-parameter fam ily of spheres, 66-69 of a two-param eter family of planes, 162, 224, 426, 439; of geodesies, 221; of a twoparameter family of spheres, 391, 444 * parametric, 53; , ; ; ; ; Equations, of a curve, 52, 53, 54 1, 2, 52, 1, 2, 3, 21; of a surface, ; ; 250 geodesies on a, 224, 268, 318, 322 fundamental property, 244; of a congruence (see Congruence) Dextrorsum, 19 Differential parameters, of the first order, total curvature, 156, ; ; 84-88, 90, 91, 120, 160, 166, 186 of the second order, 88-91, 160, 165, 166, 186 Diui (spherical representation of asymp totic lines), 192; (surf aces of Liouville), 214 (ruled TF-surfaces), 299 Dini, surface of, 291, 318 Director-cone of a ruled surface, 141 Director-developable of a surface of ; Equidistance, curves of, 456 Equidistantial system, 187, 203 Equivalent representation of two sur faces, 113, 188 Euler, equation of, 124, 221 Evolute, of a curve, 43, 45-47 of a surface, 180, 415 (see Surface of center) of the quadrics, 234 mean, of a surface, 165, 166, 372 ; ; ; F. & SeeE See . ; // Seee Monge, 305 Directrix of a ruled surface, 241 Dobriner (surfaces with spherical lines of curvature), 315 Dupin (triply orthogonal systems), 449 Dupin, indicatrix of, 124-126, 129, 150 cyclide of (see Cyclide) theorem of Malus and, 403 ; ; jE, &&gt; F, G, definition, 70 for the moving trihedral, 174 definition, 141 for the moving trihedral, 174 e,/,/, flr, definition, 393 ; &&gt; ^ ; Edge of regression, 43, 60, 69 * Family, one-parameter, of surfaces, 59, 446, 447, 451, 452, 457-461; of planes, 61-64, 69, 442, 463 of spheres, 66-69, 309, 319, 463 of curves, 78-80 of geo desies, 216, 221 Family, two-parameter, of planes, 162, 224, 426, 439 of spheres, 391, 444 Family of Lame", 461, 463, 464 Focal conic, 226, 234, 313, 314 Focal planes, 400, 401, 409, 416 Focal points, 398, 399, 425 Focal surface, of a congruence, 400 reduces to a curve, 406, 412 funda mental quantities, 409-411 develop able, 412; met by developables in lines of curvature, 414 of a pseudospherical congruence, 416; infinitesi mal deformation of, 420 intersect, 423 ; ; ; ; ; ; ; ; ; For references such as Equations of Codazzi, see Codazzi. 4TO Form INDEX Helicoid, general, 146-148 parameter meridian of, 140 geodesies, of, 140 surfaces of center of, 151, 209 149, 186 pseudospherical, 291 is a IP-sur ; ; ; of a curve, 18 Frenet-Serret formulas, 17 Fundamental equations of a congruence, 406, 407 Fundamental quadratic form, of a sur of a surface, second, face, first, 7 1 115; of a congruence, 393 Fundamental quantities, of the first order, 71; of the second order, 115 Fundamental theorem, of the theory of curves, 24 of the theory of surfaces, 159 ; ; ; ; 300 minimal, 329, 331 appli cable to a hyperboloid, 347 Helicoid, right, 146, 148, 203, 247, 250, 260, 267, 330, 347, 381, 422 Helix, circular, 2, 41, 45, 203 cylindri face, ; ; ; ; G. & g. See 8ee See e 60 ; E cal, 20, 21, 29, 30, 47, 64 Henneberg, surface of, 267 Hyperbolic point, 125, 200 Hyperbolic type, of pseudospherical sur Gauss (parametric form of equations), representation), 141 (total curvature of a surface), 155 (geodesic parallels), 200; (geodesic cir cles) 207 (area of geodesic triangle) (spherical ; ; ; face, 273 of surface of Bianchi,371, 379 Hyperboloid, equations, 228 fundamen tal quantities, 228-230; evolute of, 234 of revolution, 247, 348 lines of deformation of, 347, striction, 268 348. See Quadrics ; ; ; ; ; 209 Gauss, equations of, 154, 155, 187 Generators, of a developable surface, 41 of a surface of translation, 198 of a ruled surface, 241 Geodesic circles, 207 Geodesic contingence, angle of, 212 Geodesic curvature, 132, 134, 135, 13(5, 140, 213, 223 radius of, 132, 150, 151, center of, 132, 225, 174, 170, 209, 411 curves of 423 invariance of, 135 constant, 137, 140, 187, 223, 319 Geodesic ellipses and hyperbolas, 213215, 225 Geodesic parallels, 207 Geodesic parameters, 207 Geodesic polar coordinates, 207-209, 230, 276 Geodesic representation, 225, 317 Geodesic torsion, 137-140, 174, 176 radius of, 138, 174, 176 Geodesic triangle, 209, 210 ; Indicatrix, of Dupin (seel)upin); spheri cal (.see Spherical) Infinitesimal deformation of a surface, ; 373, 385-387 generatrices, 373, 420 of a right helicoid, 381 of ruled sur in which lines of curva faces, 381 ture are preserved, 387, 391; of the focal surfaces of a TF-congruence, 420 Intrinsic equations of a curve, 23, 29, ; ; ; ; ; 30, 30 ; ; ; Invariants, differential, 85-90 of a dif ferential equation, 380, 385, 406 Inversion, definition, 190 preserves lines of curvature, 190 preserves an isotherm ic system of lines of curva ture, 391 preserves a triply orthog onal system, 403. See Transformation ; ; ; ; ; by reciprocal radii Involute, of a curve, 43-45, 311 of a surface, 180, 184, 300 Isometric parameters. See Isothermic ; parameters ,- * Geodesies, definition, 133 plane, 140 equations of, 204, 205, 215-219; on surfaces of negative curvature, 211 on surfaces of Liouville, 218, 219 Goursat, surfaces of, 306, 372 ; ; ; Isometric representation, 100, 113 Isothermal-conjugate systems of curves, 198-200; spherical representation, 202 formed of lines of curvature, 147, 203, on associate surfaces, 300 233, 278 Isothermal -orthogonal system. See Iso ; ; Guichard (spherical representation of the developables of a congruence), 409; (congruences of Ribaucour), 421 Guichard, congruences of, 414, 415, 417, 422, 442 I/, definition, //, definition, 71 142 of, 278, 279, thermic orthogonal system Isothermic orthogonal systems, 93-98, 209, 252, 254 formed of lines of curva ture (see Isothermic surface) Isothermic parameters, 93-97, 102 Isothermic surface, 108, 159, 232, 253, 269, 297, 387-389, 391, 425, 465 ; Isotropic congruence, 412, 413, 416, 422- Hamilton, equation of, 397 Hazzidakis, transformation 338 424 Isotropic developable, 72, 171, 412, 424 Isotropic plane, 49 p. 467. * See footnote, INDEX Jacob! (geodesic lines), 217 Joachimsthal (geodesies and lines of curvature on central quadrics), 240 Joachimsthal, theorem of, 140 surfaces of, 308, 309, 319 ; 471 Meridian curve on a surface, 260 Meusnier, theorem of, 118 Middle envelope of a congruence, 413, 415 Middle point of a line of a congruence, Kummer Lame (rectilinear congruences), 392 Lagrange (minimal surfaces), 251 85 equations of, 449 family of, 401, 463, 464 Lelieuvre, formulas of, 193, 195, 417, 419, 420, 422 Lie (surfaces of translation), 197, 198 (double minimal surfaces), 259 (lines of curvature of JF-surfaces), 293 Lie, transformation of, 289, 297 Limit point, 396, 399 Limit surface, 389 Line, singular, 71 *Line of curvature, definition, 121, 122, 128; equation of, 121, 171, 247; par normal cur ametric, 122, 151, 186 vature of, 121, 131 geodesic torsion of, 139 geodesic, 140 two surfaces inter secting in, 140 spherical representa (differential parameters), ; 399 Middle surface of a congruence, 399, 401, 408, 413, 421-424 Minding (geodesic curvature), 222, 223 Minding, problem of, 321, 323, 326 ; Lame", method of, 344 Minimal curves, 6, ; 47, 49, 255, 257 on a surface, 81, 82, 85, 91, 254-265, 318, 391 on a sphere, 81, 257, 364-366 ; ; 390 ; Minimal straight lines, 48, 49, 260 Minimal surface, definition, 129, 251; asymptotic lines, 129, 186, 195, 254, 257, 269 spherical representation, 143, 251-254; ruled, 148; helicoidal, of revolution, 160 149, 330, 331 parallel plane sections of, 160 mini mal lines, 177, 186, 254-265; lines of curvature, 186, 253, 257, 264, 269 double, 258-260 algebraic, 260-262 evolute, 260, 372 adjoint, 254, 263, ; ; ; ; ; ; ; ; ; ; ; ; ; tion of, 143, 148, 150; osculating plane, 148; plane, 149, 150, 201, 305-314, 319, 320, 463 plane in both systems, 269, 300-304, 319, 320 spherical, 149, 314-317, 319, 320, 465 circular, 149, 310-314, 316, 446; on an isothermic surface, 389, Line of striction, 243, 244, 248, 268, 348, 351, 352, 369, 401, 422 Linear element, of a curve, 4, 5 of a surface, 42, 71, 171; of the spherical ; ; ; ; 267, 377; associate, 263, 267, 269, 330, 381; of Scherk, 260; of Henneberg, 267; of Enneper, 269; deformation of, 264, 327-329, 349, 381 ; determi ; geodesies, 267 Molding surface, definition, 302 equa tions of, 307, 308 lines of curvature, of, 265, ; ; nation 266 307, 308, 320 applicable, 319, 338 associate to right helicoid, 381 nor mal to a congruence of Ribaucour, ; ; ; 422 Molding surfaces, a family 465 of Lame" of, representation, 141, 173, 393; reduced form, 353 of space, 447 Lines of length zero. See Minimal lines Lines of shortest length, 212, 220 Liouville (form of Gauss equation), 187 (angle of geodesic contingence), 212 Liouville, surfaces of, 214, 215, 218, 232 ; ; (equations of a surface), 64 (molding surfaces), 302 Monge, surfaces of, 305-308, 319 Moving trihedral for a curve, 30-33 applications of, 33-36, 39, 40, 64-68 Moving trihedral for a surface, 166-170 Monge ; ; ; Loxodromic curve, 140, 209 78, 108, 112, 120, 131, rotationsof, 169; applications of, 171183, 281-288, 336-338, 352-364, 426-442 Mainardi, equations of, 156 of Normal, principal, allel to definition, 12; 16, par Malus and Dupin, theorem of, 403 v. Mangoldt (geodesies on surfaces a plane, ; 21 Normal congruence gruence) of lines (see Con Mean Mean Mean positive curvature), 212 curvature, 123, 126, 145 evolute, 165, 166, 372 ruled surfaces of a congruence, Normal curvature Curvature of curves (see Congruence) of a surface. See 422, 423, 425 Mercator chart, 109 Meridian, of a surface of revolution, 107; of a helicoid, 146 * Normal plane to a curve, 8, 15, 65 Normal section of a surface, 118, 234 Normal to a curve, 12 Normal to a surface, 57, 114, 117, 120, 121, 141, 195 p. 467. See footnote, 472 INDEX ; Normals, principal, which are principal normals of another curve, 41 which are binormals of another curve, 51 Order of contact, 8, 21 Orthogonal system of curves, ; 75, 77, 80-82, 91, 119, 129, 177, 187; par ametric, 75, 93, 122, 134 geodesies, 187 isothermic (see Iso thermic) Orthogonal trajectories, of a one-param of a eter family of planes, 35, 451 family of curves, 50, 79, 95, 112, 147, of a family of geodesies, 149, 150 216 of a family of surfaces, 446, 451, 452, 456, 457, 460, 463, 464 ; ; Point of a surface, singular, 71 elliptic, 125, 200 hyperbolic, 125, 200 para middle bolic, 125; focal (see Focal) (see Middle); limit (see Limit) Polar developable, 64, 65, 112, 209 Polar line of a curve, 15, 38, 46 Principal directions at a point, 121 Principal normal to a curve. See Normal Principal planes of a congruence, 396, 397 ; ; ; ; Principal radii of normal curvature, 120, 291, 450 lit), ; Principal surfaces of a congruence, 390398, 408 Projective transformation, preserves os culating planes, 49 preserves asymp totic lines and conjugate systems, 202 Pseudosphere, 274, 290 Pseudospherical congruence, 415, 416, 464 normal, 184 ; ; Osculating circle, 14, 21, 65 Osculating plane, definition, 10 ; ; ; ; equa tion of, 11 stationary, 18 meets the curve, 19 passes through a fixed point, 22 orthogonal trajectories of, 35 of edge of regression, 57 of an asymp of a geodesic, 133 totic line, 128 Osculating planes of two curves parallel, ; ; ; ; ; 50 Osculating sphere, 37, 38, 47, 51, 65 Parabolic point on a surface, 125 Parabolic type, of pseudospherical sur of surfaces of Bianchi, faces, 274 ; Pseudospherical surface, definition, 270 asymptotic lines, 190, 290, 414 lines of curvature, 190, 203, 280, 320 geo defor desies, 275-277, 283, 317, 318 mation, 277, 323 transformations of, 280-290, 318, 320, 370, 45(5 of Dini, 291, 318; of Enneper, 317, 820; evo lute, 318 involute, 318 surfaces with ; ; ; ; ; ; ; ; 370, 371, 442, 443, 445 Paraboloid, a right conoid, 56 tangent plane, 112 asymptotic lines, 191, 233; a surface of translation, 203 equa fundamental quanti tions, 230, 330 lines of curvature, 232, 240 ties, 231 evolute of, 234 of normals to a ruled line of striction, 268 surface, 247 ; ; the same spherical representation of their lines of curvature as, 320, 371, 437, 439, 443, 444. See Surface of ; ; ; ; ; constant total curvature Pseudospherical surface of revolution, of elliptic of hyperbolic type, 273 of parabolic type, 274 type, 274 Pseudospherical surfaces, a family of Lam6 of, 452-456, 464 ; ; ; ; deformation of, 348, 349, 367-369, 372 congruence of tangents, 401. See Quadrics on a surface Parallel, geodesic, 86, 207 ; ; of revolution, 107 Parallel curves, 44 lines Parallel surface, definition, 177 of curvature, 178 fundamental quan cur tities, 178; of surface of constant vature, 179 of surface of revolution, ; Quadratic form. See Fundamental Quadrics, confocal, 226,401 fundamen tal quantities, 229 lines of curvature, asymptotic lines, 233 233, 239, 240 geodesies, 234-236, 239, 240 associate normals to, 422. surfaces, 390, 391 See Ellipsoid, Hyperboloid, Paraboloid ; ; ; ; ; ; ; ; 185 Parallel surfaces, a family of Lame" of, 446 Parameter, definition, 1 of distribution, 245, 247, 268, 348, 424, 425 Parametric curves, 54, 55 Parametric equations. See Equations Plane curve, condition for, 2, 16 curv ature, 15 equations, 28, 49 intrinsic ; ; ; ; equations, 36 Plane curves forming a conjugate sys tem,. 224 Plane lines of curvature. See Lines of curvature Representation, conformal (see Conformal); isometric, 100, 113; equiv alent, 113, 188; Gaussian, 141; conformal-con jugate, 224 geodesic, 225, 317 spherical (see Spherical) Revolution, surfaces of. See Surface Ribaucour (asymptotic lines on surfaces of center), 184 (cyclic systems of equal circles), 280; (limit surfaces), 389 (middle envelope of an isotropic (cyclic systems), congruence), 413 (deformation of the 426, 428, 432 of the planes of a cyclic envelope (cyclic systems system), 429, 430 associated with a triply orthogonal system), 457 ; ; ; ; ; ; ; INDEX Ribaucour, congruence of, 420-422, 424, 425, 435, 442, 443 triple systems of, 452, 455, 463 Riccati equation, 25, 26, 50, 248, 429 Rodrigues, equations of, 122 * Ruled surface, definition, 241 of tan gents to a surface, 188 generators, ; ; ; 473 Spherical representation of an axis of a moving trihedral, 354 241; directrix, 241; linear element, line of 241, 247 director-cone, 241 striction, 243, 244, 248, 268, 348, 351, 352, 309, 401, 422 central point, 243 central plane, 244 parameter of dis tribution, 245, 247, 208, 348, 424, 425 doubly, 234; normals to, 195, 247; total tangent plane, 246, 247, 268 ; ; ; ; ; Spherical surface, definition, 270; par allels to, 179 of revolution, 270-272 geodesies, 275-279, 318 deformation, 323 lines of curvature, 278 276, invo transformation, 278-280, 297 of Enneper, 317 surface lute, 300 with the same spherical representation of its lines of curvature as, 338. See Surface of constant total curvature Spherical surfaces, a family of Lame" of, ; ; ; ; ; ; ; ; ; 463 Spiral surface, definition, 151 gener lines of curvature, 151 ation, 151 minimal lines, 151 asymptotic lines, 151 geodesies, 219 deformation, 349 ; ; ; ; curvature, 247 asymptotic lines, 248250 mean curvature, 249 lines of curvature, 250, 268 conjugate, 268 deformation, 343-348, 350, 367; spher ical indicatrix of, 351; infinitesimal of a congruence, deformation, 381 393-395, 398, 401, 422, 423. See Right conoid, Hyperboloid, Paraboloid ; ; ; ; ; ; ; ; ; Scheffers (equations of a curve), 28 Scherk, surface of, 260 Schwarz, formulas of, 264-267, 269 Singular line of a surface, 71 Singular point of a surface, 71 Sinistrorsum, 19 minimal Sphere, equations, 62, 77, 81 conformal representation, lines, 81 ; ; 109-111 equivalent representation, 113 fundamental quantities, 116, 171; principal radii, 120; asymptotic lines, 223, 422 Spheres, family of. See Family Spherical curve, 36, 38, 47, 50, 149, 314-316, 317, 319, 320, 465 Spherical indicatrix, of the tangents to a curve, 9, 13, 50, 177 of the binormals to a curve, 50, 177 of a ruled surface, 351 Spherical representation of a congruence, definition, 393 principal surfaces, 397, 408 developables, 409, 412-414, 422, 432-435, 437, 441 Spherical representation of a surface, fundamental quan definition, 141 lines tities, 141-143, 160-165, 173; ; ; ; Stereographic projection, 110, 112 Superosculating circle, 21 Superosculating lines on a surface. 187 t Surface, definition, 53 Surface, limit, 389 met \ Surface of center, definition, 179 by developables in a conjugate sys tem, 180, 181 fundamental quantities, 181, 182 total curvature, 183 asymp totic lines, 183, 184 lines of curva a curve, 186, 188, 308ture, 183, 184 314 developable, 186, 305-308 Surface of constant mean curvature, definition, 179 parallels to, 179 lines of curvature, 296-298 transforma tion, 297 deformation, 298 minimal curves, 318 Surface of constant total curvature, area of geodesic tri definition, 179 lines of angle, 219 geodesies, 224 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; curvature, 317; asymptotic lines, 317; See spherical representation, 372. Pseudospherical surface and Spheri cal surface ; ; ; of curvature, 143, 148, 150, 151, 188, 201, 253, 279, 280, 292, 296, 301, 302, 308, 314, 315, 320, 371, 387, 437, 442445 asymptotic lines, 144, 148, 191195, 254, 340, 390, 414 area of closed portion, 145 conjugate system, 200202, 257, 385 ; ; ; Surface of reference, 392 Surface of revolution, definition, 107 fundamental quantities, 107, 147 loxodromic curve (see Loxodromic) deformation, 108, 112, 147, 149, 260, 276, 277, 283, 326-331, 341, 349-350, 362-364, 369, 370, 372, 444; partic ular, 111, 160, 320 equivalent repre sentation, 113 lines of curvature, 126 parallel sur asymptotic lines, 131 faces, 185 geodesies, 20.5, 209, 224 Surface of translation, definition, 197, 198; equations, 197; asymptotic lines, 198; generators, 198, 203; deformation, 349, 350 associate surface, 381, 390; ; ; ; ; ; ; ; ; ; reference is to nondevelopable ruled surfaces. For developable ruled surfaces, see Developables. t For references such as Surface of Bianchi, see Bianchi. I Surfaces of center of certain surfaces are referred to under these surfaces. * Tim 474 congruence of tangents, 406 surface of a &gt;F-congruence, ; INDEX middle 422, 424 Surface with plane lines of curvature. See Lines of curvature Surface with spherical lines of curvature. See Lines of curvature Surface with the same spherical repre sentation of its lines of curvature as a pseudospherical surface. See Pseudospherical surface Surface with the same spherical repre sentation of its lines of curvature as a spherical surface. See Spherical sur face * Transformation, of curvilinear coordi of rectangular nates, 53-55, 73, 74 ; coordinates, 72 ; by reciprocal ; radii, 104, 196, 203 (see Inversion) ive (see Projective) project- Surfaces of revolution, a family of of, 451 Lame" Triply orthogonal system of surfaces, definition, 447; associated with a cyc lic system, 440 fundamental quan with one family of tities, 447-451 surfaces of revolution, 451, 452 of Kibaucour, 452, 403 of Bianchi, 452of Weingarten, 455, 454, 404, 465 transformation of, 462, 456, 403, 404 463 with one family of molding sur faces, 405 of cyclides of Dupin, 405 of isothermic surfaces, 405 ; ; ; ; ; ; ; ; ; Tangent plane to a surface, definition, 50, 114; equation, 57; developable sur meets the face, 67; distance to, 114 ; ; ; Umbilical point of a surface, definition, 120 of quadrics, 230, 232, 234, 230238, 240, 207 ; characteristic of, 126 surface, 123 is the osculating plane of asymptotic 128 Tangent surface of a curve, 41-44, 57; applicable to the plane, 101, 150 Tangent to a curve, 6, 7, 41), 50, 51) spherical indicatrix of, 9, 13, 50, 177 Tangent to a surface, 112 Tangential coordinates, 103, 104, 201 Tetrahedral surface, definition, 207 asymptotic lines, 207 deformation, 341 Tetrahedral surfaces, triple system of, 465 Tore, 124 Torsion, geodesic, 137-140, 174, 170 Torsion of a curve, definition, 10 radius of, 16, 17, 21; of a plane curve, 10 sign of, 19; constant, 60; of asymp totic line, 140 Tractrix, equations, 35 surface of revo line, ; Variation of a function, 82, 83 Voss, surface of, 341, 390, 415, 442, 443 W-congruence, 417-420, 422, 424 fundamental &gt;F-surface, definition, 291 quantities, 291-293 particular, 291, ; ; ; 300, 318, 319; spherical representation, 292; lines of curvature, 293; evolute, 294, 295, 318, 319; of Weingarten, 298, 424; ruled, 299, 319 Weierstrass (equations of a minimal sur face), 200 (algebraic minimal sur faces), 201 ; ; Weingarten 103 las), ; (tangential (geodesic ellipses coordinates). and hyperbo ; ; ; 214; (&gt;F-surfaces), 291, 292, 294 (infinitesimal deformation), 374, 387 (lines of curvature on an isothermic ; ; ; lution of, 274, 290 ; helicoid whose meridian is a, 291 as surface), 389 Weingarten, surface of, 298, 424 method of, 353-372 triple system of, 455, 450, 403, 464 ; * For references such Transformation of BJickluud, see Bitcklund. " A*tr* --- ^/Stotistks/Compiiter Science Library " BERKELEY LIBRARIES r UBfURY
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.
Below is a small sample set of documents:
Ohio State - PHILOSOPHY - 650
Ohio State - PHILOSOPHY - 650
N ICOLAS B OURBAKIElements of MathematicsAlgebra IChapters 1 - 3HERMANN, PUBLISHERS I N ARTS A N D SCIENCE293 rue Lecourbe, 75015 Paris, FrancevvADDISON- WESLEY PUBLISHING COMPANYAAdvanced Book Program Reading, MassachusettsOriginally published
Ohio State - PHILOSOPHY - 650
The Accelerated Learning HandbookA Creative Guide to Designing and Delivering Faster, More Effective Training Programsby Dave MeierMcGraw-Hill New York San Francisco Washington, D.C. Auckland Bogota Caracas Lisbon London Madrid Mexico City Milan Montre
Ohio State - IS - 443
Questioning the VeilThis page intentionally left blankQuestioning the VeilOpen Letters to Muslim WomenMarnia LazregPrinceton University Press Princeton and OxfordCopyright 2009 by Princeton University Press Published by Princeton University Press, 4
Ohio State - PHILOSOPHY - 650
Gender and Self in IslamGender and Self in Islam examines the theological, cultural, and social roots of hierarchical gender system in the Muslim communities and its impact on the constitution of the self. It traces the historical and contemporary patter
Ohio State - PHILOSOPHY - 650
Also available from Continuum:Donna Haraway: Live Theory , Jo se p h Fredric Jameson: Live,Schneider Theory I an Buchanan Gayatri Chakravorty Spivak: Live Theory, Mark Sanders Helene Cixo us : Live Theory, I an Blythe and Susan Sellers Jacques Derrida:
Ohio State - PHILOSOPHY - 650
BADIOU'S BEING AND EVENTA Reader's Guide CHRISTOPHER NORRIScontinuum."Continuum International Publishing GroupThe Tower Building 11 York Road London SEI 7NX80 Maiden Lane New York, Suite 704 NY 10038www.continuumbooks.comChristopher Norris 2009A
Ohio State - PHILOSOPHY - 650
PSYCHOANALYSISAND THE CHALLENGEOF ISLAMFethi BenslamaTranslated by Robert BonannoUniversity of Minnesota Press Minneapolis' LondonP oetry by Rainer Maria Rilke in chapter 1 is quoted from Rainer Maria Rilke, D uino Elegies, translated by Robert Hunt
Ohio State - PHILOSOPHY - 650
01_Bhattaprelims treb1/7/0813:34Page iDangerous Brown Men01_Bhattaprelims treb1/7/0813:34Page ii01_Bhattaprelims treb1/7/0813:34Page iiiDangerous Brown MenExploiting Sex, Violence and Feminism in the War on TerrorG A R G I B H AT TA C H A R
Ohio State - PHILOSOPHY - 650
BEYONDPROBABILITYGODS MESSAGE IN MATHEMATICSSeries 1: The Opening Statement of the Quran (The Basmalah)by ABDULLAH ARIK1992 MONOTHEIST PRODUCTIONS INTERNATIONAL Tucson, ArizonaCONTENTSFOREWORD INTRODUCTION A UNIQUE BOOK A No Nonsense Scripture
Ohio State - PHILOSOPHY - 650
Codes and Curves Judy L. WalkerAuthor address: Department of Mathematics and Statistics, University of Nebraska, Lincoln, NE 68588-0323 E-mail address : [email protected] Mathematics Subject Classication. Primary 11T71, 94B27; Secondary 11D45, 11
Ohio State - PHILOSOPHY - 650
Ohio State - PHILOSOPHY - 650
This page intentionally left blank.Lacan to the LetterReading Ecrits CloselyBruce FinkUniversity of Minnesota PressMinneapolis LondonAn earlier version of chapter 6 was published as "Knowledge and Jouissance," in Reading Seminar XX: Lacan's Major Wo
Ohio State - PHILOSOPHY - 650
Ohio State - PHILOSOPHY - 650
Ohio State - PHILOSOPHY - 650
A HISTORY OF WESTERN PHILOSOPHYBertrand RussellA HISTORY OF WESTERN PHILOSOPHYand its Connection with Political and Social Circumstances from the Earliest Times to the Present DayOnly for education purpose "If you like it buy it"KarnavaLONDON GEORGE
Ohio State - PHILOSOPHY - 650
How to Develop A SUPER-POWER MEMORYb y Harry LorayneA . THOMAS & CO. PRESTONContents Foreword How Keen Is Your Observation?Does what you see register in your mind? Which light is on top of the traffic light? Is the number six on your watch dial, the A
Ohio State - PHILOSOPHY - 650
MADSENFROM CALCULUS TO COHOMOLOGY
Ohio State - PHILOSOPHY - 650
Mathematical Communications 2(1997), 1121Thirty years of shape theorySibe Mardeic sAbstract. The paper outlines the development of shape theory since its founding by K. Borsuk 30 years ago to the present days. As a motivation for introducing shape the
Ohio State - PHILOSOPHY - 650
Power Up Your Mind: Learn faster, work smarterBill LucasNICHOLAS BREALEY PUBLISHINGPower Up Your MindLearn faster, work smarterBill LucasNICHOLASBREALEYPLUBLISHINGO N D O NFirst published by Nicholas Brealey Publishing in 2001 Reprinted (twi
Ohio State - PHILOSOPHY - 650
/SYS21/INTEGRA/BST/VOL1/REVISES 21-7-2001/BSTA01.3D 1 [124/24] 26.7.2001 4:11PMBasic Ship Theory/SYS21/INTEGRA/BST/VOL1/REVISES 21-7-2001/BSTA01.3D 2 [124/24] 26.7.2001 4:11PM/SYS21/INTEGRA/BST/VOL1/REVISES 21-7-2001/BSTA01.3D 3 [124/24] 26.7.2001 4:11
Ohio State - PHILOSOPHY - 650
Ohio State - PHILOSOPHY - 650
M Y S T I C, G E O M E T E R, A N D I N T U I T I O N I S TThis page intentionally left blankM Y S T I C, G E O M E T E R, AND INTUITIONISTThe Life of L. E. J. Brouwer18811966Volume 2Hope and DisillusionDIRK VAN DALEN Department of Philosophy Utrec
Ohio State - PHILOSOPHY - 650
Ohio State - PHILOSOPHY - 650
General Certificate of Education (International) Advanced Level and Advanced Subsidiary Level MATHEMATICS 9709 For examination in June and November 2010SyllabusCIE provides syllabuses, past papers, examiner reports, mark schemes and more on the internet
Ohio State - PHILOSOPHY - 650
History of Islamic Philosophy Henry CorbinTranslated by Liadain Sherrard with the assistance of Philip SherrardKEGAN PAUL INTERNATIONALLondon and New York in association withISLAMIC PUBLICATIONSforTHE INSTITUTE OF ISMAILI STUDIESLondonThe Institut
Ohio State - PHILOSOPHY - 650
BadiouThis page intentionally left blankBadioua subject to truthPeter HallwardForeword by Slavoj iekUniversity of Minnesota PressMinneapolis / LondonCopyright 2003 by the Regents of the University of Minnesota All rights reserved. No part of this
Ohio State - PHILOSOPHY - 650
The world is in a state of constant upheaval and the pressures of our daily life cause us constant stress and strain. We find it difficult to control our thoughts, to still our minds, and to achieve an inner calm. This book, which has run into many editio
Ohio State - PHILOSOPHY - 650
Ohio State - PHILOSOPHY - 650
REVELATIONS REVELATIONS OF OF THE UNSEEN(FUTUH AL GHAIB)ABD AL-QADIR AL-JILANI1CONTENTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 Page On the essential tasks of every believ
Ohio State - PHILOSOPHY - 650
Ohio State - PHILOSOPHY - 650
Ohio State - PHILOSOPHY - 650
Morehead State - IET - 19575
Homework 3 Due date: 02/09/2010, 10:00 PM EST Maximum points: 101.(4 points) Suppose that we are testing H0: = 0 versus H1: > 0 with a sample size of n = 15. Calculate bounds on the P-value for the following observed values of the test statistic: p-Valu
Morehead State - IET - 19575
Homework 2 Due date: 02/01/2010, 10:00 PM EST Maximum points: 101.(5 points) Suppose that we are testing H0: = 0 versus H1: 0. Calculate the P-value for the following observed values of the test statistic: From pages 47 & 612 Given (a, (b) (c) (d) (e) Z
Morehead State - IET - 19575
Homework 1 IET 421 Due date: 01/22/2010, 10:00 PM EST1.(5 points) Suppose that you want to investigate the factors that potentially affect cooking rice. (a) What would you use as a response variable in this experiment? How would you measure the response
Aarhus Universitet - MIT - 2
A ir pollut icion in M exicoM exico Cit y once t opped list s of places wit h t he wor st air pollut ion in t he w or ld. Alt hough effor t s t o cur b emissions have impr oved t he sit uat ion, t iny par t icles called aer osols st ill clog t he air . N
Southwestern - BIO - BIOL 100.7
Sample Assessment Materials September 2007GCE BiologyEdexcel Advanced Subsidiary GCE in Biology (8BI01)First examination 2009Edexcel Advanced GCE in Biology (9BI01)First examination 2010Edexcel GCE e-SpecYour free e-SpecEverything you need in one
Southwestern - BIO - BIOL 100.7
Sample Assessment Materials For international centres onlyGCE BiologyEdexcel Advanced Subsidiary GCE in Biology (8BI07)First examination 2009Edexcel Advanced GCE in Biology (9BI07)First examination 2010International Alternative to Internal Assessmen
Southwestern - BIO - BIOL 100.7
Write your name hereSurname Other namesCentre NumberCandidate NumberEdexcel GCEBiologyAdvanced Subsidiary Unit 1: Lifestyle, Transport, Genes and HealthTuesday 12 January 2010 Morning Time: 1 hour 30 minutesYou do not need any other materials.Pap
Southwestern - BIO - BIOL 100.7
Write your name hereSurname Other namesCentre NumberCandidate NumberEdexcel GCEBiologyAdvanced Subsidiary Unit 1: Lifestyle, Transport, Genes and HealthThursday 8 January 2009 Morning Time: 1 hour 15 minutesYou do not need any other materials.Pap
Southwestern - BIO - BIOL 100.7
Write your name hereSurname Other namesCentre NumberCandidate NumberEdexcel GCEBiologyAdvanced Subsidiary Doughboy Unit 2: Development, Plants and the EnvironmentTuesday 19 January 2010 Afternoon Time: 1 hour 30 minutesYou do not need any other ma
Massey Palmerston North - COMPUTER S - 159233
159233: Assignment 1 2010Due: Worth: Minimum required mark: Penalty per day late: 20/4/2010 20% 8/20 1 markEvolution has a dark side. In the real world, it operates on finches, eyeballs, and pentadactyl limbs, but in the world of fry, evolution works on
UC Irvine - MATH - 93849
Wisconsin Milwaukee - EE - 890
%1DParticleinaBox% % clearall clc closealln=100;0umberofdiscretizedsegments a=10*10^9;%lengthofthebox[m] deltax=a/n;%legnthofdiscretizedsegment x=0:deltax:a; u=0;%potentialenergy h=6.624*10^34;%planckconstant[J*s] hbar=h/(2*pi);%reducedplanckconstant m=9
University of Advancing Technology - FINANCE - 342
TABLE OF CONTENTSI. THE AUDITING PROFESSION. 1. The Demand for Audit and Other Assurance Services. 2. The CPA Profession. 3. Audit Reports. 4. Professional Ethics. 5. Legal Liability. THE AUDIT PROCESS. 6. Audit Responsibilities and Objectives. 7. Audit
University of Advancing Technology - FINANCE - 342
CHAPTER 2Multiple-Choice Questions1. easy a Which one of the following is not one of the three General Standards? a. Proper planning and supervision. b. Independence of mental attitude. c. Adequate training and proficiency. d. Due professional care. Whi
University of Advancing Technology - FINANCE - 342
CHAPTER 3Note to Instructors: Unless otherwise indicated in the text of a question, please assume that a question applies to audits of public and private companies. Questions that relate only to publiccompany matters will be noted by the phrase (Public)
University of Advancing Technology - FINANCE - 342
CHAPTER 5Multiple-Choice Questions1. easy d While performing services for their clients, professionals have a duty to provide a level of care which is: a. free from judgment errors. b. superior. c. greater than average. d. reasonable. Auditors who fail
University of Advancing Technology - FINANCE - 342
CHAPTER 8Multiple-Choice Questions1. easy a Which of the following is not one of the three main reasons why the auditor should properly plan engagements? a. To enable proper on-the-job training of employees. b. To enable the auditor to obtain sufficient
University of Advancing Technology - FINANCE - 342
CHAPTER 9Multiple-Choice Questions1. easy a If it is probable that the judgment of a reasonable person would have been changed or influenced by the omission or misstatement of information, then that information is, by definition of FASB Statement No. 2:
University of Advancing Technology - FINANCE - 342
CHAPTER 10Multiple-Choice Questions1. easy a Which of the following is responsible for establishing a private companys internal control? a. Management. b. Auditors. c. Management and auditors. d. Committee of Sponsoring Organizations. Which of the follo
University of Advancing Technology - FINANCE - 342
CHAPTER 11Multiple-Choice Questions1. easy b Which of the following best defines fraud in a financial statement auditing context? a. Fraud is an unintentional misstatement of the financial statements. b. Fraud is an intentional misstatement of the finan
University of Advancing Technology - FINANCE - 342
CHAPTER 12Multiple-Choice Questions1. easy d IT has several significant effects on an organization. Which of the following would not be important from an auditing perspective? a. Organizational changes. b. The visibility of information. c. The potential
University of Advancing Technology - FINANCE - 342
CHAPTER 13Multiple-Choice Questions1. easy b A listing of all the things which the auditor will do to gather sufficient, competent evidence is the: a. audit strategy. b. audit program. c. audit procedure. d. audit risk model. Shown below (1 through 5) a
University of Advancing Technology - FINANCE - 342
CHAPTER 14Multiple-Choice Questions1. easy d Which of the following is not an account affected by the sales and collection cycle? a. Cash b. Accounts receivable c. Allowance for doubtful accounts d. Gross margin Which of the following is not one of the
University of Advancing Technology - FINANCE - 342
CHAPTER 15Multiple-Choice Questions1. easy b A sample in which the characteristics of the sample are the same as those of the population is a(n): a. variables sample. b. representative sample. c. attributes sample. d. random sample. When the auditor dec
McMaster - ECON - 1bb3
McMaster - ECON - 1bb3
McMaster - ECON - 1bb3
McMaster - ECON - 1bb3 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9319483637809753, "perplexity": 1292.1963652613858}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163044524/warc/CC-MAIN-20131204131724-00060-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://www.aanda.org/articles/aa/abs/2005/09/aa2038/aa2038.html | Free access
Issue A&A Volume 431, Number 3, March I 2005 Page(s) 793 - 812 Section Cosmology (including clusters of galaxies) DOI http://dx.doi.org/10.1051/0004-6361:20042038
A&A 431, 793-812 (2005)
DOI: 10.1051/0004-6361:20042038
## Properties of Ly emitters around the radio galaxy MRC 0316-257 ,
B. P. Venemans1, H. J. A. Röttgering1, G. K. Miley1, J. D. Kurk2, C. De Breuck3, R. A. Overzier1, W. J. M. van Breugel4, C. L. Carilli5, H. Ford6, T. Heckman6, L. Pentericci7 and P. McCarthy8
1 Sterrewacht Leiden, PO Box 9513, 2300 RA, Leiden, The Netherlands
e-mail: [email protected]
2 INAF, Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125, Firenze, Italy
3 European Southern Observatory, Karl Schwarzschild Straße 2, 85748 Garching, Germany
4 Lawrence Livermore National Laboratory, PO Box 808, Livermore CA, 94550, USA
5 NRAO, PO Box 0, Socorro NM, 87801, USA
6 Dept. of Physics & Astronomy, The Johns Hopkins University, 3400 North Charles Street, Baltimore MD, 21218-2686, USA
7 Dipartimento di Fisica, Università degli studi Roma Tre, via della Vasca Navale 84, Roma, 00146, Italy
8 The Observatories of the Carnegie Institution of Washington, 813 Santa Barbara Street, Pasadena CA, 91101, USA
(Received 21 September 2004 / Accepted 21 October 2004 )
Abstract
Observations of the radio galaxy MRC 0316-257 at z = 3.13 and the surrounding field are presented. Using narrow- and broad-band imaging obtained with the VLT, 77 candidate Ly emitters with a rest-frame equivalent width of >15 Å were selected in a ´ field around the radio galaxy. Spectroscopy of 40 candidate emitters resulted in the discovery of 33 emission line galaxies of which 31 are Ly emitters with redshifts similar to that of the radio galaxy, while the remaining two galaxies turned out to be [ ] emitters. The Ly profiles have widths (FWHM) in the range of 120-800 km s -1, with a median of 260 km s -1. Where the signal-to-noise was large enough, the Ly profiles were found to be asymmetric, with apparent absorption troughs blueward of the profile peaks, indicative of absorption along the line of sight of an mass of at least M. Besides that of the radio galaxy and one of the emitters that is a QSO, the continuum of the emitters is faint, with luminosities ranging from 1.3 L* to <0.03 L*. The colors of the confirmed emitters are, on average, very blue. The median UV continuum slope is , bluer than the average slope of LBGs with Ly emission ( ). A large fraction of the confirmed emitters ( 2/3) have colors consistent with that of dust-free starburst galaxies. Observations with the Advanced Camera for Surveys on the Hubble Space Telescope show that the emitters that were detected in the ACS image have a range of different morphologies. Four Ly emitters ( 25%) were unresolved with upper limits on their half light radii of kpc, three objects ( 19%) show multiple clumps of emission, as does the radio galaxy, and the rest ( 56%) are single, resolved objects with kpc. A comparison with the sizes of Lyman break galaxies at suggests that the Ly emitters are on average smaller than LBGs. The average star formation rate of the Ly emitters is 2.6 M as measured by the Ly emission line or <3.9 M as measured by the UV continuum. The properties of the Ly galaxies (faint, blue and small) are consistent with young star forming galaxies which are still nearly dust free.
The volume density of Ly emitting galaxies in the field around MRC 0316-257 is a factor of 3.3+0.5-0.4 larger compared with the density of field Ly emitters at that redshift. The velocity distribution of the spectroscopically confirmed emitters has a dispersion of 640 km s -1, corresponding to a FWHM of 1510 km s -1, which is substantially smaller than the width of the narrow-band filter ( km s -1). The peak of the velocity distribution is located within 200 km s -1 of the redshift of the radio galaxy. We conclude that the confirmed Ly emitters are members of a protocluster of galaxies at . The size of the protocluster is unconstrained and is larger than Mpc 2. The mass of this structure is estimated to be > M and could be the progenitor of a cluster of galaxies similar to e.g. the Virgo cluster.
Key words: galaxies: active -- galaxies: high-redshift -- galaxies: evolution -- galaxies: clusters: general -- cosmology: observations -- cosmology: early Universe | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8962082266807556, "perplexity": 4093.1034968949302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368699238089/warc/CC-MAIN-20130516101358-00024-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://dsp.stackexchange.com/questions/61820/are-all-lti-systems-invertible-if-not-what-is-a-good-counterexample/61834 | # Are all LTI systems invertible? If not, what is a good counterexample?
I have been trying to figure this out for a while now. Everywhere I have looked I could easily find examples of invertible LTI systems, but I could not find any counterexamples. Can anybody shed some light on this for me?
• What do you mean by invertible? Is this continuous or discrete? – copper.hat Nov 11 '19 at 5:07
• i would say that, if stability remains a requirement for an LTI system to be invertable, then any continuous-time LTI system, $H(s)$, with any zeros in the right half-plane is not invertable. nor would any discrete-time LTI system, $H(z)$, having zeros outside the unit circle be invertable. – robert bristow-johnson Nov 11 '19 at 19:12
• @copper.hat: They probably mean a system that can transform the output back into the input. – user541686 Nov 11 '19 at 22:50
You need to define what you mean by "invertible". Do you mean invertible by a causal and stable system? If yes, then any system that is not minimum-phase is not invertible (because the inverse system can't be causal and stable).
Example of a system that cannot be inverted by a causal and stable system: a simple delay $$y(t)=x(t-T)$$, $$T>0$$, could only be inverted by a non-causal system.
If by "invertible" you mean a system that can be inverted by a system that is possibly not causal and/or not stable, then it's still straightforward to find systems that can't be inverted according to this broader criterion: just take a system with one or more zeros in its frequency response. The information about the input signal at the frequencies where the system's frequency response is zero cannot be recovered by any other system.
Example of a system that cannot be inverted by any system: an ideal low-pass filter cannot be inverted by any system, because any information above the filter's cut-off frequency is lost and cannot be recovered.
EDIT: the latter definition of invertibility (where causality and stability of the inverse system are ignored) is just the definition of an injective mapping. In terms of systems this means that no two distinct input signals generate the same output signal. This category is what is also described in Laurent Duval's answer. Note, however, that in practice we usually prefer inverse systems that can also be realized, i.e., we require stability and causality, and then the first definition of invertibility given above is appropriate.
A necessary condition for invertibility is that any output has only one possible input (or injectivity, as proposed in comments). Since we are looking at counterexamples, we can look at when this condition is not satisfied.
The null system, that turns every signal into a zero flat line, is not invertible, but a bit trivial.
A system that computes a discrete derivative, eg with impulse response $$[1 \, -1]$$, is not invertible in that sense, as for any signal $$s[n]$$, and constant $$c$$, all signals $$s[n]+c$$ have the same output discrete derivative.
On the theoretical side, since LTI systems are characterized by their eigenvectors/eigenvalues, an LTI system with a zero eigenvalue lacks one necessary condition for being invertible.
Matt's answer (and his former one How to determine if the system is invertible) is more detailed in terms of "realizability".
• I'm sure you know this, but: what you're describing is actually just injectivity. For invertibility you also need surjectivity, though for linear ℝⁿ-endomorphisms that already follows from injectivity. – leftaroundabout Nov 11 '19 at 10:30
• You are very right. I made classical the shortcut that an injective function is bijective on its image Is every injective function invertible? – Laurent Duval Nov 11 '19 at 10:38
Whether LTI or not all systems are invertible if
unique (distinct) inputs produce unique (distinct) outputs
Causality and stability are later concerns for making sense of the obtained inverse system.
For example the inverse to the delay system
$$y[n] = x[n-d]$$
is
$$y[n] = x[n+d]$$
Which is clearly noncausal for $$d > 0$$, and is not realizable in real-time processing. But this system will be perfectly implemented in on offline audio processing console. Furthermore it can even be implemented in real-time image processing where the index is not time but space and causality does not restrict realizbility.
If you are into stable realizations (which makes sense), then your inverse system should include the unit circle in its Fourier transform. One simple conclusion for FIR filters is that if the forward filters frequency responses include zeros on the unit circle then the inverse filter will be unstable.
The following non-LTI system
$$y[n] = g[n] x[n]$$
is invertible iff $$g[n] \neq 0$$ for all $$n$$. Otherwise it's not invertible if for a set of $$n$$ $$g[n] = 0$$.
The nonlinear system
$$y[n] = x[n]^2$$
is not invertible, whereas the nonlinear system
$$y[n] = x[n]^3$$
is invertible.
For LTI systems with rational transfer functions
$$H(Z) = \frac{B(z)}{A(z)}$$
the inverse system will be
$$H_i(z) = \frac{1}{H(z)} = \frac{A(z)}{B(z)}$$
Poles and zeros are replaced and stability and causality of inverse system is analysed based on these new poles.
In addition to all the answers that are correct in a mathematical sense, in a practical sense, a system whose frequency response goes below some finite but small-enough value will not be usefully invertable, even if a simple mathematical analysis would suggest that it is.
In frequency-domain terms, the frequency response of a system's inverse will have gain equal to the reciprocal of the system gain at each frequency point. In practical terms, once this gain gets too great, all you will be doing is amplifying noise rather than actually contributing to a reasonable estimate of the input signal.
• and these kind of considerations open a door of discussions into the huge body of system inversion... – Fat32 Nov 11 '19 at 20:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 18, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8816056251525879, "perplexity": 504.6107707355335}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400212959.12/warc/CC-MAIN-20200923211300-20200924001300-00036.warc.gz"} |
https://brilliant.org/problems/doesnt-matter-if-you-add-or-multiply/ | # One Equation, Five Variables?
$a + b + c + d + e = a \times b \times c \times d \times e$
How many unordered 5-tuples of positive integers are there which satisfy the above equation?
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9734970927238464, "perplexity": 2638.9478000084678}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038065903.7/warc/CC-MAIN-20210411233715-20210412023715-00580.warc.gz"} |
https://inquiryintoinquiry.com/2015/10/01/forgetfulness-of-purpose-%E2%80%A2-4/ | Forgetfulness Of Purpose • 4
Ashby now invites us to consider a series of games, beginning as follows.
11/3. Play and outcome. Let us therefore forget all about regulation and simply suppose that we are watching two players, R and D, who are engaged in a game. We shall follow the fortunes of R, who is attempting to score an a. The rules are as follows. They have before them Table 11/3/1, which can be seen by both:
$\begin{array}{cc|ccc} \multicolumn{5}{c}{\text{Table 11/3/1}} \\[4pt] & & & R & \\ & & \alpha & \beta & \gamma \\ \hline & 1 & b & a & c \\ D & 2 & a & c & b \\ & 3 & c & b & a \end{array}$
D must play first, by selecting a number, and thus a particular row. R, knowing this number, then selects a Greek letter, and thus a particular column. The italic letter specified by the intersection of the row and column is the outcome. If it is an a, R wins; if not, R loses.
I’ll pause the play here and give readers a chance to contemplate strategies.
Reference
• Ashby, W.R. (1956), An Introduction to Cybernetics, Chapman and Hall, London, UK. Republished by Methuen and Company, London, UK, 1964. Online.
This entry was posted in Anamnesis, Ashby, C.S. Peirce, Cybernetics, Memory, Peirce, Pragmata, Purpose, Systems Theory and tagged , , , , , , , , . Bookmark the permalink.
2 Responses to Forgetfulness Of Purpose • 4
1. The Hawk says:
Two points worth considering, should you find the time: 1) look at Russell Ackoff’s book “On Purposeful Systems,” which Eric Trist dealt with via humour via asking Russ: “Yes Russ, but what about porpoiseful systems?”, sending Russ to the world of systems thinking to find larger life. If you are still unsure of this point of Russ needing to move on take a look at page 94 of Russ’s book to see how he proposes to measure love, in that all purpose requires measurement. then, 2) I’d go easy on Ashby’s concepts of gaming, as he favoured strategic thinking. As my corporate advisement since 1992 argues: “get over strategic thinking.” A much more exciting and successful approach is via taking note of Anatole Rappaport’s attitude to gaming, e.g., Prisoner’s Dilemma. Their are alternatives to earth as humans have come to define it …ha..ha..
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 1, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8621712327003479, "perplexity": 4263.884617546782}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703531702.36/warc/CC-MAIN-20210123001629-20210123031629-00029.warc.gz"} |
http://www.abstract-talk.org/wp/?tag=max-flow | # Distributed Minimum Cut Approximation
Abstract:
We study the problem of computing approximate minimum edge cuts by distributed algorithms. We present two randomized approximation algorithms that both run in a standard synchronous message passing model where in each round, $O(log n)$ bits can be transmitted over every edge (a.k.a. the CONGEST model). The first algorithm is based on a simple and new approach for analyzing random edge sampling, which we call random layering technique. For any any weighted graph and any $\epsilon \in (0, 1)$, the algorithm finds a cut of size at most $O(\epsilon^{-1}\lambda)$ in $O(D) + \tilde{O}(n^{1/2 + \epsilon})$ rounds, where $\lambda$ is the minimum-cut size and the $\tilde{O}$-notation hides poly-logarithmic factors in $n$. In addition, using the outline of a centralized algorithm due to Matula [SODA '93], we present a randomized algorithm to compute a cut of size at most $(2+\epsilon)\lambda$ in $\tilde{O}((D+\sqrt{n})/\epsilon^5)$ rounds for any $\epsilon>0$. The time complexities of our algorithms almost match the $\tilde{\Omega}(D + \sqrt{n})$ lower bound of Das Sarma et al. [STOC '11], thus leading to an answer to an open question raised by Elkin [SIGACT-News '04] and Das Sarma et al. [STOC '11].
To complement our upper bound results, we also strengthen the $\tilde{\Omega}(D + \sqrt{n})$ lower bound of Das Sarma et al. by extending it to unweighted graphs. We show that the same lower bound also holds for unweighted multigraphs (or equivalently for weighted graphs in which $O(w\log n)$ bits can be transmitted in each round over an edge of weight $w$). For unweighted simple graphs, we show that computing an $\alpha$-approximate minimum cut requires time at least $\tilde{\Omega}(D + \sqrt{n}/\alpha^{1/4})$.
Guest: Mohsen Ghaffari (Electrical Engineering and Computer Science Department, Massachusetts Institute of Technology)
Host: Merav Parter
# Approximate Duality of Multicommodity Multiroute Flows and Cuts: Single Source Case
## Petr Kolman and Christian Scheideler
Abstract:
Given an integer h, a graph $G = (V, E)$ with arbitrary positive edge capacities and k pairs of vertices $(s_1,t_1),(s_2, t_2),\dots,(s_k,t_k)$, called terminals, an h-route cut is a set $F \subset E$ of edges such that after the removal of the edges in F no pair $s_i-t_i$ is connected by h edge-disjoint paths (i.e., the connectivity of every $s_i-t_i$ pair is at most $h-1$ in $(V, E \setminus F))$. The h-route cut is a natural generalization of the classical cut problem for multicommodity flows (take h = 1). The main result of this paper is an $O(h^{5}2^{2h}(h+log k)^2)$-approximation algorithm for the minimum h-route cut problem in the case that $s_1 = s_2 =\dots= s_k$, called the single source case. As a corollary of it we obtain an approximate duality theorem for multiroute multicommodity flows and cuts with a single source. This partially answers an open question posted in several previous papers dealing with cuts for multicommodity multiroute problems.
Guest: Christian Scheideler
Host: Shantanu Das | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 9, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9215931296348572, "perplexity": 579.9327078891214}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247504790.66/warc/CC-MAIN-20190221132217-20190221154217-00301.warc.gz"} |
http://math.stackexchange.com/questions/308652/what-does-this-notation-mean-fu | # What does this notation mean? F|U
If $F$ is a function and $U$ is a set, then what does $F|U$ mean?
In http://ocw.mit.edu/courses/mathematics/18-101-analysis-ii-fall-2005/lecture-notes/lecture6.pdf, the Inverse Function Theorem, the notation is used
Also, the file in stating the Lemma 2.14 states that $A \in U$, what does this mean? A set is an element of another set? What is $A$?
EDIT: Could someone kindly explain to me what is going on in Lemma 2.16. How does $\delta$ come into play? I dont see the role in the lemma. I also don't see why this proves one-to-one
-
Where in the file? – Asaf Karagila Feb 20 '13 at 1:11
As a note, this notation is also quite common in linear algebra to denote essentially the same thing: The restriction of a linear operator to an invariant subspace. – EuYu Feb 20 '13 at 1:15
I really really want to interpret your question a different way... – John Moeller Feb 20 '13 at 1:19
Also, for your last question, sets can be elements of other sets. That's not a problem. In fact you can encode most mathematics into sets, so you can always work under the scary assumption that everything is a set. – Asaf Karagila Feb 20 '13 at 1:19
It means $f$ restricted to the subset $U$. In the paper you look for a local diffeomorphism, so you restrict the function to a suitable subset.
Edit to address later added part of your question "What is $A$": $A$ is a typo. It should read $a$.
-
If $F: D \rightarrow V$ is a function and $U \subseteq D$, then $F|U$ is the function $f: U \rightarrow V: u \mapsto F(u)$.
-
So all is saying that this $F$ is really just $f: U to V$ map? – love Feb 20 '13 at 1:17
Yes, just $F$ but its domain restricted to $U$. – sxd Feb 20 '13 at 2:13
It is important to remember a function has three parts: a domain, a range and a rule tying each element of the domain to the range. So if $f: X\rightarrow Y$, and $A\subset X$, $f|A$ is just the function $f: A\rightarrow Y$ so that $(f|A)(x) = f(x)$ for all $x\in A$. When you change the range or domain of a function, you change the function. It's not just the rule!
-
$F\mid U$ is the function $F$ restricted to the set $U$, where $U$ is a subset of a larger domain on which $F$ is defined. The image of $F\mid U$ is often denoted by $F[U] = \{f(x): x \in U\}$.
If $A \in U$, then we are talking about $A$ being an element of the subset/set $U$, an element which may or may not be a set itself.
-
You will also sometimes see the notation $f\mid_U$ to denote the restriction of a function $f$ to the subset $U$. – amWhy Feb 20 '13 at 1:23
Also, sometimes there is a little hook on the bar (which I prefer): $f\!\upharpoonright U$ or $f\!\upharpoonright_U$. – Kundor May 1 '14 at 21:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9378613233566284, "perplexity": 235.15198315346822}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207925274.34/warc/CC-MAIN-20150521113205-00044-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-8-polynomials-and-factoring-mid-chapter-quiz-page-510/26 | ## Algebra 1: Common Core (15th Edition)
$15x+10$.
First, find the total area of the outer rectangle. $A=a\cdot b$ ...the area of a rectangle. ...substitute $(x+6)$ for $a$ and $(x+3)$ for $b$ in the formula. $A=(x+6)(x+3)$ ...use the FOIL method. $A=(x)(x)+(x)(3)+(6)(x)+(6)(3)$ ...simplify $A=x^{2}+3x+6x+18$ ...add like terms. $\color{red}{A=x^{2}+9x+18}$ ... (area of the outer rectangle.) Now find the area of the inner rectangle $A=a\cdot b$ ...the area of a rectangle. ...substitute $(x-2)$ for $a$ and $(x-4)$ for $b$ in the formula. $A=(x-2)(x-4)$ ...use the FOIL method. $A=(x)(x)+(x)(-4)+(-2)(x)+(-2)(-4)$ ...simplify $A=x^{2}-4x-2x+8$ ...add like terms. $A=\color{red}{x^{2}-6x+8}$ ... (area of the inner rectangle.) Finally, find the area of the shaded region. Area of shaded region=Area of outer rectangle - Area of inner rectangle $A=x^{2}+9x+18-(x^{2}-6x+8)$ $=x^{2}+9x+18-x^{2}+6x-8$ ...add like terms. $=\color{red}{15x+10}$ The area of the shaded region is $15x+10$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.809860110282898, "perplexity": 813.1005132556037}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038082988.39/warc/CC-MAIN-20210415005811-20210415035811-00049.warc.gz"} |
https://community.filemaker.com/thread/133237 | # Container with calculation resulting in file path does not display PDF on Windows
Question asked by [email protected] on Nov 10, 2010
Latest reply on Dec 8, 2010 by philmodjunk
### Summary
Container with calculation resulting in file path does not display PDF on Windows
FileMaker Pro
11.02
Window 7
### Description of the issue
I have created a container field with a calculation that resolves into a file path for a pdf in the same folder. The path is
image:filename.pdf
This works fine when used on a mac but when the files are located on a PC the display says "The file cannot be displayed:filename.pdf".
Where there is no file the message is "The file cannot be found:filename.pdf". Which is correct on both platforms.
Can't the PC display a PDF? If the file is a .jpg or other image it works. Does this mean Windows does not consider PDF files an image file?
I have tried a complete path and using imagewin etc but nothing works to display a PDF in a container by resolving a calculation into a file path.
Help | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.866569459438324, "perplexity": 2348.169028600044}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889567.48/warc/CC-MAIN-20180120102905-20180120122905-00289.warc.gz"} |
https://www.physicsforums.com/threads/how-close-to-light-speed-can-you-theoretically-get.218450/ | # How close to light speed can you theoretically get?
1. Feb 27, 2008
### Meatbot
Can you go so fast that after say one second, light has traveled less than a planck length further than you did (with respect to an outside observer of course)?
Is c the actual speed limit, or is the speed limit slightly less than c?
Maybe I'm not stating this properly and forgive me if not, but I think you know what I mean.
Last edited: Feb 27, 2008
2. Feb 27, 2008
### americanforest
As your speed increases your inertia also increases and it becomes harder and harder to accelerate you further.
$$m=m_0\gamma$$
$$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
$$F=ma=am_0\gamma$$
Last edited: Feb 27, 2008
3. Feb 28, 2008
### 1effect
A massive object can never achieve c.
Assume that the total energy at rest is $$E_0=m_0c^2$$
The energy when the object reached speed $$v$$ is $$E_1=\gamma m_0c^2$$
The total work expended is $$\Delta W =E_1-E_0=(\gamma-1)m_0c^2$$
For $$v->c$$ $$\Delta W$$ goes to infinity.
4. Feb 28, 2008
### pam
There is no known limit to gamma.
The Planck length is not a limit on anything.
5. Feb 28, 2008
### nanobug
Yes.
'c' is an unattainable limit for objects whose mass is not zero.
6. Feb 28, 2008
### Fredrik
Staff Emeritus
I think that's an interesting question actually.
The Planck length and related quantities aren't present in the theory of special relativity, so the answer within the framework of SR is clearly that the speed limit is exactly c.
Light travels 299792458 meters in one second. You're asking if it's possible to travel more than 299792458-lP in one second, in the universe we live in (as opposed to the one described by SR, where it certainly is possible since there's no Planck length). There's nothing special about a second, so we should be able to replace "one second" with any other unit of time in your question and still get the same answer. Let's choose "one Planck time". Since the speed of light is one Planck length in one Planck time, your question becomes "is it possible to travel more than zero Planck lengths in one Planck time"?
It's funny that when you break it down like that, it appears that 0 and c are the only possible speeds, but we know that's not the case, so there's definitely something strange going on here. Maybe speed in a quantum theory of space-time is the probability that we will "jump" a Planck length in a Planck time.
So I don't think anyone really knows the answer to your question, since there's no complete quantum theory of gravity. (A quantum theory of gravity would almost certainly also be a quantum theory of space-time). I wonder if the candidate theories like strings and loop quantum gravity have a clear answer to this question. Perhaps someone will tell us that in this thread. (Wink wink, nudge nudge).
Last edited: Feb 29, 2008
7. Feb 28, 2008
### Mentz114
Frederik,
trenchant analysis. A new Zeno paradox maybe ?
8. Feb 29, 2008
### DrGreg
Just imagine you are inside a spaceship travelling at the fastest possible speed less than c. You stand up and try to walk forward. Would you find some mysterious force preventing you from moving and thus breaking the "speed limit"? Of course not. So there can't be such a fastest speed.
I'm no expert on quantum theory, but I don't think it is right to think of the Planck length as being "the smallest possible distance". It's more like "the smallest distance you can measure" (and even that's probably an over-simplification).
Also, in quantum theory, it is usual to measure momentum rather than speed. There is no theoretical momentum limit.
You might get a better answer by asking this question in the Quantum Physics forum.
In the real Universe, there is a practical upper limit. The faster you go, the more energy you need, so eventually you would run out. So, to give a ludicrous example, your kinetic energy could never exceed the total energy of the whole Universe!
9. Feb 29, 2008
### Meatbot
That's pretty much what I was getting at, but you expressed it much more eloquently. It seemed that something odd was going on with this, but I didn't know how to express it. Nice answer. Perhaps 0 and c ARE the only speeds and it only appears that they aren't.
Formulated another way, is it possible to move 1/2 a planck length from your current position?
Last edited: Feb 29, 2008
10. Feb 29, 2008
### jlorda
How can mass of an object be = to zero?
How can the mass of an object be = to zero ? If mass is zero would it still exist? How can nothing be something? Does this mean that light can not be a particle ?
11. Feb 29, 2008
### nanobug
An object with zero mass may only exist if traveling at the speed of light. In this case, the object would show a nonzero relativistic mass equal to its kinetic energy. Example: photons.
12. Feb 29, 2008
### jlorda
So are you saying that it does have a relative mass? I'm not sure what you are saying.
13. Feb 29, 2008
### nanobug
It has a mass equivalent to it's kinetic energy, per Einstein's famous E=mc^2. If the kinetic energy is E then the relativistic mass of a massless object is m=E/c^2.
http://en.wikipedia.org/wiki/Mass_in_special_relativity
14. Feb 29, 2008
### jlorda
Relativistic mass is just another name for the energy? according to Wikipedia. So we know that mass is an expression of energy from e=mc^2? So if an object has mass of 0 then
0 = E/c^2 = ? Im trying to make sense of this.
15. Mar 1, 2008
### americanforest
$$m_{0}^{2}c^{4}\gamma^{2}=E^2=p^{2}c^{2}+m_{0}^{2}c^{4}$$
If mass=0 then energy equals momentum times the speed of light.
Last edited: Mar 1, 2008
16. Mar 1, 2008
### _Mayday_
17. Mar 2, 2008
### Fredrik
Staff Emeritus
Unfortunately questions like that are only well-defined within the framework of a theory, and we still don't have the theory we would need to even ask that question in a way that makes sense mathematically.
(The concept of "position" is well-defined e.g. when we're talking about classical point particles moving in a space-time that can be represented mathematically by a smooth manifold, but there's no reason to believe that space and time in the actual universe is anything like a smooth manifold on small scales).
Similar Discussions: How close to light speed can you theoretically get? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9037380814552307, "perplexity": 613.3343044242505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689373.65/warc/CC-MAIN-20170922220838-20170923000838-00561.warc.gz"} |
https://www.maplesoft.com/support/help/maple/view.aspx?path=Units/Natural/int | Units[Natural] - Maple Programming Help
Home : Support : Online Help : Science and Engineering : Units : Environments : Natural : Units/Natural/int
Units[Natural]
int
definite and indefinite integration in the Natural Units environment
Calling Sequence int(expr, x) int(expr, x=a..b)
Parameters
expr - algebraic expression x - name or name multiplied by a unit a, b - algebraic expressions
Description
• In the Natural Units environment, the int function integrates an expression with respect to a name that can have a unit. The result is the integral of the expression, with respect to the variable of integration, with a unit, the integrand unit multiplied by the variable of integration unit if any.
• Any endpoints must be unit-free and are assumed to have the units of the variable of integration.
• For other properties, see the global function int.
Examples
> $\mathrm{with}\left(\mathrm{Units}\left[\mathrm{Natural}\right]\right):$
> $-3.532{x}^{2}W$
${-}{3.532}{}{{x}}^{{2}}{}⟦{W}⟧$ (1)
> $\mathrm{int}\left(,xs\right)$
${-}{1.177333333}{}{{x}}^{{3}}{}⟦{J}⟧$ (2)
> $32{x}^{2}\mathrm{ft}+7x\mathrm{inch}+45m$
$\left(\frac{{6096}}{{625}}{}{{x}}^{{2}}{+}\frac{{889}}{{5000}}{}{x}{+}{45}\right){}⟦{m}⟧$ (3)
> $\mathrm{int}\left(,xm\right)$
$\left(\frac{{2032}}{{625}}{}{{x}}^{{3}}{+}\frac{{889}}{{10000}}{}{{x}}^{{2}}{+}{45}{}{x}\right){}⟦{{m}}^{{2}}⟧$ (4)
> $4{x}^{4}-3x+2$
${4}{}{{x}}^{{4}}{-}{3}{}{x}{+}{2}$ (5)
> $\mathrm{int}\left(,xs\right)$
$\left(\frac{{4}}{{5}}{}{{x}}^{{5}}{-}\frac{{3}}{{2}}{}{{x}}^{{2}}{+}{2}{}{x}\right){}⟦{s}⟧$ (6) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 13, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.97348552942276, "perplexity": 1201.6323793262427}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370520039.50/warc/CC-MAIN-20200404042338-20200404072338-00273.warc.gz"} |
https://astarmathsandphysics.com/a-level-physics-notes/materials/2837-the-anomalous-expansion-of-water.html?tmpl=component&print=1&page= | ## The Anomalous Expansion of Water
Most liquids contract when they solidify. The atoms in the liquid form bonds and move closer together.
If water at 0 degrees celsius is frozen – it becomes ice – then it expands and becomes less dense.
This is why ice floats – it is less dense than water. Pure water at 0 degrees Celsius has a density ofbut pure ice at 0 degrees celsius has a density of
Most liquids expand when they are heated. The molecules move faster, jostle each other more and are further apart on average. Water is strange in this respect.
If water at 0 degrees celsius is heated, it contracts and becomes less dense. It keeps contracting when heated until 4 degrees celsius, then it starts expanding.
This is important for life on Earth. It means the bottom of a lake is the last part to freeze, so fish can usually survive the winter. This may have been very important during ice ages, or the supposed peropd in the Earth's history when it was almost completely frozen – only a ten mile stretch either side of the equator remaining liquid.
It will also have meant liquid would have flowed more readily. Rain would not have collected on the surface of ice. Being denser, it would have flowed off the ice and collected eventually in the ocean. This may have made the survival of life possible in the above mentioned period when the Earth was almost completely frozen. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8726828694343567, "perplexity": 955.5337855354369}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105195.16/warc/CC-MAIN-20170818233221-20170819013221-00258.warc.gz"} |
http://www.physicsforums.com/showthread.php?s=443f0b2b5b60627d5498a5bfd0e6d26b&p=4656562 | # Lorentz transformation of delta function
by Chenkb
Tags: delta, delta function, function, lorentz, lorentz boost, transformation
P: 19 For two body decay, in CM frame, we know that the magnitude of the final particle momentum is a constant, which can be described by a delta function, ##\delta(|\vec{p^*}|-|\vec{p_0^*}|)##, ##|\vec{p_0^*}|## is a constant. When we go to lab frame (boost in z direction), what's the Lorentz transformation of the delta function? regards!
P: 263 What do you mean by "which can be described by a delta function" ?
P: 19
Quote by maajdl What do you mean by "which can be described by a delta function" ?
I mean that we can use a delta function to fix the momentum i.e. p=p0*.
Maybe my example of two body decay is not so suitable, but my question is just for mathematics, that is the Lorentz transformation of ##\delta(|\vec{p}|-|\vec{p_0^*}|)## | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9726601243019104, "perplexity": 748.9903514601262}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00378-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/289097/bell-numbers-how-to-put-egf-eex-1-into-a-series | Bell Numbers: How to put EGF $e^{e^x-1}$ into a series?
I'm working on exponential generating functions, especially on the EGF for the Bell numbers $B_n$.
I found on the internet the EGF $f(x)=e^{e^x-1}$ for Bell numbers. Now I tried to use this EGF to compute $B_3$ (should be 15). I know that I have to put the EGF into a series and have a look at the coefficients. Using $e^{f(x)}=1+f(x)+\frac{f(x)^2}{2!}+\frac{f(x)^3}{3!}+\ldots$ I get
\begin{eqnarray*} e^{e^x-1}&=&1+(e^x-1)+\frac{(e^x-1)^2}{2!}+\frac{(e^x-1)^3}{3!}\\ &=&1+e^x-1+\frac{e^{x^2}-2e^x+1}{2!}+\frac{e^{x^3}-3e^{x^2}+3e^x-1}{3!}\\ &=&e^x+\frac{e^{x^2}}{2!}-e^x+\frac{1}{2!}+\frac{e^{x^3}}{3!}-\frac{e^{x^2}}{2!}+\frac{e^{x}}{2!}-\frac{1}{3!}\\ &=&\frac{e^{x}}{2!}+\frac{e^{x^3}}{3!}-\frac{1}{2!}+\frac{1}{3!}\\ &=&\frac{1}{2!}e^x+\frac{1}{3!}e^{x^3}-\frac{1}{3}\\ &=&\frac{1}{2!}\left( 1+x^2+\frac{x^4}{2!}+\frac{x^8}{3!} \right)+\frac{1}{3!}\left( 1+x^3+\frac{x^6}{2!}+\frac{x^9}{3!}\right)\\ &=&\frac{1}{2}+\frac{x^2}{2}+\frac{x^4}{4}+\frac{x^8}{24}+\frac{1}{6}+\frac{x^3}{3!}+\frac{x^6}{12}+\frac{x^9}{36}\\ &=&\frac{1}{2}+\frac{x^2}{2}+\frac{x^3}{6}+\ldots \end{eqnarray*}
I think I can stop here, because the coefficient in front of $\frac{x^3}{3!}$ is not $15$.
Perhaps someone can help me out and give a hint to find my mistake?
-
$\left(\exp(x)\right)^2 = \exp(2 x) \not= \exp(x^2)$ – Sasha Jan 28 '13 at 17:25
$B_3=5$ and $B_4=15$. – Michael Hardy Jan 28 '13 at 17:26
@Michael Hardy: Yes you're right, but doesn't matter, because the coefficient in front of $\frac{x^4}{4!}$ is 15 neither. – ulead86 Jan 28 '13 at 17:27
@Sasha Thanks, I'll try it again. – ulead86 Jan 28 '13 at 17:28
It's probably easiest to expand the exponential in the exponent first, since that will lead to a finite number of terms to be evaluated:
\begin{align}e^{(e^x-1)} &= \exp\left(x+\frac{x^2}{2}+\frac{x^3}{6}+O(x^4)\right)\\ &=1+\left(x+\frac{x^2}{2}+\frac{x^3}{6}+O(x^4)\right) +\frac{1}{2}\left(x+\frac{x^2}{2}+\frac{x^3}{6}+O(x^4)\right)^2 +\frac{1}{6}\left(x+\frac{x^2}{2}+\frac{x^3}{6}+O(x^4)\right)^3+O(x^4)\\ &=1+\left(x+\frac{x^2}{2}+\frac{x^3}{6}+O(x^4)\right)+\frac{1}{2}\left(x^2+2(x)\left(\frac{x^2}{2}\right)+O(x^4)\right)+\frac{1}{6}\left(x^3+O(x^4)\right)\\ &=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^2}{2}+\frac{x^3}{2}+\frac{x^3}{6}+O(x^4)\\ &=1+x+x^2+\frac{5x^3}{6}+O(x^4) \end{align} From which the coefficients can be read off straightforwardly.
-
Thats nuch easier. Tanks a lot. – ulead86 Jan 28 '13 at 21:02
\begin{align} e^{e^x-1}&= 1+(e^x-1)+\frac{(e^x-1)^2}{2!}+\frac{(e^x-1)^3}{3!}+\cdots+\frac{(e^x-1)^n}{n!}+ \cdots \\[8pt] &= \cdots\cdots+\frac{e^{nx}+xe^{(n-1)x}+\binom n2 e^{(n-2)x}+\cdots+ne^x + 1}{n!}+ \cdots\cdots \end{align} One of the terms in the expansion of $(e^x-1)^n$ is $e^x$. When that is expanded, one of the terms will be $x^4/4!$. No matter how big $n$ is, you don't run out of terms involving $x^4$. So you don't just get finitely many terms involving $x^4$ and add up their coefficients to see if you get $\dfrac{15x^4}{4!}$. Rather, you get an infinite series.
The Wikipedia article titled Exponential formula gives a power series expansion of $$e^{f(x)}= \cdots+\frac{b_n x^n}{n!}+\cdots$$ when the power series expansion of $$f(x) = a_1+\frac{a_2 x^2}{2!}+\cdots+\frac{a_n x^n}{n!}+\cdots$$ is known. Notice this from the article: $$b_3 = a_3+3a_2 a_1 + a_1^3$$ because there is one partition of the set $\{ 1, 2, 3 \}$ that has a single block of size $3$, there are three partitions of $\{ 1, 2, 3 \}$ that split it into a block of size $2$ and a block of size $1$, and there is one partition of $\{ 1, 2, 3 \}$ that splits it into three blocks of size $1$. Apply that to $4$: $$b_4 = a_4 + 4a_3a_1 + 3a_2^2+6a_2a_1^2 + a_1^4$$ since there is one partition of the set $\{1,2,3,4\}$ that has a single block of size 4; there are four partitions into a block of size $3$ and a block of size $1$; three are three partitions into two blocks of size $2$; there are six partitions into a block of size $2$ and two blocks of size $1$; and there is one partition into four blocks of size $1$.
Wikipedia's article titled Dobinski's formula treats the expansion of each individual Bell number as an infinite series.
There is also Faà di Bruno's formula one the derivatives of composite functions.
-
Ok, but how do I find out the 3rd Bell number with EGF then? – ulead86 Jan 28 '13 at 17:41
I've added a link to a Wikipedia article that deals with this. – Michael Hardy Jan 28 '13 at 17:41
\begin{eqnarray*} e^{e^x-1}&=&1+(e^x-1)+\frac{(e^x-1)^2}{2!}+\frac{(e^x-1)^3}{3!}\\ &=&1+e^x-1+\frac{e^{2x}-2e^x+1}{2!}+\frac{e^{3x}-3e^{2x}+3e^x-1}{3!}\\ &=&e^x+\frac{e^{2x}}{2!}-e^x+\frac{1}{2!}+\frac{e^{3x}}{3!}-\frac{e^{2x}}{2!}+\frac{e^{x}}{2!}-\frac{1}{3!}\\ &=&\frac{e^{x}}{2!}+\frac{e^{3x}}{3!}-\frac{1}{2!}+\frac{1}{3!}\\ &=&\frac{1}{2!}e^x+\frac{1}{3!}e^{3x}-\frac{1}{3}\\ &=&\frac{1}{2!}\left( 1+x+\frac{x^2}{2!}+\frac{x^3}{3!} \right)+\frac{1}{3!}\left( 1+3x+\frac{9x^2}{2!}+\frac{27x^3}{3!}\right)-\frac{1}{3}\\ &=&\frac{1}{2}+\frac{x}{2}+\frac{x^2}{4}+\frac{x^3}{24}+\frac{1}{6}+x+\frac{3x^2}{4}+\frac{9x^3}{12}-\frac{1}{3}\\ &=&\frac{2}{3}+\frac{3x}{2}+\frac{5x^2}{4}+\frac{9x^3}{12}+\ldots \end{eqnarray*}
-
This gives the wrong value for every coefficient. – Steven Stadnicki Jan 28 '13 at 18:10
You probably don't want to use the series expansion for the exponential as you've done, because you will receive contributions to each coefficient from all orders of this expansion, so you'll still have infinite sums left to evaluate. Instead, you should use the fact that term $n$ in a sequence is given by $F^{(n)}(0)$, where $F(x)$ is the sequence's exponential generating function (and $F^{(n)}$ is the $n$-th derivative of $F$). If $F(x)=e^{f(x)}$, then its derivatives are given by $$F(0)=e^{f(0)} \\ F'(0)=f'(0)e^{f(0)} \\ F''(0)=(f''(0)+f'(0)^2)e^{f(0)} \\ F'''(0)=(f'''(0)+3f''(0)^2 f'(0) + f'(0)^3)e^{f(0)} \\ F^{(4)}(0)=(f^{(4)}(0)+f'''(0)f'(0)+6f'''(0)f''(0)f'(0)+3f''(0)^2+3f''(0)f'(0)^2+f'(0)^4)e^{f(0)}$$ and so on. If $f(x)=e^x-1$, then $e^{f(0)}=1$ and $f^{(n)}(0)=1$ for any $n \ge 1$. Adding up the terms, then, you have $B_0=B_1=1$, $B_2=2$, $B_3=5$, $B_4=15$, and so on.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9401801824569702, "perplexity": 291.3071309090848}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049281978.84/warc/CC-MAIN-20160524002121-00053-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/336391/relation-between-the-decomposition-invariants-of-a-projective-reduced-curve-and | # Relation between the decomposition invariants of a projective reduced curve and its normalization
Let $$X$$ be a reduced projective scheme over $$k$$ which is of pure dimension 1. Let $$\pi: X \to \mathbb{P}_k^1$$ be a finite (hence affine, surjective and flat) morphism of schemes having degree $$n$$. Since $$X$$ is Cohen-Macaulay, $$\pi_*\mathcal{O}_X$$ is a free $$\mathcal{O}_{\mathbb{P}_k^1}$$-module of finite rank $$n$$ and hence decomposes into a direct sum of Serre's twisted sheaves: $$\pi_*\mathcal{O}_X \cong \bigoplus_{i=1}^n \mathcal{O}_{\mathbb{P}_k^1}(|\mathcal{O}_X|_i)$$ The integers $$|\mathcal{O}_X|_1 \geq \ldots \geq |\mathcal{O}_X|_n$$ are uniquely determined by $$\mathcal{O}_X$$. Now we have a similar situation for the structure sheaf of any irreducible component $$X_i$$ of $$X$$: The closed immersion $$j_i: X_i \to X$$ is a finite morphism and hence $$\pi_i = \pi \circ j_i: X_i \to X \to \mathbb{P}_k^1$$ is also finite of degree $$n_i < n$$ if $$X_i \neq X$$. Hence $$(\pi_i)_*\mathcal{O}_{X_i} \cong \pi_* ((j_i)_* \mathcal{O}_{X_i}) \cong \pi_* (\mathcal{O}_X / \mathcal{I}_i) \cong \bigoplus_{j=1}^{n_i} \mathcal{O}_{\mathbb{P}_k^1}(|\mathcal{O}_{X_i}|_j)$$ where $$\mathcal{I}_{X_i}$$ denotes the ideal sheaf cutting out $$X_i$$ in $$X$$.
Since $$X$$ is reduced, we have a finite morphism $$X' = \bigoplus_{i=1}^m X'_i \to X$$ where $$X'$$ is the normalization of $$X$$ (and $$X_i'$$ is the normalization of the component $$X_i$$) and a corresponding injective morphism $$\mathcal{O}_X \hookrightarrow \bigoplus_{i=1}^n \mathcal{O}_{X_i}/\mathcal{I}_i$$ with finite index $$\chi(\mathcal{S})$$ where $$\mathcal{S}$$ makes the following sequence exact $$0 \to \mathcal{O}_X \hookrightarrow \bigoplus_{i=1}^n \mathcal{O}_{X_i}/\mathcal{I}_i \to \mathcal{S} \to 0.$$ It is not hard to see that $$\pi_* \mathcal{O}_X \hookrightarrow \pi_*\left( \bigoplus_{i=1}^n \mathcal{O}_{X_i}/\mathcal{I}_i\right) \cong \bigoplus_{i=1}^{m} \bigoplus_{j=1}^{n_i} \mathcal{O}_{\mathbb{P}_k^1}(|\mathcal{O}_{X_i}|_j).$$
My question is: What are the relations between $$L_{X'} := (|\mathcal{O}_{X_i}|_j)_{i,j}$$ and $$L_X := (|\mathcal{O}_{X}|_\ell)_\ell$$ both arranged in descending order? To be more specific: Does $$L_{X'}[i] - L_X[i] \in O\left( \frac{\chi(\mathcal{S})}{n}\right)$$ hold, i.e. are the differences balanced. Does someone know a good read for this kind of situations or any references at all? Any good idea is also welcome.
What I do know so far:
1. $$L_{X} \leq L_{X'}$$, that is for all $$i=1,\ldots,n : L_{X}[i] \leq L_{X'}[i]$$,
2. $$\sum_{\ell=1}^n |\mathcal{O}_{X}|_\ell = \chi(\mathcal{O}_X) -n$$,
3. $$\sum_{i,j} |\mathcal{O}_{X_i}|_j = \sum_{i=1}^m (\chi(\mathcal{O}_{X_i}) -n_i) = \chi(\mathcal{O}_X) + \chi(\mathcal{S}) -n$$
4. Combining 2. and 3. we have: $$\sum_{i=1}^n L_{X'}[i] - L_X[i] = \chi(\mathcal{S})$$.
OK, I think there is no bound of the type you want. Namely, choose $$X$$ to be the nodal curve obtained by glueing $$n$$ copies $$X_1, \ldots, X_n$$ of the base $$\mathbf{P}^1$$ by glueing points as follows: first glue $$X_3, \ldots, X_n$$ each to $$X_1$$ in a single point, next glue $$X_2$$ to $$X_1$$ in $$N$$ distinct points. We assume the points we are glueing at map to $$n - 2 + N$$ pairwise distinct points in $$\mathbf{P}^1$$.
First we observe that $$|\mathcal{O}_{X_i}|_j = 0$$ always.
For a scheme $$Z$$ over $$\mathbf{P}^1$$ denote $$\mathcal{O}_Z(i)$$ the pullback of $$\mathcal{O}_{\mathbf{P}^1}(i)$$ to $$Z$$.
Denote $$X' \subset X$$ the union of $$X_1$$ and $$X_2$$. A computation show that $$H^1(X', \mathcal{O}_{X'}(N - 2))$$ is nonzero. Since $$\mathcal{O}_X(N - 2) \to \mathcal{O}_{X'}(N - 2)$$ is surjective and since $$X$$ is a curve, we see that $$H^1(\mathcal{O}_X(N - 2))$$ is nonzero. This implies that $$|\mathcal{O}_X|_n \leq -N$$.
• If I could follow your instructions correctly, then the curve $X'$ you described is of the form $X' = V_+(F') \subset \mathbb{P}_k^2$ with $F'$ being a homogeneous polynomial of degree $N+1$. And we obtain $X$ as $V_+(F) \subset \mathbb{P}_k^2$ where $F'$ divides $F$. In your notation, we thus have $\deg F = \deg F' + n$. – windsheaf Jul 23 at 9:46
• (continued) The intended bound is $-\chi(\mathcal{S})/\deg \pi = -\chi(\mathcal{O}_X)/\deg F$. Now since $X$ is a local complete intersection, we have $-\chi(\mathcal{O}_X) = p_a(X) -1 = \frac{1}{2}(\deg F -1)(\deg F-2)-1 \in O(\deg F^2)$. Moreover, $\deg \pi = \deg F$ and thus the intended bound is in $O(\deg F)$. But your argument is that we deduce $|X|_n \leq - \deg F' = \deg F - n \leq -\chi(\mathcal{S})/\deg \pi$ which is still in the intended order. – windsheaf Jul 23 at 9:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 65, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9940186738967896, "perplexity": 91.634601768643}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541301598.62/warc/CC-MAIN-20191215042926-20191215070926-00456.warc.gz"} |
https://mantis.crd.co/ | MANTIS RESC CRD?!?!
welcome to my starter friendly resource carrd! currently its only me (reij) as the admin so if you have any questions, pop down into my other resource carrd!! :D
sorry if my tuts are kinda hard to understand ehe,, im not very good at coding myself either
all of these codes work with pro standard!! (actually built for pro standard cuz thats the only one i have)
notice: again this aint my main resource crd so there isnt a cbox. if u have questions visit me here!!
(url change because i was too late in subscribing back to pro AHAJDH)
scrollboxes!
text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text
code here
flowers
text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text
code here
biscuit
text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text
code here
chocolate
text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text
code here
blue lace
text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text
code here
lace border by @baeyhkun
text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text
code here
borderless
text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text
code here
rounded
text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text
code here
simple border
text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text
code here
doily
text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text
code here
text!
peepaw
code here
peepaw
code here
peepaw
code here
peepaw
code here
peepaw
code here
peepaw
code here
peepaw
code here
peepaw
code here
peepaw
code here
images!
code here
code here
code here
tutorials!
If you want me to add more tutorials, drop down to noodles.crd.co!!
harroo!!! im reij. am seasian and i like meat and potatoes LMFAO also my carrd
1. get the scrollbox code you wanna put the img in (in this case i used a borderless one)2. put this code between where it says <div id="embed name"> and </div><img src="image.jpg">but replace the 'image.jpg' with the link of the image of your choice. it can be png or gif!3. voila (be sure its the image address you're copying!)
If you want your link to be open on another tab1. put this code between where it says <div id="embed name"> and </div><a href="link" target="_blank">
</a>
If you want to link another section in your carrd1. put this code between where it says <div id="embed name"> and </div><a href=“link”>
</a>
same thing just replace the 'link' with the link of the section of your choice and 'image link' with the image of your choice!2. voila (note that you dont have to paste your whole link like carrd.crd.co/#section01, you can also just paste the #section01 by itself!)
1. put this code between where it says <div id="embed name"> and </div><img src=image link>replace the 'image link' with the pixel of your choice!2. voila
1. alrite, so find the font that you'd like to use! (in this case im using the rainyhearts font)2. copy this first part of the code to be right underneath the <style> tag@font-face {
font-family:'whatever font you chose';
src: url(https://dl.dropbox.com/whatever font you're using.ttf);
}
replace the pseudocode with the font link of your choice!3. add this code to be grouped with the font sizes, color and stuff..font-family:'whatever font you chose';4. voila (you dont have to type the code down because some resc crds already give it to you nicely packaged!)
1. put this code between where it says <div id="embed name"> and </div><a href=link here>text here</a>replace the 'link here' with the link of your choice! and the 'text here' with any info you want!2. voila
stuff you needa know in general:<br> (aka line break element) = gives you a line break in your text. works like spacing paragraphs irl! you can also double up on them to increase the gap size between text<div> or </div> or anything that has div in it = it defines a section of code either starting or ending!the / = if paired with a tag, normally means that tag has been closed or finished! e.g: </div>, </style>, </a>the code insided the <style> tag = its css code that determines the aesthetic look of your code!
example: #scroll {height: 250px;}
<a> = defines a hyperlink<style> = where the css code is storedtext formatting:<b> or <strong> = bolded text
<i> = italics
<sub> = subscript
<sup> = superscript
more terms here since these are the main ones! | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9966598153114319, "perplexity": 104.30115409646972}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948817.15/warc/CC-MAIN-20230328073515-20230328103515-00336.warc.gz"} |
https://root-forum.cern.ch/t/tefficieny/14321 | TEfficieny
Hi,
here my problem.
I am creating Histograms for different MC dijet samples (J0-J6) individually. The problem now with creating the Efficiency is that obviously in the different samples the statistics is different to a point where there are no entries. I have a feeling that J0 and J1 samples (very high weights but no statistics) contribute as huge errors to my efficiency curves.
If I only for example take the J3 sample I have errors of about 2 % if I add the J0 and J1 (basically no entries in the interesting region) my errors are about 40%.
I am using the “mode” option. Without it not only the errors are going to 50% but also the whole efficiency curve is at pushed to 0.5.
In addition I was wondering if there is a way to not fill the bins where none of the samples has an entry because as it is right now they are filled with a value about .5 and an error of .3
thanks for the help
Michael
OK,
so I think I know where the problem is. I understand that there is a norm factor (basically takes into account that one entry with a weight 1000 needs a huge error)(norm = sum(weights)/sum(weights^2)). What I did now is that I took out the weight sum contributing to the total norm in case an entry (denominator) is zero.
I am not sure if that is absolutely correct but otherwise the errors are overestimated I believe
–Michael
Any suggestions? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8568770885467529, "perplexity": 671.675292262744}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103920118.49/warc/CC-MAIN-20220701034437-20220701064437-00431.warc.gz"} |
http://tex.stackexchange.com/questions/75421/left-aligning-equations?answertab=votes | # Left-aligning equations
How do I left-align selective equation blocks, i.e. not centered ?
I know how to do it for ALL equation blocks in the document setting
\usepackage[fleqn]{amsmath}
\setlength{\mathindent}{0pt}
in the preamble, but I only want to do this for some equation blocks.
\documentclass[pdftex,12pt,a4paper,english,dutch,leqno]{article}
\usepackage{babel}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[dvipsnames,table]{xcolor}
\usepackage{color}
\definecolor{vanillegeel}{RGB}{255,240,216}
\usepackage{empheq}
\begin{document}
De raaklijn aan $(x_{0},f(x_{0}))$ wordt algemeen bepaald met :
\begin{empheq}[box={\fboxsep=3pt\colorbox{vanillegeel}}]{align*}
\textnormal{$\begin{cases}z=f(x)\text{ is continu in }x_{0}\text{.}\\ f'(x_{0})=0\text{ : }z=f(x_{0})\text{ is de horizontale raaklijn.}\\ \lim_{x\to x_{0}} f'(x)=\pm\infty\text{ : }x=x_{0}\text{ is de verticale raaklijn.}\\ f'(x_{0})\in\mathbb{R}_{0}\text{ : }z-f(x_{0})=f'(x_{0})\cdot(x-x_{0})\text{ is de raaklijn.\hspace{-12pt}}\end{cases}$}\end{empheq}
\end{document}
-
+1 for the background color! Really nice for the eyes. – morbusg Oct 5 '12 at 13:30
Place '&' at the beginning of each line inside the empheq environment and do not use the cases environment. This is the easiest way to get what you want. – bobb_the_builder Jul 17 '15 at 16:23
Don't use align*.
De raaklijn aan $(x_{0},f(x_{0}))$ wordt algemeen bepaald met:
\begin{flushleft}
\colorbox{vanillegeel}{%
$\displaystyle \begin{cases} z=f(x) \text{ is continu in } x_{0}. \\ f'(x_{0})=0 \text{ : } z=f(x_{0}) \text{ is de horizontale raaklijn.} \\ \lim_{x\to x_{0}} f'(x)=\pm\infty \text{ : } x=x_{0} \text{ is de verticale raaklijn.} \\ f'(x_{0})\in\mathbb{R}_{0} \text{ : } z-f(x_{0})=f'(x_{0})\cdot(x-x_{0}) \text{ is de raaklijn.} \end{cases}$\kern-\nulldelimiterspace\kern-2\arraycolsep}
\end{flushleft}
-
What does \kern-\nulldelimiterspace\kern-2\arraycolsep} do in this ? – Petoetje59 Oct 5 '12 at 14:28
@Petoetje59 It's a substitute of your \hspace{-12pt} based on document parameters, rather than "calculation by eye". – egreg Oct 5 '12 at 14:45
Not allowing you to get the unsung hero badge! ;) +1. – Harish Kumar Oct 5 '12 at 14:48
@HarishKumar Darn! – egreg Oct 5 '12 at 14:49
How come \begin{align*} doesn't work but flushleft does? – Look behind you Sep 1 '15 at 8:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9066392779350281, "perplexity": 4413.064033170467}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860124045.24/warc/CC-MAIN-20160428161524-00032-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://converter.ninja/length/decimeters-to-kilometers/222-dm-to-km/ | # 222 decimeters in kilometers
## Conversion
222 decimeters is equivalent to 0.0222 kilometers.[1]
## Conversion formula How to convert 222 decimeters to kilometers?
We know (by definition) that: $1\mathrm{dm}=0.0001\mathrm{km}$
We can set up a proportion to solve for the number of kilometers.
$1 dm 222 dm = 0.0001 km x km$
Now, we cross multiply to solve for our unknown $x$:
$x\mathrm{km}=\frac{222\mathrm{dm}}{1\mathrm{dm}}*0.0001\mathrm{km}\to x\mathrm{km}=0.0222\mathrm{km}$
Conclusion: $222 dm = 0.0222 km$
## Conversion in the opposite direction
The inverse of the conversion factor is that 1 kilometer is equal to 45.045045045045 times 222 decimeters.
It can also be expressed as: 222 decimeters is equal to $\frac{1}{\mathrm{45.045045045045}}$ kilometers.
## Approximation
An approximate numerical result would be: two hundred and twenty-two decimeters is about zero point zero two kilometers, or alternatively, a kilometer is about forty-five point zero four times two hundred and twenty-two decimeters.
## Footnotes
[1] The precision is 15 significant digits (fourteen digits to the right of the decimal point).
Results may contain small errors due to the use of floating point arithmetic. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 6, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9542005658149719, "perplexity": 2333.0232575834666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104669950.91/warc/CC-MAIN-20220706090857-20220706120857-00604.warc.gz"} |
https://www.groundai.com/project/interference-alignment-in-dense-wireless-networks/ | Interference Alignment in Dense Wireless Networks
Interference Alignment in Dense Wireless Networks
Urs Niesen U. Niesen is with the Mathematics of Networks and Communications Research Department, Bell Labs, Alcatel-Lucent. Email: [email protected]
Abstract
We consider arbitrary dense wireless networks, in which nodes are placed in an arbitrary (deterministic) manner on a square region of unit area and communicate with each other over Gaussian fading channels. We provide inner and outer bounds for the -dimensional unicast and the -dimensional multicast capacity regions of such a wireless network. These inner and outer bounds differ only by a factor , yielding a fairly tight scaling characterization of the entire regions. The communication schemes achieving the inner bounds use interference alignment as a central technique and are, at least conceptually, surprisingly simple.
Capacity scaling, interference alignment, multicast, multicommodity flow, opportunistic communication, wireless networks.
I Introduction
Interference alignment is a recently introduced technique to cope with the transmissions of interfering users in wireless systems see [1, 2, 3]. In this paper, we apply this technique to obtain fairly precise (up to factor) information-theoretic scaling results for the unicast and multicast capacity regions of dense wireless networks.
I-a Related Work
The study of scaling laws for wireless networks, describing the system performance in the limit of large number of users, was initiated by Gupta and Kumar in [4]. They analyzed a network scenario in which nodes are placed uniformly at random on a square of area one (called a dense network in the following) and are randomly paired into source-destination pairs with uniform rate requirement. Under a so-called protocol channel model, in which only point-to-point communication is allowed and interference is treated as noise, they showed that the largest uniformly achievable per-node rate scales as up to a polylogarithmic factor in . Achievability was shown using a multi-hop communication scheme combined with straight-line routing. Different constructions achieving slightly better scaling laws, i.e., improving the polylogarithmic factor in , were subsequently presented in [5, 6].
These results are in some sense negative, in that they show that with current technology, captured by the protocol channel model assumption, the per-node rate in large wireless networks decreases with increasing network size even if the deployment area is kept constant. An immediate question is therefore if this negative result is due to the protocol channel model assumption or if there is a more fundamental reason for it. To address this question, several authors have considered an information-theoretic approach to the problem, in which the channel is simply assumed to be a Gaussian fading channel without any restrictions on the communication scheme [7, 8, 9, 10, 11]. We shall refer to this as the Gaussian fading channel model in the following. These works construct cooperative communication schemes and show that they can significantly outperform multi-hop communication in dense networks. In particular, Özgür et al. showed in [11] that in Gaussian fading dense wireless networks with randomly deployed nodes and random source-destination pairing, the maximal uniformly achievable per-node rate scales like111The notation is used to indicate that the maximal uniformly achievable per-node rate is upper bounded by and lower bounded by . Similar expressions will be used throughout this section. for any . In other words, in dense networks222We point out that the situation is quite different in extended networks, in which nodes are placed on a square of area . Here network performance depends on the path-loss exponent , governing the speed of decay of signal power as a function of distance. For small , cooperative communication is order optimal, whereas for large , multi-hop communication is order optimal [12, 13, 14, 15, 16, 17, 18, 11, 19]., cooperative communication can increase achievable rates to almost constant scaling in —significantly improving the scaling resulting from the protocol channel model assumption. The scaling law was subsequently tightened to in [20, 21].
While these results removed the protocol channel model assumption made in [4], they kept the assumptions of random node placement and random source-destination pairing with uniform rate. Wireless networks with random node placement and arbitrary traffic pattern have been analyzed in [22, 23] for the protocol channel model and in [24] for the Gaussian fading channel model. On the other hand, wireless networks with arbitrary node placement and random source-destination pairing with uniform rate have been investigated in [25] for the protocol channel model and in [20] for the Gaussian fading channel model. While methods similar to the ones developed in [25] can also be used to analyze wireless networks with arbitrary node placement and arbitrary traffic pattern under the protocol channel model, the performance of such general networks under a Gaussian channel model (i.e., an information-theoretic characterization of achievable rates) is unknown.
Finally, it is worth mentioning [26, 27], which derive scaling laws for large dense interference networks. In particular, [27] considers a dense random node placement with random source-destination pairing. However, the model there is an interference channel as opposed to a wireless network as modeled in the works mentioned above. In other words, the source nodes cannot communicate with each other, and similarly the destination nodes cannot communicate with each other. This differs from the model adopted in this paper and the works surveyed so far, in which no such restrictions are imposed. For such interference networks, [27] derives the asymptotic sum-rate as the number of nodes in the network increases.
I-B Summary of Results
In this paper, we consider the general problem of determining achievable rates in dense wireless networks with arbitrary node placement and arbitrary traffic pattern. We assume a Gaussian fading channel model, i.e., the analysis is information-theoretic, imposing no restrictions on the nature of communication schemes used. We analyze the -dimensional unicast capacity region , and the -dimensional multicast capacity region of an arbitrary dense wireless network. describes the collection of all achievable unicast traffic patterns (in which each message is to be sent to only one destination node), while describes the collection of all achievable multicast traffic patterns (in which each message is to be sent to a set of destination nodes). We provide explicit approximations and of and in the sense that
^Λ\textupUC(n) ⊂Λ\textupUC(n)⊂K1log(n)^Λ\textupUC(n), ^Λ\textupMC(n) ⊂Λ\textupMC(n)⊂K2log(n)^Λ\textupMC(n),
for constants not depending on . In other words, and approximate the unicast and multicast capacity regions and up to a factor . This provides tight scaling results for arbitrary traffic pattern and arbitrary node placement.
The results presented in this paper improve the known results in several respects. First, as already pointed out, they require no probabilistic modeling of the node placement or traffic pattern, but rather are valid for any node placement and any traffic pattern and include the results for random node placement and random source-destination pairing with uniform rate as a special case. Second, they provide information-theoretic scaling results that are considerably tighter than the best previously known, namely up to a factor here as compared to in [11] and in [20, 21]. Moreover, the results in this paper provide an explicit expression for the pre-constant in the term that is quite small, and hence these bounds yield good results also for small and moderate sized wireless networks. Third, the achievable scheme used to prove the inner bound in this paper is, at least conceptually, quite simple, in that the only cooperation needed between users is to perform interference alignment. This contrasts with the communication schemes achieving near linear scaling presented so far in the literature, which require hierarchical cooperation and are harder to analyze.
I-C Organization
The remainder of this paper is organized as follows. Section II introduces the network model and notation. Section III presents the main results of this paper. Section IV describes the communication schemes used to prove achievability. Section V contains proofs, and Sections VI and VII contain discussions and concluding remarks.
Ii Network Model and Notation
Let
A≜[0,1]2
be a square of area one, and consider nodes (with ) placed in an arbitrary manner on . Let be the Euclidean distance between nodes and , and define
rmin(n)≜n1/2minu≠vru,v.
The minimum separation between nodes in the node placement is then . Note that for a grid graph, and with high probability for nodes placed uniformly and independently at random on . In general, we have
rmin(n)≤4/√π<3, (1)
and, while the results presented in this paper hold for any , the case of interest is when decays at most polynomially with , i.e., for some constant . Note that we do not make any probabilistic assumptions on the node placement, but rather allow an arbitrary (deterministic) placement of nodes on . In particular, the arbitrary node placement model adopted here contains the random node placement model as a special case. The arbitrary node placement model is, however, considerably more general since it allows for classes of node placements that only appear with vanishing probability under random node placement (e.g., node placements with large gaps or isolated nodes).
We assume the following complex baseband-equivalent channel model. The received signal at node at time is given by
yv[t]≜∑u≠vhu,v[t]xu[t]+zv[t],
where is the channel gain from node to node , is the signal sent by node , and is additive receiver noise at node , all at time . The additive noise components are assumed to be independent and identically distributed (i.i.d.) circularly-symmetric complex Gaussian random variables with mean zero and variance one. The channel gain has the form
hu,v[t]≜r−α/2u,vexp(√−1θu,v[t]), (2)
where is the path-loss exponent. As a function of and , the phase shifts are assumed to be i.i.d. uniformly distributed over . As a function of time , we only assume that varies in a stationary ergodic manner as a function of for every . Note that the distances between the nodes do not change as a function of time and are assumed to be known throughout the network. The phase shifts are assumed to be known at time at every node in the network. Together with the knowledge of the distances , this implies that full causal channel state information (CSI) is available throughout the network. We impose a unit average power constraint on the transmitted signal at every node in the network.
The phase-fading model (2) is adopted here for consistency with the capacity-scaling literature. All results presented in this paper can be extended to Rayleigh fading, see Section VI-C.
A unicast traffic matrix associates with every node pair the rate at which node wants to transmit a message to node . The messages corresponding to distinct pairs are assumed to be independent. Note that we allow the same node to be source for several destinations , and the same node to be destination for several sources . The unicast capacity region is the closure of the collection of all achievable unicast traffic matrices . Knowledge of the unicast capacity region provides hence information about the achievability of any unicast traffic matrix .
A multicast traffic matrix associates with every pair of node and subset the rate at which node wants to multicast a message to the nodes in , i.e., every node wants to receive the same message from . The messages corresponding to distinct pairs are again assumed to be independent. Note that we allow the same node to be source for several multicast groups , and the same subset of nodes to be multicast group for several sources . The multicast capacity region is the closure of the collection of all achievable multicast traffic matrices . Observe that unicast traffic is a special case of multicast traffic, and hence is a -dimensional “slice” of the -dimensional region .
The next example illustrates the definitions of unicast and multicast traffic.
Example 1.
Consider and . Assume node wants to transmit a message to node at a rate of bit per second, and a message to node at rate bits per second. Node wants to transmit a message at rate bits per second to node . The messages are assumed to be independent. This traffic requirement can be described by a unicast traffic matrix with , , , and for all other pairs. Note that node is source for and , and that node is destination for and . Note also that node is neither a source nor a destination for any communication pair, and can hence be understood as a helper node.
Assume now node wants to transmit the same message to both and at rate bit per second, and a private message to only node at rate bits per second. Moreover, node wants to transmit the same message to both and at rate bits per second. The messages are assumed to be independent. This traffic requirement can be described by a multicast traffic matrix with , , , and for all other pairs. Note that is source for two multicast groups and , and that is multicast group for two sources and . ∎
Throughout, we denote by and the logarithms with respect to base and , respectively. To simplify notation, we suppress the dependence on within proofs whenever this dependence is clear from the context.
Iii Main Results
We now present the main results of this paper. Section III-A provides a scaling characterization of the unicast capacity region , and Section III-B provides a scaling characterization of the multicast capacity region of a dense wireless network. Section III-C contains example scenarios illustrating applications of the main theorems.
Iii-a Unicast Traffic
Define
^Λ\textupUC(n)≜{λ\textupUC∈\mathdsRn×n+:∑w≠uλ\textupUCu,w≤1 ∀u∈V(n), ∑u≠wλ\textupUCu,w≤1 ∀w∈V(n)}.
is the collection of all unicast traffic matrices such that for every node in the network the total traffic
∑w≠uλ\textupUCu,w
from is less than one, and such that for every node in the network the total traffic
∑u≠wλ\textupUCu,w
to is less than one.
The next theorem shows that is a tight approximation of the unicast capacity region of the wireless network.
Theorem 1.
For all , , and node placement with minimum node separation ,
2−α/2^Λ\textupUC(n)⊂Λ\textupUC(n)⊂log(n2+α/2r−αmin(n))^Λ\textupUC(n).
Assuming that decays no faster than polynomial in (see the discussion in Section II), Theorem 1 states that approximates up to a factor . In other words, provides a scaling characterization of the unicast capacity region . This scaling characterization is considerably more general than the standard scaling results, in that it holds for any node placement and provides information on the entire -dimensional unicast capacity region (see Fig. 1). In particular, define
ρ⋆λ\textupUC(n)≜max{ρ:ρλ\textupUC∈Λ\textupUC(n)}
to be the largest multiple such that is achievable. Then, for any arbitrary node placement and arbitrary unicast traffic matrix , Theorem 1 determines up to a multiplicative gap of order uniform in . This contrasts with the standard scaling results, which provide information on only for a uniform random node placement and a uniform random unicast traffic matrix (constructed by pairing nodes randomly into source-destination pairs with uniform rate).
Theorem 1 also reveals that the unicast capacity region of a dense wireless network has a rather simple structure in that it can be approximated up to a factor by an intersection of half-spaces. Each of these half-spaces corresponds to a cut in the wireless network, bounding the total rate across this cut. While there are such cuts in the network, Theorem 1 implies that only a small fraction of them are of asymptotic relevance. From the definition of , these are precisely the cuts involving just a single node (with traffic flowing either into or out of that node).
Iii-B Multicast Traffic
Let
^Λ\textupMC(n)≜{λ\textupMC∈\mathdsRn×2n+:∑W⊂V(n):W∖{u}≠∅λ\textupMCu,W≤1 ∀u∈V(n), ∑u≠w∑W⊂V(n):w∈Wλ\textupMCu,W≤1 ∀w∈V(n)}. (3)
Similarly to defined in Section III-A, the region is the collection of multicast traffic matrices such that for every node in the network the total traffic
∑W⊂V(n):W∖{u}≠∅λ\textupMCu,W
from is less than one, and such that for every node in the network the total traffic
∑u≠w∑W⊂V(n):w∈Wλ\textupMCu,W
to is less than one.
The next theorem shows that is a tight approximation of the multicast capacity region of the wireless network.
Theorem 2.
For all , , and node placement with minimum node separation ,
2−1−α/2^Λ\textupMC(n)⊂Λ\textupMC(n)⊂log(n2+α/2r−αmin(n))^Λ\textupMC(n).
Assuming as before that decays no faster than polynomial in , Theorem 2 asserts that approximates up to a factor . In other words, as in the unicast case, we obtain a scaling characterization of the multicast capacity region . Again, this scaling characterization is considerably more general than standard scaling results, in that it holds for any node placement and provides information about the entire -dimensional multicast capacity region . Define, as for unicast traffic matrices,
ρ⋆λ\textupMC(n)≜max{ρ:ρλ\textupMC∈Λ\textupMC(n)}
to be the largest multiple such that is achievable. Then Theorem 2 allows, for any arbitrary node placement and arbitrary multicast traffic matrix , to determine up to a multiplicative gap of order uniform in . In particular, no probabilistic assumptions about the structure of or are necessary.
As with , Theorem 2 implies that the multicast capacity region of a dense wireless network is approximated up to a factor by an intersection of half spaces. In other words, we are approximating a region of dimension (i.e., exponentially big in ) through only a linear number of inequalities. As in the case of unicast traffic, each of these inequalities corresponds to a cut in the wireless network, and it is again the cuts involving just a single node that are asymptotically relevant.
Iii-C Examples
This section contains several examples illustrating various aspects of the capacity regions and their approximations . Example 2 compares the scaling laws obtained in this paper with the ones obtained using hierarchical cooperation as proposed in [11]. Example 3 discusses symmetry properties of and . Example 4 provides a traffic pattern showing that the outer bounds in Theorems 1 and 2 are tight up to a constant factor.
Example 2.
(Random source-destination pairing)
Consider a random node placement with every node placed independently and uniformly at random on . Assume we pair each node with a node chosen independently and uniformly at random. Denote by the resulting source-destination pairs. Note that each node is source exactly once and destination on average once. Each source wants to transmit an independent message to at rate (depending on , but not on ). The question is to determine , the largest achievable value of . This question was considered in [11], where it was shown that, with probability as and for every ,
Ω(n−ε)≤ρ⋆(n)≤O(nε). (4)
The lower bound is achieved by a hierarchical cooperation scheme, and we denote its rate by .
We now show that using the results presented in this paper these bounds on can be significantly sharpened. Set for and for all other entries of . is then given by
ρ⋆(n)=max{ρ:ρλ\textupUC∈Λ\textupUC(n)}.
Setting
^ρ⋆(n)≜max{^ρ:^ρλ\textupUC∈^Λ\textupUC(n)},
we obtain from Theorem 1 that
2−α/2^ρ⋆(n)≤ρ⋆(n)≤log(n2+α/2r−αmin(n))^ρ⋆(n). (5)
It remains to evaluate . By construction of , we have
maxu∈V(n)∑w≠uλ\textupUCu,w=1.
Moreover, by [28],
\mathdsP(12≤lnln(n)ln(n)maxw∈V(n)∑u≠wλ\textupUCu,w≤2)≥1−o(1).
Using the definition of , this yields that
lnln(n)2ln(n)≤^ρ⋆(n)≤2lnln(n)ln(n) (6)
with high probability.
Recall that the minimum distance between nodes is , and that, for a random node placement, with high probability as (see, e.g., [11, Theorem 3.1]). Hence (5) and (6) show that that for random node placement and random source-destination pairing
2−1−α/2lnln(n)ln(n)≤ρ⋆(n)≤(4+3α)log(e)lnln(n) (7)
with probability as . The lower bound is achieved using a communication scheme presented in Section IV-B based on interference alignment, and we denote its rate by .
Comparing (7) and (4), we see that the scaling law obtained here is significantly sharper, namely up to a factor here as opposed to a factor for any in [11]. Moreover, (7) provides good estimates for any value of , whereas (4) is only valid for large values of , with a pre-constant in that increases rapidly as (see [21, 29] for a detailed discussion on the dependence of the pre-constant on ). For a numerical example, Table I compares per-node rates of the hierarchical cooperation scheme of [11] (more precisely, an upper bound to it, with optimized parameters as analyzed in [21]) with the per-node rates obtained through interference alignment as proposed in this paper. For the numerical example, we choose .
We point out that the per-node rate decreases as the number of nodes increases only because of the random source-destination pairing. In fact, if the nodes are paired such that each node is source and destination exactly once, then the interference alignment based scheme achieves a per-node rate , i.e., the per-node rate does not decay to zero as . ∎
Example 3.
(Symmetry of and )
Theorems 1 and 2 provide some insight into (approximate) symmetry properties of the unicast and multicast capacity regions and . Indeed, their approximations and are invariant with respect to node positions (and hence, in particular, also invariant under permutation of nodes).
More precisely, consider a unicast traffic matrix . For a permutation of the nodes set
~λ\textupUCu,w≜λ\textupUCπ(u),π(w).
Then if and only if . Hence Theorem 1 yields that if , then
2−α/2log−1(n2+α/2r−αmin(n))~λ\textupUC∈Λ\textupUC(n).
Similarly, let be a multicast traffic matrix, and define
~λ\textupMCu,W≜λ\textupMCπ(u),π(W),
where, for , . Theorem 2 implies that if , then
2−1−α/2log−1(n2+α/2r−αmin(n))~λ\textupMC∈Λ\textupMC(n).
In other words, the location of the nodes in a dense wireless network (with decaying at most polynomially in ) affects achievable rates at most up to a factor . This contrasts with the behavior of extended wireless networks, where node locations crucially affect achievable rates [20]. ∎
Example 4.
(Tightness of outer bounds)
We now argue that the outer bounds in Theorems 1 and 2 are tight up to a constant factor in the following sense. There exists a constant such that for every we can find traffic matrices and on the boundary of the outer bound in Theorems 1 and 2 such that and . Or, more succinctly, there exists a constant such that
Λ\textupUC(n)∖Klog(n2+α/2r−αmin(n))^Λ\textupUC(n) ≠∅, Λ\textupMC(n)∖Klog(n2+α/2r−αmin(n))^Λ\textupMC(n) ≠∅.
This shows that the gap between the inner and outer bounds in Theorems 1 and 2 is due to the use of the interference alignment scheme to prove the inner bound, and that to further decrease this gap a different achievable scheme has to be considered. Throughout this example, we assume for some constant .
Choose a node , and let, for each ,
λ\textupUCu,w≜{1n−1if w=w⋆,0otherwise.
Note that . Under this traffic matrix , each node has an independent message for a common destination node .
If we ignore the received signals at all nodes and transmit no signal at , we transform the wireless network into a multiple access channel with users. Since for any , each node can reduce its power such that the received power at node is equal to . In this symmetric setting, the equal rate point of the capacity region of the multiple access channel has maximal sum rate, and hence each node can reliably transmit its message to at a per-node rate of
1n−1log(1+(n−1)2−α/2) ≥1n−1log(n2−α/2) =1n−1(1−α2log(n))log(n).
Thus, for ,
12log(n)λ\textupUC∈Λ\textupUC(n). (8)
On the other hand, using the assumption ,
log(n2+α/2r−αmin(n))<(2+α(1/2+κ))log(n),
and hence
(9)
Therefore, setting
K≜(4+α(1+2κ))−1>0,
we obtain from (8) and (9) that
Λ\textupUC(n)∖Klog(n2+α/2r−αmin(n))^Λ\textupUC(n)≠∅.
In words, at least along one direction in , the outer bound in Theorem 1 is loose by at most a constant factor.
Since is a -dimensional “slice” of the -dimensional region , the same result follows for as well. ∎
Iv Communication Schemes
This section describes the communication schemes achieving the inner bounds in Theorems 1 and 2. Both schemes use the idea of interference alignment as a building block, which is recalled in Section IV-A. The communication scheme for unicast traffic is introduced in Section IV-B and the scheme for multicast traffic in Section IV-C.
Iv-a Interference Alignment
Interference alignment is a technique introduced recently in [1, 2]. The technique is best illustrated with an example taken from [3]. Assume we pair the nodes into source-destination pairs such that each node in is source and destination exactly once. Consider the channel gains and for two different times and . Assume we could choose and such that and for all . By adding up the received symbols and , destination node obtains
ywi[t1]+ywi[t2]=hui,wi[t1](xui[t1]+xui[t2])+zwi[t1]+zwi[t2].
Thus, by sending the same symbol twice (i.e., ), every source node is able to communicate with its destination node at essentially half the rate possible without any interference from other nodes.
Using this idea and the symmetry and ergodicity of the distribution of the channel gains, the following result is shown in [3].
Theorem 3.
For any source-destination pairing such that and for , the rates
λ\textupUCui,wj={12log(1+2|hui,wi|2)if i=j,0otherwise,
are achievable, i.e., .
For a source-destination pairing as in Theorem 3, construct a matrix such that
Sui,wj={1if i=j,0otherwise.
Note that is a permutation matrix, and we will call such a traffic pattern a permutation traffic. Using and ,
12log(1+2r−αui,wi)≥12log(1+21−α/2)≥2−α/2,
and hence Theorem 3 provides an achievable scheme showing that . In other words, Theorem 3 shows that, for every permutation traffic, a per-node rate of is achievable. In the next two sections, we will use this communication scheme for permutation traffic as a building block to construct communication schemes for general unicast and multicast traffic.
Iv-B Communication Scheme for Unicast Traffic
Consider a general unicast traffic matrix . If happens to be a scalar multiple of a permutation matrix, then Theorem 3 provides us with an achievable scheme to transmit according to . In order to apply Theorem 3 for general , we need to schedule transmissions into several slots such that in each slot transmission occurs according to a permutation traffic. This transforms the original problem of communicating over a wireless network into a problem of scheduling over a switch with input and output ports and traffic requirement .
This problem has been widely studied in the literature. In particular, using a result from von Neumann [30] and Birkhoff [31] (see also [32] for the application to switches) it can be shown that for any there exist a collection of schedules (essentially permutation matrices, see the proof in Section V-A for the details) and nonnegative weights summing to one such that
∑iωiSi=λ\textupUC.
This suggests the following communication scheme. Split time into slots according to the weights . In the slot corresponding to , send traffic over the wireless network using interference alignment for the schedule . In other words, we time share between the different schedules according to the weights .
We analyze this communication scheme in more detail in Section V-A. In particular, we show that it achieves any point in . Combined with a matching outer bound, we show that this scheme is optimal for any unicast traffic pattern up to a factor .
Recall from Example 3 that the capacity region is approximately symmetric with respect to permutation of the traffic matrix. This implies that the rate achievable for any permutation traffic is approximately the same. While the decomposition of the traffic matrix into schedules is not unique, this invariance suggests that it does not matter too much which decomposition is chosen. The situation is different for Rayleigh fading (as opposed to phase fading considered here), where different decompositions can be used for opportunistic communication. This approach is explored in detail in Section VI-C.
Iv-C Communication Scheme for Multicast Traffic
We now turn to multicast traffic. Given the achievable scheme presented for unicast traffic in Section IV-B reducing the problem of communication over a wireless network to that of scheduling over a switch, it is tempting to try the same approach for multicast traffic as well. Unfortunately, scheduling of multicast traffic over switches is considerably more difficult than the corresponding unicast version (see, for example, [33] for converse results showing the infeasibility of multicast scheduling over switches with finite speedup). We therefore adopt a different approach here. The proposed communication scheme is reminiscent of the two-phase routing scheme of Valiant and Brebner [34].
Consider a source node that wants to multicast a message to destination group . The proposed communication scheme operates in two phases. In the first phase, the node splits its message into parts of equal length. It then sends one (distinct) part over the wireless network to each node in . Thus, after the first phase, each node in has access to a distinct fraction of the original message. In the second phase, each node in sends its message parts to all the nodes in . Thus, at the end of the second phase, each node in can reconstruct the entire message. All pairs operate simultaneously within each phase, and contention within the phases is resolved by appropriate scheduling (see the proof in Section V-B for the details).
A different way to look at this proposed communication scheme is as follows. Consider the nodes in , and construct a graph with for some additional node and with if either or . In other words, is a “star” graph with central node (see Fig. 2). We assign to each edge an edge capacity of one. The proposed communication scheme for the wireless network can then be understood as a two layer architecture, consisting of a physical layer and a network layer. The physical layer implements the graph abstraction , and the network layer routes data over .
In Section V-B, we show that the set of rates that can be routed over contains . We then argue that if , then , i.e., if messages can be routed over the graph at rates , then almost the same rates are achievable in the wireless network. Combining this with a matching outer bound, we show that the proposed communication scheme is optimal for any multicast traffic pattern up to a factor .
V Proofs
This section contains the proofs of Theorem 1 (in Section V-A) and Theorem 2 (in Section V-B).
V-a Proof of Theorem 1
We start with the proof of the outer bound in Theorem 1. For subsets , , denote by the capacity of the multiple-input multiple-output (MIMO) channel between nodes in and nodes in . Applying the cut-set bound [35, Theorem 14.10.1] to the sets , , we obtain
∑u≠wλ\textupUCu,w≤C({w} | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9521307349205017, "perplexity": 411.4553635046298}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572484.20/warc/CC-MAIN-20190916035549-20190916061549-00120.warc.gz"} |
https://tantalum.academickids.com/encyclopedia/index.php/Levi-Civita_connection | # Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). The fundamental theorem of Riemannian geometry states that there is unique connection which satisfies these properties.
In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection. The coordinate-space expression of the connection are called Christoffel symbols.
## Formal definition
Let [itex](M,g)[itex] be a Riemannian manifold (or pseudo-Riemannian manifold) then an affine connection [itex]\nabla[itex] is Levi-Civita connection if it satisfy the following conditions
1. Preserves metric, i.e., for any vector fields [itex]X[itex], [itex]Y[itex], [itex]Z[itex] we have [itex]Xg(Y,Z)=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)[itex], where [itex]Xg(Y,Z)[itex] denotes the derivative of function [itex]g(Y,Z)[itex] along vector field [itex]X[itex].
2. Torsion-free, i.e., for any vector fields [itex]X[itex] and [itex]Y[itex] we have [itex]\nabla_XY-\nabla_YX=[X,Y][itex], where [itex][X,Y][itex] are the Lie brackets for vector fields [itex]X[itex] and [itex]Y[itex].
## Derivative along curve
Levi-Civita connection defines also a derivative along curves, usually denoted by [itex]D[itex].
Given a smooth curve [itex]\gamma[itex] on [itex](M,g)[itex] and a vector field [itex]V[itex] on [itex]\gamma[itex] its derivative is defined by
[itex]\frac{D}{dt}V=\nabla_{\dot\gamma(t)}V.[itex]
• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy
Information
• Clip Art (http://classroomclipart.com) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9729738235473633, "perplexity": 3060.729717681492}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103877410.46/warc/CC-MAIN-20220630183616-20220630213616-00536.warc.gz"} |
https://mathhelpboards.com/threads/riemannian-metric-question.4727/ | # Riemannian metric question
• Banned
• #1
#### Poirot
##### Banned
Feb 15, 2012
250
I'm interested in part iv) on the attachment. This is my work so far:
e=(1,0) and e'=(0,1) form a basis of the tangent space at any point z=(x,y). Making the identification (x,y)->x+iy, we get g(e,e')=0 and g(e,e)=g(e',e')=$\frac{1}{im(z)^2}$.
a(t)=z+t and b(t)=z+it are generating curves for e,e' respectively.
(lets call the function f)
$f(z)=\frac{z}{|z|^2}$ so $f(a(t))=\frac{z+t}{|z+t|^2}$. I need to find f'(a(t)) to proceed. How can I cope with differentiating the modulus of a complex number z? Thanks
#### Attachments
• 70.7 KB Views: 13
#### ZaidAlyafey
##### Well-known member
MHB Math Helper
Jan 17, 2013
1,667
Re: riemannian metric question
I don't know about metric spaces , but I know about complex analysis ... To differentiate a function a necessary requirement is to satisfy the cauchy-riemann equation .. suppose that $$\displaystyle f(z)=|z|$$ this function is clearly not differentiable
$$\displaystyle f(z)=\sqrt{x^2+y^2}$$
By the cauchy-reimann equation we must have $$\displaystyle \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$$
Which is clearly not satisfied for $$\displaystyle |z|$$
The function you are trying to differentiate seems a function of several variables ? , are you differentiating with respect to t ? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9685227870941162, "perplexity": 1626.1516736312244}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487613453.9/warc/CC-MAIN-20210614201339-20210614231339-00575.warc.gz"} |
http://www.hw2sw.com/2011/06/29/capacitor-on-breadboard/6/ | Bonus track for survivors. Mathematical skills!
Did you ask yourself why charging/discharging graphs have that kind of time behavior?
Integral operation has the answer (symbol: $\int$). What does it do?
Even if we consider them almost stationary - no change in time - in reality we're treating time-dependent sizes; assuming that this sizes change every infinitesimal instant, we use integral operation to rebuild the whole behavior.
Help yourself thinking you're on a roll-coaster, watching constantly to ground, trying to measure the distance at every moment from starting to ending position; now sum all your computations and integral result is ready.
So considering our initial circuit operating with constant sizes (not varying in time), and assuming that starting C voltage is zero we can write equations in which power supply is equal to sum of voltage-differences both on R and C:
$V=\Delta V_{R} + \Delta V_{C} = \Delta V_{R} + \frac{\Delta Q}{C}$
We derive (other mathematical operation) first and last equation members; in more human words we examine the equation on single generic quick instants, same kind of above "infinitesimal instant":
$0 = \delta V_{R} + \delta V_{C} = R\frac{di(t)}{dt} + \frac{i(t)}{C}$
Constant size like our power supply V are zero after derivation, because this measures the size's variation degree: a constant simply doesn't vary.
Now we have a first-order differential equation with constant coefficients.
Resolving it means that we rebuild the whole current behavior:
$\int_{0}^{t}\frac{1}{i(\tau)}di(\tau) = -\frac{1}{RC}\int_{0}^{t}d\tau \Longrightarrow ln|i(t)|_{0}^{t} = -\frac{1}{RC}\tau|_{0}^{t} \Longrightarrow ln\frac{i(t)}{i(0)} = -\frac{1}{RC}t \Longrightarrow i(t) = i(0)e^{-\frac{t}{RC}}$
We found mathematical explanation of current behavior. What about voltage?
Consider that a voltage present on an electrical item at a certain instant is given by:
1. starting voltage $V_{C0}$ (zero for our capacitor);
2. remaining voltage charge for C.
Translated becomes
$v_{C}(t) = \Delta V_{C} + V_{C0}=\frac{1}{C}\int_{0}^{t}e^{-\frac{\tau}{RC}}d\tau + 0$
We know explicit expression of i(t) so substituting we get
$v_{C}(t) = \frac{i(0)}{C}\int_{0}^{t}e^{-\frac{\tau}{RC}}d\tau = -i(0)\cdot Re^{-\frac{\tau}{RC}}|_{0}^{t} = V_{0}(1-e^{\frac{t}{RC}})$
Finally we can justify exponential behavior of both voltage and current in capacitor charging (similar with discharging)
• $i(t) = i(0)e^{-\frac{t}{RC}}$
• $v(t) = V_{0}(1-e^{\frac{t}{RC}})$
[Tip: inserting C before R you get same results, but only for constant sizes (in DC, direct coupling); in AC (alternate coupling) this is not more possible.]
Here we're playing with quite low component sizes (capacitor used in test is 10 micro-Farad, $10^{-5}\mu F$; not so negligible indeed) and you can get charged C in your hands safely: generally it's recommended to take care and get capacitors with protection or not by its terminals at least. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 10, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8793913125991821, "perplexity": 2566.773613370764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038101485.44/warc/CC-MAIN-20210417041730-20210417071730-00202.warc.gz"} |
https://chat.stackexchange.com/transcript/106623?m=59764970 | 2:54 AM
4
In the book "Understanding Analysis, second edition" by Stephen Abbot, the unboundedness of the set of natural number $\mathbb{N}$ is proven as the following proof: Assume, for contradiction, that $\mathbb{N}$ is bounded above. By the Axiom of Completeness (AoC), $\mathbb{N}$ should then have a ...
3:19 AM
1
If $a, b, c, d$ are positive real numbers such that $c^2+d^2=(a^2+b^2)^2$, prove that $$\frac{a^2}{c}+\frac{b^2}{d} \geq1$$ and equality hold iff $ad=bc$.
4:08 AM
2
Let $X \in R^n$ be a compact convex set, and $f:X \to X$ be a continuous function. Then, can we say that from all $x_0 \in X$, the fixed-point iterations $x_{k+1}=f(x_k)$ to converge to some fixed-point $\bar{x}(x_0) \in X$? If not, what are the conditions that $f$ must satisfy such that the iter...
1 hour later…
5:35 AM
1
Let $\{x_1,...,x_n\}$ be a set of convex points labeled in a cyclic order. I am trying to show that the following structure is equivalent with an affinely regular polygon: Fix $j$ for some $j\in \{1,...,n\}$. Then we know the following to be true: $\{x_j,x_{j-1}\}\parallel \{x_{j+1},x_{j-2}\}$ mo...
@Feeds Answers to this question are eligible for a +100 reputation bounty. polygonlink1 wants to draw more attention to this question.
3
X is a non-negative random variable, with $\mathbb{E}(X) < \infty$. My goal is to show this inequality: $$\sqrt{1+(\mathbb{E}(X))^2} \leq \mathbb{E}(\sqrt{1+X^2})$$ x² is a convex function, so with the Jensen inequality I get that: \sqrt{1+(\mathbb{E}(X))^2} \leq \sqrt{1+\mathbb{E}(X^2)} = \sq... 3 hours later… 8:30 AM 1 I have the following CDF \begin{align*} F(x)=&P \left ({|h|^{2} \le \frac { x \left ({1 + |g|^{2} \rho _{2} }\right)}{\phi \rho _{1}} }\right), \end{align*} where the RHS is found to be \begin{align*} P \left ({|h|^{2} \le \frac { x \left ({1 + |g|^{2} \rho _{2} }\right)}{\phi \rho _{1}} }\right)... 8:53 AM @HNQmath.se Removed from HNQ by closure. @Feeds Answers to this question are eligible for a +50 reputation bounty. nashynash is looking for an answer from a reputable source. 3 I come across this problem which is about bifurcation. I am trying to take all the cases. I am expecting Hopf bifurcation to occur here but the last case I could not find the fixed point. Could you please help me? Consider the vector field on plane \begin{align} \dot{x} &= x - xy +1\\ \label{sys1... @Feeds Answers to this question are eligible for a +200 reputation bounty. Free Books wants to draw more attention to this question: > I would like some hint about how to solve the last case. 1 hour later… 10:20 AM 2 I came across the following question just now, A triangle \Delta ABC is drawn such that \angle{ACB} = 30^o and side length AC = 9*\sqrt{3} If side length AB = 9, how many possible triangles can ABC exist as? Here is a diagram for reference: Here is what I did: I used the Law of Sine... 1 If x \in [a,b], then I want to find the optimal upper bound of the product |x - a| |x - b| \leq M $$It seems obvious that$$ |x - a| |x - b| \leq |b-a|^2 $$however it seems that the optimal upper bound is in fact$$ |x-a||x-b| \leq \frac{|b-a|^2}{4} $$Does anyone know how to prove this? 3 I'm studying with the book 'Numerical Linear Algebra' written by L. N. Trefethen, and I wrote code regarding the text below: Here is a numerical example. Let A be a 200 x 200 matrix whose entries are independent samples from the real normal distribution of mean 2 and standard deviation 0.5/\sqr... 10:45 AM 11:21 AM 1 Definition : A curve \omega : [0,1]\to X is defined absolutely continuous whenever there exists g\in L^1([0,1]) such that d(\omega(t_0),\omega(t_1))\le\int_{t_0}^{t_1}g(s)ds for every t_0<t_1. I would like to show that, according to the above definition, the graph of the Cantor function... 11:51 AM @Feeds Answers to this question are eligible for a +50 reputation bounty. edamondo wants to draw more attention to this question. @HNQmath.se Removed from HNQ by adding MathJax. 12:49 PM 2 I need some help with a question. I have to calculate$$\lim_{x \to 3}\frac{x^2}{x - 3}\int_3^x \frac{\sin t}{t}dt.$$If I'm not wrong, we can write$$\sin(x) = \sum_{n = 0}^{\infty}\frac{x^{2n+1}}{(2n+1)!} \Longrightarrow \frac{\sin(x)}{x}=\sum_{n = 0}^{\infty}\frac{x^{2n}}{(2n+1)!},$$then$$...
1 hour later…
1:50 PM
1
Think of $\Sigma$ as a covariance matrix, or any positive semidefinite matrix. Let $A(\lambda)$ be a $n \times n$ positive semidefinite matrix with $\lambda > 0$ and the following specifications to its inverse: $(A(\lambda))^{-1}_{jk}=\begin{cases}\Sigma_{jk}+\lambda, & j=k\\ \Sigma_{jk}+\lambda ... 2:09 PM @Feeds Answers to this question are eligible for a +200 reputation bounty. SABOY is looking for an answer from a reputable source. 2:28 PM 3 I am trying to evaluate $$\iint_{R} x+y \:d A$$, where$R$is the region formed by the vertices $$(0,0),(5,0),\left(\frac{5}{2}, \frac{5}{2}\right) \text { and }\left(\frac{5}{2},-\frac{5}{2}\right)$$. My try: Here is the picture of the region which has two triangular regions. Let the top traing... 2:39 PM 2:54 PM 1 hour later… 4:22 PM 1 Let$(\Omega,\mathcal A,\operatorname P)$be a probability space,$E$be a normed$\mathbb R$-vector space and$(X_t)_{t\ge0}$be an$E$-valued càdlàg Lèvy process on$(\Omega,\mathcal A,\operatorname P)$. How can we prove that there is a (unique) transition kernel$\pi$from$(\Omega,\mathcal A...
4:44 PM
2
I was wondering if the following integral has a closed-form solution? $$I(x) = \int J_0(x)\sin(ax)\mathrm{d}x$$ where $a$ is a constant. I know the answer for the case when $a=1$, see here. I tried the similar method in that link but I was stuck. Integrating by parts yields I(x) =x J_0(x)\sin(... 5:20 PM @Feeds Answers to this question are eligible for a +50 reputation bounty. 0xbadf00d is looking for a canonical answer. @Feeds Answers to this question are eligible for a +50 reputation bounty. Jiaxin Zhong wants to draw more attention to this question. -1 I have 6 (A…F) noisy 3D normal vectors <x_hat, y_hat, z_hat > and noisy point cloud points <x, y, z> that form a cube and are related by the following vector operations: A cross B = B cross C = C cross D = D cross A A cross E = E cross C = F cross A = C cross F A=-C B=-D E=-F (A cross B) dot E = ... 1 hour later… 6:51 PM 0 Take the density of a generalized student-t, i.e., \begin{align*} p( y_t | \sigma , \mu, \nu ) = \frac {\Gamma (\frac {\nu +1}{2})}{\Gamma (\frac {\nu }{2}){\sqrt {\pi \nu }}{\sigma }\,}\left(1+{\frac {1}{\nu }}\left({\frac {y_t- \mu }{\sigma }}\right)^{2}\right)^{-\frac {\nu +1}{2}} \end{alig... 7:04 PM @Feeds Answers to this question are eligible for a +50 reputation bounty. DanGoodrick wants to draw more attention to this question: > Please let me know how I can improve the question if it isn't clear. @Feeds Answers to this question are eligible for a +50 reputation bounty. Monolite wants to draw more attention to this question. 7:27 PM 3 Original Problem: Let z be a complex number. The number 1 is written on a board. You perform a series of moves, where in each move you may either replace the number w written on the board with zw$or replace the number$w$with a different complex number$w'$so that$$\max(\lvert\oper... 2 hours later… 9:04 PM 4 Let U be the set of all sets. Define a partial ordering on U by inclusion: A≤B iff A ⊆ B for A, B ∈ U. Consider a chain C of U under this partial ordering: C : A1 ≤ A2 ≤ A3 ≤ · · · . Define B = ∪i⩾1A{i}. Clearly, B ∈ U and it is an upper bound of the chain C. Hence, Zorn’s Lemma implies that U ha... 4 How can I determine if this two figures are homeomorphic? I'm guessing they're not homeomorphic. I have tried using cut points but from what I understand both figures have the same number of cut points. I can see that in the first picture the circle in the center is connected to the four other ci... 9:57 PM 4 Let$G= \langle g_1, g_2 \rangle$be a finite group. Let$k$be a finite field with${\rm char}(k)=p>0$such that$p \mid |G|$. Let the$kG$-module$M$be a MeatAxe-module in GAP. The generators of$M$are given by the two matrices$m_1$and$m_2$, respectively, which reflect the actions of$g_1$... 2 hours later… 11:59 PM 2 Let$A, B$, and$C$be three points in$\mathbb{R}^2$such that$A = (x_1,y_1), B=(x_2,y_2)$, and$C= (x_3,y_3)$where$x_1<x_2<x_3$and$y_1 > y_2 > y_3$. In other words,$B$is to the upper left of$C$and$A$is to the upper left of$B$. What is the curve$y\$ of fastest descent that contains t... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9716140627861023, "perplexity": 979.3661406944877}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663021405.92/warc/CC-MAIN-20220528220030-20220529010030-00494.warc.gz"} |
https://readingfeynman.org/tag/state-vectors/ | # Quantum math: states as vectors, and apparatuses as operators
Pre-script (dated 26 June 2020): Our ideas have evolved into a full-blown realistic (or classical) interpretation of all things quantum-mechanical. In addition, I note the dark force has amused himself by removing some material. So no use to read this. Read my recent papers instead. 🙂
Original post:
I actually wanted to write about the Hamiltonian matrix. However, I realize that, before I can serve the plat de résistance, we need to review or introduce some more concepts and ideas. It all revolves around the same theme: working with states is like working with vectors, but so you need to know how exactly. Let’s go for it. 🙂
In my previous posts, I repeatedly said that a set of base states is like a coordinate system. A coordinate system allows us to describe (i.e. uniquely identify) vectors in an n-dimensional space: we associate a vector with a set of real numbers, like x, y and z, for example. Likewise, we can describe any state in terms of a set of complex numbers – amplitudes, really – once we’ve chosen a set of base states. We referred to this set of base states as a ‘representation’. For example, if our set of base states is +S, 0S and −S, then any state φ can be defined by the amplitudes C+ = 〈 +S | φ 〉, C0 = 〈 0S | φ 〉, and C = 〈 −S | φ 〉.
We have to choose some representation (but we are free to choose which one) because, as I demonstrated when doing a practical example (see my description of muon decay in my post on how to work with amplitudes), we’ll usually want to calculate something like the amplitude to go from one state to another – which we denoted as 〈 χ | φ 〉 – and we’ll do that by breaking it up. To be precise, we’ll write that amplitude 〈 χ | φ 〉 – i.e. the amplitude to go from state φ to state χ (you have to read this thing from right to left, like Hebrew or Arab) – as the following sum:
So that’s a sum over a complete set of base states (that’s why I write all i under the summation symbol ∑). We discussed this rule in our presentation of the ‘Laws’ of quantum math.
Now we can play with this. As χ can be defined in terms of the chosen set of base states too, it’s handy to know that 〈 χ | i 〉 and 〈 i | χ 〉 are each other’s complex conjugates – we write this as: 〈 χ | i 〉 = 〈 i | χ 〉* – so if we have one, we have the other (we can also write: 〈 i | χ 〉* = 〈 χ | i 〉). In other words, if we have all Ci = 〈 i | φ 〉 and all Di = 〈 i | χ 〉, i.e. the ‘components’ of both states in terms of our base states, then we can calculate 〈 χ | φ 〉 as:
〈 χ | φ 〉 = ∑ Di*Ci = ∑〈 χ | i 〉〈 i | φ 〉,
provided we make sure we do the summation over a complete set of base states. For example, if we’re looking at the angular momentum of a spin-1/2 particle, like an electron or a proton, then we’ll have two base states, +ħ/2 and +ħ/2, so then we’ll have only two terms in our sum, but the spin number (j) of a cobalt nucleus is 7/2, so if we’d be looking at the angular momentum of a cobalt nucleus, we’ll have eight (2·j + 1) base states and, hence, eight terms when doing the sum. So it’s very much like working with vectors, indeed, and that’s why states are often referred to as state vectors. So now you know that term too. 🙂
However, the similarities run even deeper, and we’ll explore all of them in this post. You may or may not remember that your math teacher actually also defined ordinary vectors in three-dimensional space in terms of base vectors ei, defined as: e= [1, 0, 0], e= [0, 1, 0] and e= [0, 0, 1]. You may also remember that the units along the x, y and z-axis didn’t have to be the same – we could, for example, measure in cm along the x-axis, but in inches along the z-axis, even if that’s not very convenient to calculate stuff – but that it was very important to ensure that the base vectors were a set of orthogonal vectors. In any case, we’d chose our set of orthogonal base vectors and write all of our vectors as:
A = Ax·e1 + Ay·e+ Az·e3
That’s simple enough. In fact, one might say that the equation above actually defines coordinates. However, there’s another way of defining them. We can write Ax, Ay, and Az as vector dot products, aka scalar vector products (as opposed to cross products, or vector products tout court). Check it:
A= A·e1, A= A·e2, and A= A·e3.
This actually allows us to re-write the vector dot product A·B in a way you’ve probably haven’t seen before. Indeed, you’d usually calculate A·B as |A|∙|B|·cosθ = A∙B·cosθ (A and B is the magnitude of the vectors A and B respectively) or, quite simply, as AxB+ AyB+ AzBz. However, using the dot products above, we can now also write it as:
We deliberately wrote B·A instead of Abecause, while the mathematical similarity with the
〈 χ | φ 〉 = ∑〈 χ | i 〉〈 i | φ 〉
equation is obvious, B·A = A·B but 〈 χ | φ 〉 ≠ 〈 φ | χ 〉. Indeed, 〈 χ | φ 〉 and 〈 φ | χ 〉 are complex conjugates – so 〈 χ | φ 〉 = 〈 φ | χ 〉* – but they’re not equal. So we’ll have to watch the order when working with those amplitudes. That’s because we’re working with complex numbers instead of real numbers. Indeed, it’s only because the A·B dot product involves real numbers, whose complex conjugate is the same, that we have that commutativity in the real vector space. Apart from that – so apart from having to carefully check the order of our products – the correspondence is complete.
Let me mention another similarity here. As mentioned above, our base vectors ei had to be orthogonal. We can write this condition as:
ei·ej = δij, with δij = 0 if i ≠ j, and 1 if i = j.
Now, our first quantum-mechanical rule says the same:
〈 i | j 〉 = δij, with δij = 0 if i ≠ j, and 1 if i = j.
So our set of base states also has to be ‘orthogonal’, which is the term you’ll find in physics textbooks, although – as evidenced from our discussion on the base states for measuring angular momentum – one should not try to give any geometrical interpretation here: +ħ/2 and +ħ/2 (so that’s spin ‘up’ and ‘down’ respectively) are not ‘orthogonal’ in any geometric sense, indeed. It’s just that pure states, i.e. base states, are separate, which we write as: 〈 ‘up’ | ‘down’ 〉 = 〈 ‘down’ | ‘up’ 〉 = 0 and 〈 ‘up’ | ‘up’ 〉 = 〈 ‘down’ | ‘down’ 〉 = 1. It just means they are just different base states, and so it’s one or the other. For our +S, 0S and −S example, we’d have nine such amplitudes, and we can organize them in a little matrix:
In fact, just like we defined the base vectors ei as e= [1, 0, 0], e= [0, 1, 0] and e= [0, 0, 1] respectively, we may say that the matrix above, which states exactly the same as the 〈 i | j 〉 = δij rule, can serve as a definition of what base states actually are. [Having said that, it’s obvious we like to believe that base states are more than just mathematical constructs: we’re talking reality here. The angular momentum as measured in the x-, y- or z-direction, or in whatever direction, is more than just a number.]
OK. You get this. In fact, you’re probably getting impatient because this is too simple for you. So let’s take another step. We showed that the 〈 χ | φ 〉 = ∑〈 χ | i 〉〈 i | χ 〉 and B·= ∑(B·ei)(ei·A) are structurally equivalent – from a mathematical point of view, that is – but B and A are separate vectors, while 〈 χ | φ 〉 is just a complex number. Right?
Well… No. We can actually analyze the bra and the ket in the 〈 χ | φ 〉 bra-ket as separate pieces too. Moreover, we’ll show they are actually state vectors too, even if the bra, i.e. 〈 χ |, and the ket, i.e. | φ 〉, are ‘unfinished pieces’, so to speak. Let’s be bold. Let’s just cut the 〈 χ | φ 〉 = ∑〈 χ | i 〉〈 i | χ 〉 by writing:
Huh?
Yes. That’s the power of Dirac’s bra-ket notation: we can just drop symbols left or right. It’s quite incredible. But, of course, the question is: so what does this actually mean? Well… Don’t rack your brain. I’ll tell you. We define | φ 〉 as a state vector because we define | i 〉 as a (base) state vector. Look at it this way: we wrote the 〈 +S | φ 〉, 〈 0S | φ 〉 and 〈 −S | φ 〉 amplitudes as C+, C0, C, respectively, so we can write the equation above as:
So we’ve got a sum of products here, and it’s just like A = Ax·e+ Ay·e2 + Az·e3. Just substitute the Acoefficients for Ci and the ebase vectors for the | i 〉 base states. We get:
| φ 〉 = |+S〉 C+ + |0S〉 C0 + |+S〉 C
Of course, you’ll wonder what those terms mean: what does it mean to ‘multiply’ C+ (remember: C+ is some complex number) by |+S〉? Be patient. Just wait. You’ll understand when we do some examples, so when you start working with this stuff. You’ll see it all makes sense—later. 🙂
Of course, we’ll have a similar equation for | χ 〉, and so if we write 〈 χ | i 〉 as Di, then we can write | χ 〉 = ∑ | i 〉〈 χ | i 〉 as | χ 〉 = ∑ | i 〉 Di.
So what? Again: be patient. We know that 〈 χ | i 〉 = 〈 i | χ 〉*, so our second equation above becomes:
You’ll have two questions now. The first is the same as the one above: what does it mean to ‘multiply’, let’s say, D0* (i.e. the complex conjugate of D0, so if D= a + ib, then D0* = a − ib) with 〈0S|? The answer is the same: be patient. 🙂 Your second question is: why do I use another symbol for the index here? Why j instead of i? Well… We’ll have to re-combine stuff, so it’s better to keep things separate by using another symbol for the same index. 🙂
In fact, let’s re-combine stuff right now, in exactly the same way as we took it apart: we just write the two things right next to each other. We get the following:
What? Is that it? So we went through all of this hocus-pocus just to find the same equation as we started out with?
Yes. I had to take you through this so you get used to juggling all those symbols, because that’s what we’ll do in the next post. Just think about it and give yourself some time. I know you’ve probably never ever handled such exercise in symbols before – I haven’t, for sure! – but it all makes sense: we cut and paste. It’s all great! 🙂 [Oh… In case you wonder about the transition from the sum involving i and j to the sum involving i only, think about the Kronecker expression: 〈 j | i 〉 = δij, with δij = 0 if i ≠ j, and 1 if i = j, so most of the terms are zero.]
To summarize the whole discussion, note that the expression above is completely analogous with the B·= BxA+ ByA+ BzAformula. The only difference is that we’re talking complex numbers here, so we need to watch out. We have to watch the order of stuff, and we can’t use the Dnumbers themselves: we have to use their complex conjugates Di*. But, for the rest, we’re all set! 🙂 If we’ve got a set of base states, then we can define any state in terms of a set of ‘coordinates’ or ‘coefficients’ – i.e. the Ci or Di numbers for the φ or χ example above – and we can then calculate the amplitude to go from one state to another as:
In case you’d get confused, just take the original equation:
The two equations are fully equivalent.
[…]
So we just went through all of the shit above so as to show that structural similarity with vector spaces?
Yes. It’s important. You just need to remember that we may have two, three, four, five,… or even an infinite number of base states depending on the situation we’re looking at, and what we’re trying to measure. I am sorry I had to take you through all of this. However, there’s more to come, and so you need this baggage. We’ll take the next step now, and that is to introduce the concept of an operator.
Look at the middle term in that expression above—let me copy it:
We’ve got three terms in that double sum (a double sum is a sum involving two indices, which is what we have here: i and j). When we have two indices like that, one thinks of matrices. That’s easy to do here, because we represented that 〈 i | j 〉 = δij equation as a matrix too! To be precise, we presented it as the identity matrix, and a simple substitution allows us to re-write our equation above as:
I must assume you’re shaking your head in disbelief now: we’ve expanded a simple amplitude into a product of three matrices now. Couldn’t we just stick to that sum, i.e that vector dot product ∑ Di*Ci? What’s next? Well… I am afraid there’s a lot more to come. For starters, we’ll take that idea of ‘putting something in the middle’ to the next level by going back to our Stern-Gerlach filters and whatever other apparatus we can think of. Let’s assume that, instead of some filter S or T, we’ve got something more complex now, which we’ll denote by A. [Don’t confuse it with our vectors: we’re talking an apparatus now, so you should imagine some beam of particles, polarized or not, entering it, going through, and coming out.]
We’ll stick to the symbols we used already, and so we’ll just assume a particle enters into the apparatus in some state φ, and that it comes out in some state χ. Continuing the example of spin-one particles, and assuming our beam has not been filtered – so, using lingo, we’d say it’s unpolarized – we’d say there’s a probability of 1/3 for being either in the ‘plus’, ‘zero’, or ‘minus’ state with respect to whatever representation we’d happen to be working with, and the related amplitudes would be 1/√3. In other words, we’d say that φ is defined by C+ = 〈 +S | φ 〉, C0 = 〈 0S | φ 〉, and C = 〈 −S | φ 〉, with C+ = C0 = C− = 1/√3. In fact, using that | φ 〉 = |+S〉 C+ + |0S〉 C0 + |+S〉 C− expression we invented above, we’d write: | φ 〉 = (1/√3)|+S〉 + (1/√3)|0S〉 C0 + (1/√3)|+S〉 C or, using ‘matrices’—just a row and a column, really:
However, you don’t need to worry about that now. The new big thing is the following expression:
〈 χ | A | φ〉
It looks simple enough: φ to A to χ. Right? Well… Yes and no. The question is: what do you do with this? How would we take its complex conjugate, for example? And if we know how to do that, would it be equal to 〈 φ | A | χ〉?
You guessed it: we’ll have to take it apart, but how? We’ll do this using another fantastic abstraction. Remember how we took Dirac’s 〈 χ | φ 〉 bra-ket apart by writing | φ 〉 = ∑ | i 〉〈 i | φ 〉? We just dropped the 〈 χ left and right in our 〈 χ | φ 〉 = ∑〈 χ | i 〉〈 i | φ 〉 expression. We can go one step further now, and drop the φ 〉 left and right in our | φ 〉 = ∑ | i 〉〈 i | φ 〉 expression. We get the following wonderful thing:
| = ∑ | i 〉〈 i | over all base states i
With characteristic humor, Feynman calls this ‘The Great Law of Quantum Mechanics’ and, frankly, there’s actually more than one grain of truth in this. 🙂
Now, if we apply this ‘Great Law’ to our 〈 χ | A | φ〉 expression – we should apply it twice, actually – we get:
As Feynman points out, it’s easy to add another apparatus in series. We just write:
Just put a | bar between B and A and apply the same trick. The | bar is really like a factor 1 in multiplication. However, that’s all great fun but it doesn’t solve our problem. Our ‘Great Law’ allows us to sort of ‘resolve’ our apparatus A in terms of base states, as we now have 〈 i | A | j 〉 in the middle, rather than 〈 χ | A | φ〉 but, again, how do we work with that?
Well… The answer will surprise you. Rather than trying to break this thing up, we’ll say that the apparatus A is actually being described, or defined, by the nine 〈 i | A | j 〉 amplitudes. [There are nine for this example, but four only for the example involving spin-1/2 particles, of course.] We’ll call those amplitudes, quite simply, the matrix of amplitudes, and we’ll often denote it by Aij.
Now, I wanted to talk about operators here. The idea of an operator comes up when we’re creative again, and when we drop the 〈 χ | state from the 〈 χ | A | φ〉 expression. We write:
So now we think of the particle entering the ‘apparatus’ A in the state ϕ and coming out of A in some state ψ (‘psi’). We can generalize this and think of it as an ‘operator’, which Feynman intuitively defines as follows:
The symbol A is neither an amplitude, nor a vector; it is a new kind of thing called an operator. It is something which “operates on” a state to produce a new state.”
But… Wait a minute! | ψ 〉 is not the same as 〈 χ |. Why can we do that substitution? We can only do it because any state ψ and χ are related through that other ‘Law’ of quantum math:
Combining the two shows our ‘definition’ of an operator is OK. We should just note that it’s an ‘open’ equation until it is completed with a ‘bra’, i.e. a state like 〈 χ |, so as to give the 〈 χ | ψ〉 = 〈 χ | A | φ〉 type of amplitude that actually means something. In practical terms, that means our operator or our apparatus doesn’t mean much as long as we don’t measure what comes out, so then we choose some set of base states, i.e. a representation, which allows us to describe the final state, i.e. 〈 χ |.
[…]
Well… Folks, that’s it. I know this was mighty abstract, but the next posts should bring things back to earth again. I realize it’s only by working examples and doing exercises that one can get some kind of ‘feel’ for this kind of stuff, so that’s what we’ll have to go through now. 🙂
Some content on this page was disabled on June 16, 2020 as a result of a DMCA takedown notice from The California Institute of Technology. You can learn more about the DMCA here: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9176040291786194, "perplexity": 971.9755579767151}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735860.28/warc/CC-MAIN-20200804043709-20200804073709-00511.warc.gz"} |
https://www.lessonplanet.com/teachers/muscular-system-science-9th-12th | # Muscular System
For this muscular system worksheet, high schoolers define related terms of the muscular system, they identify types of muscles, they order the events of muscle contraction and given a clue about structures and function of the muscular system they find the proper term and write it in a series of boxes. The circled letters combine to form the final term for the theory of muscle contraction. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.858722448348999, "perplexity": 1316.484444850956}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423839.97/warc/CC-MAIN-20170722002507-20170722022507-00072.warc.gz"} |
http://physics.stackexchange.com/questions/81883/2-dimensional-coulombs-law-equation | # 2 dimensional Coulomb's law equation
We can notice that in the Coulomb's law equation,
$$$$\tag{1}F=\frac{1}{4\pi\epsilon}\cdot\frac{q_1q_2}{r^2}$$$$
$4\pi r^2$ factor in the denominator expresses directly the surface of a virtual sphere with radius $r$. Actually we can look at this equation as it was for $3$ dimensional objects. If we suppose want to consider for $2$ dimensional objects, can we modify the equation as,
$$$$\tag{2}F=\frac{1}{2\pi\epsilon}\cdot\frac{q_1q_2}{r}$$$$
Here we can think of $2\pi r$ as area of virtual circle. I don't really know whether it works or not. So, can we have equation (2) as the modified equation for electrostatic force between two $2$ dimensional uniformly charged objects?
-
Physically speaking, the laws of electrodynamics are 3-dimensional and so you have to take these as starting point and see what they imply for any charge configuration of interest. A force $F$ of form $\propto\frac{1}{4\pi}\frac{1}{r^2}$ falls faster than one which goes as $\propto\frac{1}{2\pi}\frac{1}{r}$ and so without futher information, the physics which apply is the known behaviour $\propto\frac{1}{4\pi}\frac{1}{r^2}$, which you can also write as $\propto\frac{\partial}{\partial r}\left(\frac{1}{4\pi}\frac{-1}{r}\right)$
Mathematically speaking, what what you do is to compute $F\propto\text{grad}(G)$, where the force $F$ is the gradient of a potential $G$ which is given from the Poisson equation in $n$ dimensions, and where there is only one charge in the center of the coordinate system. Your two dimensional force is $F\propto \frac{1}{2\pi}\frac{1}{r}= \frac{1}{2\pi}\frac{\partial }{\partial r}\mathrm{ln}(r)$, i.e. $G= \frac{1}{2\pi}\mathrm{ln}(r)$. A list of similar potentials is given here, only the fifth of which corresponds to electrostatics in 3 dimensions:
http://en.wikipedia.org/wiki/Green%27s_function#Table_of_Green.27s_functions
-
Gauss law is the most general form of equation to describe the electric field. Columb law for an arbitrary electric field states F=q*E. Gauss law in its integral form reads
D is the electric flux density, dS is the surface normal element, rho is the charge density and dV is the volume element. What that equation physically says is, the charge confined in a volume is equal to the surface integral of flux normal to the surface of that volume. As you see it is 3D by definition as it includes volume and surface. If you tested equation 2 you wrote against Gauss law, you will see it is inconsistent. That is why equation 2 doesn't describe a point charge under any circumstance, simply because the flux across the "circle" as you described it is part of the total flux through the sphere.
As a general rule, Gauss law applies to 3D, when you want to use in 2D or 1D you should start from 3D and make necessary simplifications. For 2D usage think of it as taking a slice to convert the 3D to 2D. The law will remain the same.
For the record, equation 2 has a r-dependence that describes an infinitely long charged line. That is one of the common exercises student do in elementary electromagnetic class, which is finding the electric field of an infinitely long charged line using Gauss law.
Have a look here for general description of Gauss law. In page 6 you see the example I am speaking about.
-
Well surely you can consider it for 2 dimensional chardes, but to check it out experimentally would simply not be possible. As no charge known to us is 2 dimensional in its existence and its electric influence is also spread in the 3 dimensions we know of, experiencing and experimenting with 2d is not possible to date and hence your hypothesis can not be tested for validation.
Seeing the analogy your extrapolation seems correct and i believe similarly we can get results even for a single dimensional world or even multiple dimensional worlds. But again all of these can neither be proven nor disproven.
-
There are analogous systems in condensed matter, like Abrikosov vortices in superconductors, which have effective 2D interactions. – Michael Brown Oct 23 '13 at 9:52
I do not really know about abrikosov vortices but is that(condensed matter) only where the analogue would be applicable, nowhere else ? – Rijul Gupta Oct 23 '13 at 9:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9391329884529114, "perplexity": 332.7999969502216}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207926736.56/warc/CC-MAIN-20150521113206-00136-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://slideplayer.com/slide/267141/ | # S PEED OF G RAVITY by Robert Nemiroff Michigan Tech.
## Presentation on theme: "S PEED OF G RAVITY by Robert Nemiroff Michigan Tech."— Presentation transcript:
S PEED OF G RAVITY by Robert Nemiroff Michigan Tech
Physics X: About This Course Officially "Extraordinary Concepts in Physics" Being taught for credit at Michigan Tech o Light on math, heavy on concepts o Anyone anywhere is welcome No textbook required o Wikipedia, web links, and lectures only o Find all the lectures with Google at: "Starship Asterisk" then "Physics X" o http://bb.nightskylive.net/asterisk/viewforum.php?f=39
S PEED OF G RAVITY Does gravity propagate at the speed of light? It is clear that gravitational radiation propagates at c, although this has never been directly experimentally confirmed, since gravitational radiation has not yet been directly detected. But what about gravity itself?
S PEED OF G RAVITY S PEED OF G RAVITY : H ISTORY Newton (1680s) assumes that the speed of gravity is infinite. o This fits all contemporary observations. Laplace (1805) tries to fit Newton's theory with a wave mechanism where the speed of gravity was equal to the speed of light o Failed. Laplace then estimated that gravity moves ~106 times faster than light.
S PEED OF G RAVITY S PEED OF G RAVITY : H ISTORY Lorentz (1904) creates ether theory where gravity propagates at light speed. o Laplace problem fixed. o Precession of Mercury too small. Many more people propose many gravitational theories. Einstein proposes General Relativity (1915) o In GR gravity propagates a light speed.
A BERRATION OF L IGHT You see two stars before you in the distance. You start moving rapidly toward them. What do you see? 1. The stars appear to move apart. 2. The stars appear to move together. 3. The stars appear the same. 4. The stars get out of the way just to be safe.
A BERRATION OF L IGHT 2. The stars appear to move together. This is a known effect from special relativity. This is caused by the finite speed of light. This is in addition to the Doppler - color effect where the stars in front would appear more blue, while the stars behind would appear more red.
A BERRATION OF L IGHT You are placed in a circular orbit around the Sun. Because of aberration, does the Sun appear precisely 90 degrees from your orbital motion? 1. Yes, that is necessary for a circular orbit. 2. No, aberration makes the Sun appear slightly ahead of you. 3. No, aberration 'leaves the Sun behind' and makes the Sun appear slightly behind you.
A BERRATION OF L IGHT 2. No, aberration makes the Sun appear slightly ahead of you. The faster you orbit, the more the Sun will appear ahead of you.
A BERRATION OF L IGHT OK, the Sun appears slightly ahead of you. Does sunlight push you back, creating a "drag force" as you orbit the Sun. 1. Yes, that sounds reasonable. 2. No, that would cause the Earth to fall into the Sun. 3. No -- the Sun, being over there, cannot create a force over here.
A BERRATION OF L IGHT 1. Yes, that sounds reasonable. This is called the Poynting-Robertson effect and is a primary reason (for example), why dust particles fall into the Sun. The effect on the Earth, although real, is very small.
A BERRATION OF G RAVITY OK, the Sun appears slightly ahead of you. Does the gravity of the Sun also appear slightly ahead of you? 1. Yes, since sunlight and gravity move at the same speed. 2. No, gravity is immune to this effect. 3. Does this mean the Solar System is unstable?
A BERRATION OF G RAVITY 2. No, gravity is immune to this effect. Although few direct experiments have been done, the stability of the Earth's orbit puts limit on aberrational effects. As detailed in Carlip (2000), in Einstein's general relativity, there are velocity dependent terms that cancel the aberration effect. The calculation is complex but demanded by conservation of angular momentum, although the emission of gravitational radiation will make the cancellation inexact. Carlip (2000)
A BERRATION OF G RAVITY It is therefore possible to see someone in one direction, and feel the force of its pull from a different direction! Strange!
A BERRATION OF AN E LECTRIC F IELD Does a charged object orbiting an oppositely charged object see an aberrated electric field? 1. Yes, all electromagnetic effects will feel aberration. 2. No, electric fields are (also) immune to this effect.
A BERRATION OF AN E LECTRIC F IELD 2. No, electric fields are immune to this effect. There is no aberration of an electric field. This is an observed fact. Given Noether's theorem, any field that is invariant over in time will conserve energy (locally) and hence will not show aberration. When worked out in detail, velocity dependent terms come in to cancel the effect of the finite speed of propagation of a changing electric field.Noether's theorem
Download ppt "S PEED OF G RAVITY by Robert Nemiroff Michigan Tech."
Similar presentations | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9109829664230347, "perplexity": 2106.8512739297657}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591831.57/warc/CC-MAIN-20180720193850-20180720213850-00028.warc.gz"} |
http://www.mediander.com/connects/23231/parabola/ | In mathematics, a parabola is a plane curve, which is mirror-symmetrical, and is approximately U-shaped when oriented as shown in the diagram below (it remains a parabola if is differently oriented). It fits any of several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. MORE
Mediander uses proprietary software that curates millions of interconnected topics to produce the Mediander Topics search results. As with any algorithmic search, anomalous results may occur. If you notice such an anomaly, or have any comments or suggestions, please contact us. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8771675825119019, "perplexity": 1328.6424079368905}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948588294.67/warc/CC-MAIN-20171216162441-20171216184441-00516.warc.gz"} |
https://research.nottingham.edu.cn:443/en/publications/robust-adaptive-control-based-on-variable-boundary-for-a-twin-mot | # Robust Adaptive Control Based on Variable Boundary for a Twin-Motor Cable-Driven System
Bin Li, Liang Yan, Xiaoshan Gao, Chris Gerada
Research output: Journal PublicationArticlepeer-review
## Abstract
Cable-driven parallel mechanism has been widely studied due to its advantages of fast response and large workspace. The structural uncertainties in the system often introduce internal disturbance and unavoidably influence its motion precision. Generally, to reduce the influence, robust control is adopted and the boundary of the disturbance is regarded as a fixed value given by rough estimation. This method often causes large vibration of the motion component, which certainly compromises the dynamic performance of the system. Therefore, to solve this problem, a variable boundary analytical scheme based on the decoupling of control law and internal disturbance is proposed in this article. Specifically, sufficient condition for the solvability of disturbance boundary is proved by the boundedness of the structural matrix error. Then, the disturbance boundary is modeled analytically, so that its particular value can be determined with respect to the system status. Following that, a robust adaptive control algorithm based on the variable boundary of disturbance is developed. It is verified that the tracking error is globally uniformly bounded. The experimental results show that the proposed control method can effectively reduce the tracking errors and attenuate the tension chattering compared with the conventional robust adaptive control scheme with a roughly estimated boundary.
Original language English 7054-7063 10 IEEE Transactions on Industrial Electronics 69 7 https://doi.org/10.1109/TIE.2021.3097609 Published - 1 Jul 2022 Yes
## Keywords
• Cable-driven system
• Parameters uncertainty | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8600853681564331, "perplexity": 1108.8037526603425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335491.4/warc/CC-MAIN-20220930145518-20220930175518-00315.warc.gz"} |
https://pqnelson.wordpress.com/2012/07/08/notes-on-mathematical-writing/ | ## Notes on Mathematical Writing
1. I’m going to be posting my double integral notes, but I’d like to discuss my strategy when writing “more rigorous” mathematics. Modern mathematics consists of definitions, theorems, and proofs…so I’ll discuss the idiosyncrasies of each. (I’ll probably revise this several times…)
2. Definitions. A definition introduces a new gadget (e.g., vector spaces), some new structure (e.g., a linear operator, a norm, etc.), or some new property (e.g., invertibility).
So the typical definition would look like the following:
An “Object” consists of “Stuff” equipped with “Structure” such that these equations — called “Properties” (or Axioms) — hold.
A typological remark: sometimes axioms are given in bold. For example, consider the following snippet of a definition
…satisfying the following properties:
Disjoint Union: [snip]
Sewing: [snip]
Normalization: [snip]
This has the added bonus of allowing us to write “By sewing, this gadget then transforms Equation (blah) into Equation (snort).”
Some people prefer giving as few axioms as possible, then proving the rest. This seems pathological to me, and confuses the reader. I prefer being frank and honest, telling the reader “This gadget has the following useful properties” and in a remark note when some imply others.
3. Theorems. Euclid had a systematic manner of presenting theorems, as Proclus noted.
Knuth suggests (in rule 11) we should “try to state things twice, in complementary ways…” which should be done with theorems as well. Personally, I think writing the theorem without symbols — if possible — would be best. This would correspond to Euclid’s “Enunciation”.
After the theorem statement, we should note the proof strategy and/or key moments of the proof.
Continuous functions on a closed interval are uniformly continuous.
Theorem. If $f\colon[a,b]\to\mathbb{R}$ is continuous on every $x\in[a,b]$, then $f$ is uniformly continuous on $[a,b]$.
We will prove this directly.
Proof. Let $f$ be a continuous function on the closed interval $[a,b]$, then… [the rest of the proof is omitted].
We should make the theorem be precise, boring, and cookie-cutter in its format. But we should say, either before or after, what it really says in a frank way: “Continuous functions on a closed interval are uniformly continuous.”
4. Proofs. We should begin introducing all the necessary variables and quantities. Constantly ask yourself “What does the reader know?” Don’t be afraid to reiterate what has been done “We’ve just shown (blah), but we need to prove (oink).”
5. Examples. Examples should be worked out exercises. So it consists of two parts: (1) statement of the problem, and (2) its solution.
In physics, it consists of 4 parts, which we will not discuss now [google “Hugh and Young IDEE”].
Now the problem lies in how do we pick good examples? It’s a very hard thing to do. It should be clean, i.e. free of excessive calculations; it should be inviting, interesting to the reader; and it should be challenging, i.e. not as simple as it seems.
Harrington has Ten Commandments for mathematical writing, which I more or less agree with, they are:
Rule 1. Organize in segments. Harrington’s “segments” are what I call chunks (in my blog entries, they are the numbered collection of paragraphs, possibly with a title of some sort). It can be read comfortably from beginning to end without pausing.
So what’s a good segment? An example, a definition, a theorem and its proof. These are typical. What else? Well, motivating a concept, reflections, etc.
Rule 2. Write segments linearly. We should organize each chunk to do one thing, and only one thing. Give us an example, or a result and its proof, or a concept.
The chunk should flow linearly, making it easy to read. How can we accomplish this? As Halmos once said: organize, organize, organize. Write up an outline, with only the highlights written down. For example:
1. Definition. Continuity…
Example 1. Polynomials.
Example 2. Trig Functions.
Non-Example 3. Step function at the jump.
2. Definition. Derivative…
Example 1. Polynomials.
Example 2. Trig Functions.
Non-Example 3. Absolute value at zero.
3. Theorem. Chain rule.
Example 1. Apply it to $\cos(\exp(x^{5}))$.
4. Theorem. Product rule.
Example 1. Apply it inductively to $x^{n}$ to recover the product rule.
Rule 3. Consider a hierarchical development. Think about how the segments depend on each other. This affects how we organize our outline.
Don’t be afraid to reorganize.
Rule 4. Use consistent notation and nomenclature. Again, I don’t think I have to say anything about this, but if you use LaTeX you might want to consider using macros to specify notation. I usually write things like \def\CC{\mathbb{C}} and so on, just so I have consistent typography.
Rule 5. State results consistently. Be sure to write theorems consistently. Usually it’s of the form “If P, then Q.”
Similarly, with definitions, be consistent “A gadget consists of…equipped with…such that…”.
Rule 6. Don’t underexplain but don’t overexplain. But if in doubt, overexplain. Harrington demands you write for an audience (Halmos says “Pick someone you know, and pretend you’re explaining it to them”)…I usually write for myself, knowing I’ll forget the material and have to re-learn it quick.
Consequently, I write specifically so if someone says “I need to learn everything you know about [blank]” I can say “Give me five minutes” (so I can read my notes, and recall everything about the subject). That’s how I wrote my Feynman diagram notes.
Rule 7. Tell them what you’ll tell them. Tell the reader where you’re going. Also consider motivating the problem appropriately.
Rule 8. Use suggestive references. Don’t say “By theorem blah, we transform Equation (snort) into Equation (oink).” The reader has to flip back to find “theorem blah”, then consider carefully how to apply it. How to revise it?
One should write “Since all continuous functions on closed intervals are uniformly continuous” instead of “By theorem blah”: i.e., write the contents of the theorem instead of referring to it.
One should then write “…we obtain [from our previous equation] (oink)”.
Don’t fear repetition, but whenever you repeat yourself…reword it differently, just to break the monotony.
Rule 9. Consider examples and counterexamples. Examples should demonstrate the point, or demonstrate the bounds of some definition. Count-examples are useful for showing when some algorithm fails.
Rule 10. Use visualization when possible. You know, a picture is worth a thousand words… | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 8, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9066038727760315, "perplexity": 1713.8758107861722}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824293.62/warc/CC-MAIN-20171020173404-20171020193404-00118.warc.gz"} |
https://proofwiki.org/wiki/Mathematician:Francis_Clarke | # Mathematician:Francis H. Clarke
## Mathematician
Canadian and French mathematician, known for his contributions to nonsmooth analysis (a term that is due to him), and particularly for his theory of generalized gradients (gradients généralisés), as well as for his work in optimization, the differential equations , control theory, calculation of variations, and modeling in several application domains. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9072389602661133, "perplexity": 1462.397441528854}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347436466.95/warc/CC-MAIN-20200603210112-20200604000112-00514.warc.gz"} |
https://cdsweb.cern.ch/collection/ATLAS%20Theses?ln=pt&as=1 | # ATLAS Theses
2020-02-25
06:56
Study of the decay properties of the Higgs boson into two b quarks and upgrade of the ATLAS inner tracker / d'Eramo, Louis This thesis is focussing on the search for the Standard Model (SM) Higgs boson decaying to a pair of b-quarks in the vector boson associated production mode (VH) with the data collected with the ATLAS experiment at √s = 13 TeV corresponding to an integrated luminosity of L= 79.8 fb$^{−1}$ [...] tel-02428533. - 2019. - 340 p.
Full text - Fulltext
2020-02-24
09:41
Searches for Diboson New Physics and the L1Calo Software Development with the ATLAS Detector / Lin, Chiao-Ying The Standard Model has been a successful theory in describing the behaviour of fundamental particles, but there are still problems remaining unsolved [...] CERN-THESIS-2019-300 - 259 p.
Approve this document (restricted) - Full text
2020-02-22
07:03
A Search for New Physics in the Dilepton Channel with the ATLAS Detector at the LHC / Fitzgerald, Eric Andrew Vitus This thesis presents a search for a new, neutral heavy gauge boson decaying to lepton pairs using data from the ATLAS detector at the LHC [...] AAT-3611618 ; CERN-THESIS-2014-472. - 2014. - 296 p.
Full text
2020-02-19
06:01
Boosting to the top: measurements of boosted top quarks and Higgs bosons with the ATLAS detector at the Large Hadron Collider / Fenton, Michael Between 2015 and 2018, proton-proton collisions were performed at the highest energy ever achieved in man-made particle accelerators, with the Large Hadron Collider at CERN [...] CERN-THESIS-2019-295 - 2019. - 222 p.
Link to Glasgow U. server - Full text
2020-02-12
18:00
Searching for Supersymmetry using the Higgs Boson / Weston, Thomas The Large Hadron Collider and its experiments constitute the largest particle physics research programme to date, allowing for extensive research of the existing Standard Model and for potential evidence of physics beyond that of our current comprehension [...] CERN-THESIS-2019-286 - 180 p.
Full text
2020-02-10
20:38
Search for resonant $WZ$ production in the fully leptonic final state with the ATLAS detector / Freund, Benjamin Diboson resonance searches are an essential test of electroweak symmetry breaking theories beyond the Standard Model (SM) of particle physics [...] CERN-THESIS-2019-285 - 191 p.
Full text
2020-01-30
23:27
Search for top squark pair production and decay in $t$ and $\tilde{\chi}^{0}_{1}$, with two leptons in the final state, at the ATLAS Experiment with LHC Run 2 data / Longo, Luigi The elementary particles, together with the strong, electromagnetic and weak interactions among them, are coherently described by the Standard Model (SM) of particle physics [...] CERN-THESIS-2018-466 - 168 p.
Full text
2020-01-28
16:52
Search for flavour-changing neutral currents in processes with a single top quark in association with a photon using a deep neural network at the ATLAS experiment at $\sqrt{s}$ = 13 TeV / Gessner, Gregor In this thesis, a search for flavour-changing neutral currents in processes involving a singly produced top quark and a photon is presented [...] CERN-THESIS-2019-277 - 199 p.
Full text
2020-01-27
15:41
Developing Module Assembly and Quality Control Procedures for the HL-LHC Upgrade of the ATLAS Inner Tracker / Steentoft, Jonas To further probe the fundamental structures of matter, the Large Hadron Collider (LHC) at CERN is undergoing a programme of upgrades aimed at increasing the instantaneous luminosity by a factor 5 − 7 [...] CERN-THESIS-2019-276 - 105 p.
Full text
2020-01-24
12:06
New pixel-detector technologies for the ATLAS ITk upgrade and the CLIC vertex detector / Vicente Barreto Pinto, Mateus This thesis contains the Ph.D [...] CERN-THESIS-2019-273 - 231 p.
Full text | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9824401140213013, "perplexity": 2139.4892172396985}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146940.95/warc/CC-MAIN-20200228012313-20200228042313-00121.warc.gz"} |
http://math.stackexchange.com/questions/49907/is-the-dual-representation-of-an-irreducible-representation-always-irreducible | # Is the dual representation of an irreducible representation always irreducible?
Let $G$ be a group and let $V$ be a complex vector space which is a representation of $G$. Let's write the (left) action of $g\in G$ on $v\in V$ as $gv$.
The dual vector space of $V$ is the set of linear maps from $V$ to $\mathbb{C}$, and is written as $V^*$. I'll use the notation $(\phi,v)$ to mean $\phi(v)$ if $\phi\in V^*$ and $v\in V$.
The dual representation of $V$ is defined to be the vector space $V^*$ with the (right) $G$-action $(\phi g,v):=(\phi,gv)$ for $\phi\in V^*$, $v\in V$ and $g\in G$.
A representation of $G$ is irreducible if the only $G$-invariant subspaces are the zero subspace or the whole vector space.
I can show that if $V^*$ is an irreducible representation, then so is $V$. Indeed, if $W$ is a $G$-invariant subspace of $V$ then its annihilator $W^\perp=\{\phi\in V^*\colon w\in W\implies (\phi,w)=0\}$ is a $G$-invariant subspace of $V^*$, so either $W^\perp=V^*$, which clearly implies $W=0$, or $W^\perp=0$, which (by fiddling around with a Hamel basis) yields $W=V$.
Does the converse hold? That is, if $V$ is irreducible, must $V^*$ also be irreducible?
If $\dim V<\infty$ then $V^{**}$ is equivalent to $V$, so the answer is yes in this case. But I don't know what happens if $\dim V=\infty$.
-
If $G$ acts from the left on $V$, doesn't it act from the right on $V^*$? – Rasmus Jul 6 '11 at 18:26
I made the title more informative. Feel free to roll-back if you don't like it. – Rasmus Jul 6 '11 at 18:29
@Rasmus: That depends on your conventions: you can use the antihomomorphism $g \mapsto g^{-1}$ to turn the right action into a left one (and vice versa). – t.b. Jul 6 '11 at 18:33
@Rasmus: yes, it does seem to act from the right! I'll change the notation... but the question still seems to make sense, and be interesting. – irrep Jul 6 '11 at 18:34
No. Let $G = S_{\infty}$ denote the group of permutations of $\mathbb{N}$ which fix all but finitely many elements. $G$ is countable, so any irreducible representation has at most countable dimension. $G$ has a countable-dimensional irreducible representation $V$ given by the subspace of $\bigoplus_{i=1}^{\infty} \mathbb{C}$ of sequences adding to $0$, and $V^{\ast}$ is of uncountable dimension, so cannot be irreducible. (Explicitly, there is an obvious bilinear pairing on $V$ giving $V^{\ast}$ a proper invariant subspace isomorphic to $V$.)
Edit: Here's a cute non-constructive argument. Suppose that $G$ has an infinite-dimensional irreducible representation $V$. If $V^{\ast}$ is reducible, we're done. Otherwise, $V$ is a proper invariant subspace of $V^{\ast \ast}$ (at least given the axiom of choice), so $V^{\ast}$ is an irreducible representation whose dual is not irreducible.
This is indeed really nice. Maybe the most obvious example of a group with an infinite-dimensional irreducible representation is obtained by taking $G = \operatorname{GL}(V)$ for any infinite-dimensional vector space $V$? – Pete L. Clark Jul 6 '11 at 21:40
Answering my own comment above: $V$ is irreducible because it is spanned by the set $G(1,-1,0,0,\dots)$, and if $v\in V$ with $v\ne0$, say $v=(v_1,\dots,v_n,0,0\dots)$ then by applying an element of $G$ you can arrange that $v_1\ne v_n$. Then if $g$ is the $n$-cycle $g=(1,2,\dots,n)$ and $h=(1,2)g^{-1}$, I think that $(v_1-v_n)^{-1}(e+g+g^2+\dots+g^{n-2}+h)v = (1,-1,0,0,\dots)$, hence the smallest invariant subspace containing $v$ is $V$. – irrep Jul 7 '11 at 12:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9919634461402893, "perplexity": 98.54002299732667}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657114204.83/warc/CC-MAIN-20140914011154-00149-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://brilliant.org/problems/solve-the-mechanics/ | # solve the mechanics...
two particles move in a uniform gravitational field with an acceleration g. At initial moment the particles were located at one point and moved with velocities v1=3.0m/s and v2=4.0m/s horizontally in opposite directions. Find distance between the particles at the moment when the velocity vectors become mutually perpendicular.
× | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8718850612640381, "perplexity": 644.4707392760631}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607046.17/warc/CC-MAIN-20170522190443-20170522210443-00616.warc.gz"} |
http://tex.stackexchange.com/questions/62904/a-variant-of-includeonly | # A variant of \includeonly
Is there a way to have pdfLaTeX emit separate PDF files for each `\include`d file, and somehow massage the `\include`/`\includeonly` mechanism so that when using the `\includeonly` command those files are inserted in the resulting PDF file, but only the `\includeonly`ed files are actually recompiled?
My main use of the `\includeonly` mechanism is to speed up compilation when working on part of the document, but it results in the end PDF to not be complete —this is a bit annoying if one needs to refer to things in the excluded parts, for example. A modification of the mechanism as vaporously sketched above would solve this.
-
This will inevitably result in somewhat broken files —repeated line pages, for example, among the least evil problems— but this is only intended to be used while working on the document... – Mariano Suárez-Alvarez Jul 11 '12 at 1:50
The answer is no. – egreg Jul 11 '12 at 6:26
This is not a complete answer, but more an idea on how it might work. It is, however, way to long for a comment.
The idea is to patch the `\include` command that it internally uses `pdfpages` to input the parts that are not mentioned in `\includeonly` from some existing PDF from a previous complete build. Some clever arithmetic might be necessary to calculate the page numbers, but in principle this could work.
A big issue might be hyperrefs. All parts are compiled with the same `.aux` files, so labels and page anchors should be consistent. However, they are lost when embedding PDFs with `\includepdf`. Nevertheless, in conjuction with (2) `pax` it might be possible to fix this. From the `pdfpages` documentation:
Links and other interactive features of PDF documents When including pages of a PDF only the so called content stream of these pages is copied but no links. Up to now there are no TeX-engines (pdfTeX, XeTeX, ...) available that can copy links or other interactive features of a PDF document, too. Thus, all kinds of links1 will get lost during inclusion. (Using `\includepdf`, `\includegraphics`, or other low-level commands.) However, there’s a gleam of hope. Some links may be extracted and later reinserted by a package called pax which can be downloaded from CTAN. Have a look at it!
`pax` is a combination of a Java tool to extract link information from the PDF you intend to embed and a package that reads in this information to restore the links into embedded PDF. It is still considered as experimental and I have not tried it. However, it is written by Heiko Oberdiek, so I would be optimistic that it works.
To conclude: Given some decent TeX skills, I think that your idea is achievable. However, given the complexity and involved tool chain, I doubt that in the end this is would be a big time saver. If compilation time is the issue, I suggest to consult How can I speed up LaTeX compilation? for other optimization strategies.
-
My compilation time is very fast, actually: I am using a custom format and a few other less significant improvements (together with way too much memory and a solid state drive!) But at over a hundred pages, compilation does take noticeable time. `\includeonly` is useful, for has the annoying effect of producing incomplete files. – Mariano Suárez-Alvarez Jul 11 '12 at 7:26
If I could tell pdflatex to keep processing but not outputting any material to the output file, and then turn back outputting off, I could easily do this, by the way (together with a little `Makefile` magic, but I am already doing this —for example, the custom format is rebuilt automagically for me when the prologue of the document or any of the ancillary style files, are modified) – Mariano Suárez-Alvarez Jul 11 '12 at 7:28
I imagine simply turning `\shipout` into a noop during the time output is not wanted will not work... – Mariano Suárez-Alvarez Jul 11 '12 at 8:06
@MarianoSuárez-Alvarez: Now I am a bit confused about your actual goal. Is it to speed up or is it not to speed up? If it is, do you really think that "keep processing, but not outputting material" would significantly speed up? – Daniel Jul 11 '12 at 8:10
No. The idea is to first create a PDF file for each `\ìnclude`d file (and each of these should look exactly the same as it would look if the complete document is processed: it is for this that I'd like to turn off outputting) and then, in the «normal» runs to make `\include` insert those PDF files instead of simply ignoring them when they are not in the `\includeonly` list. The only outside help here would be to have something run once per included file to generate the partial PDF files. – Mariano Suárez-Alvarez Jul 11 '12 at 8:17 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8405133485794067, "perplexity": 923.0503506256166}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860111455.18/warc/CC-MAIN-20160428161511-00023-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://mathoverflow.net/tags?tab=popular | # Tags
A tag is a keyword or label that categorizes your question with other, similar questions. Using the right tags makes it easier for others to find and answer your question.
Type to find tags:
ag.algebraic-geometry× 9481 Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology. nt.number-theory× 6298 Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions reference-request× 5170 used if a reference is needed in a paper or textbook on a specific result. co.combinatorics× 3726 Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics. at.algebraic-topology× 3567 Homotopy, stable homotopy, homology and cohomology, homotopical algebra. gr.group-theory× 3534 Questions about the branch of abstract algebra that deals with groups. dg.differential-geometry× 3454 Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis. pr.probability× 3137 Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory. fa.functional-analysis× 2982 Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory. rt.representation-theory× 2672 Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra. ct.category-theory× 2500 Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories. ac.commutative-algebra× 2290 Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics. linear-algebra× 2222 Questions about the properties of vector spaces and linear transformations, including linear systems in general. lo.logic× 2067 first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, p… set-theory× 2044 forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set t… graph-theory× 2014 Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with … gt.geometric-topology× 1687 Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial… gn.general-topology× 1653 Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties. ca.analysis-and-odes× 1485 Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics. mg.metric-geometry× 1353 Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces. ap.analysis-of-pdes× 1260 Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics. ra.rings-and-algebras× 1223 Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. soft-question× 1209 about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. In other words, questions that… cv.complex-variables× 1187 Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves. complex-geometry× 1132 the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry. matrices× 1118 Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typi… lie-groups× 1109 Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth. real-analysis× 1063 homotopy-theory× 1054 an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homot… ds.dynamical-systems× 926 Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations. mp.mathematical-physics× 917 Mathematical methods in quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics. measure-theory× 884 Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets. homological-algebra× 862 polynomials× 819 Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please… finite-groups× 819 lie-algebras× 818 algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s… | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9174190163612366, "perplexity": 1326.3964366984096}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375098990.43/warc/CC-MAIN-20150627031818-00214-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://planetmath.org/AdditiveInverseOfASumInARing | additive inverse of a sum in a ring
Let $R$ be a ring with elements $a,b\in R$. Suppose we want to find the inverse of the element $(a+b)\in R$. (Note that we call the element $(a+b)$ the sum of $a$ and $b$.) So we want the unique element $c\in R$ so that $(a+b)+c=0$. Actually, let’s put $c=(-a)+(-b)$ where $(-a)\in R$ is the additive inverse of $a$ and $(-b)\in R$ is the additive inverse of $b$. Because addition in the ring is both associative and commutative we see that
$\displaystyle(a+b)+((-a)+(-b))$ $\displaystyle=$ $\displaystyle(a+(-a))+(b+(-b))$ $\displaystyle=$ $\displaystyle 0+0=0$
since $(-a)\in R$ is the additive inverse of $a$ and $(-b)\in R$ is the additive inverse of $b$. Since additive inverses are unique this means that the additive inverse of $(a+b)$ must be $(-a)+(-b)$. We write this as
$-(a+b)=(-a)+(-b).$
It is important to note that we cannot just distribute the minus sign across the sum because this would imply that $-1\in R$ which is not the case if our ring is not with unity.
Title additive inverse of a sum in a ring AdditiveInverseOfASumInARing 2013-03-22 15:45:02 2013-03-22 15:45:02 rspuzio (6075) rspuzio (6075) 11 rspuzio (6075) Theorem msc 16B70 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 26, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9808158874511719, "perplexity": 99.07692796281668}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573570.6/warc/CC-MAIN-20190919183843-20190919205843-00448.warc.gz"} |
http://theanalysisofdata.com/probability/5_7.html | ## Probability
### The Analysis of Data, volume 1
Important Random Vectors: Exercises
## 5.7. Exercises
1. Justify the derivations of the expectation and variance of the Bernoulli random vector.
2. Consider a multinomial vector $\bb X=(X_1,\ldots,X_n)$ and a mapping $\bb X\to Y$ defined by $Y=\sum_{i\in A} X_i$ for some $A\subset \{1,\ldots,n\}$. What is the distribution of $Y$? Write down the pmf in a compact form.
3. Characterize the elliptical contours of the multivariate Gaussian pdf with non-diagonal $\Sigma$ in terms of the eigenvalues and eigenvectors of $\Sigma$. Hint: use spectral decomposition (Proposition C.3.8) and the relationship in Proposition 5.2.4.
4. Express the exponential, Poisson RVs and the multinomial random vector as exponential family random vectors. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9583247303962708, "perplexity": 405.005835900837}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363332.1/warc/CC-MAIN-20211207014802-20211207044802-00197.warc.gz"} |
http://www.maplesoft.com/support/help/Maple/view.aspx?path=simplex/display | simplex - Maple Programming Help
Home : Support : Online Help : Mathematics : Optimization : Simplex Linear Optimization : simplex/display
simplex
display
display a Linear Program in Matrix Form
Calling Sequence display(C) display(C,[x, y, z])
Parameters
C - set of linear relations
Description
• The function display(C) constructs the matrix equation Ax rel B for the constraints defining a linear program.
• The set of linear equations C passed to display should be in the special form produced by simplex[setup].
• The command with(simplex,display) allows the use of the abbreviated form of this command.
Examples
> $\mathrm{with}\left(\mathrm{simplex}\right):$
> $\mathrm{display}\left(\left\{x+3y+z\le 0,w-2y-z\le 2\right\}\right)$
$\left[\begin{array}{rrrr}{0}& {1}& {3}& {1}\\ {1}& {0}& {-}{2}& {-}{1}\end{array}\right]{}\left[\begin{array}{c}{w}\\ {x}\\ {y}\\ {z}\end{array}\right]{\le }\left[\begin{array}{r}{0}\\ {2}\end{array}\right]$ (1) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9776315689086914, "perplexity": 3270.651567416216}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128321309.25/warc/CC-MAIN-20170627101436-20170627121436-00384.warc.gz"} |
https://www.physicsforums.com/threads/setting-up-an-integral-for-the-area-of-a-surface-of-revolution.374814/ | # Homework Help: Setting up an Integral for the area of a surface of revolution
1. Feb 2, 2010
### darkblue
1. The problem statement, all variables and given/known data
Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve y=xe-x 1=<x=<3 about the y-axis.
2. Relevant equations
S=integral from a to b x 2pix ds where ds=sqrt(1+(dy/dx)2)dx
3. The attempt at a solution
The first thing I tried to do is solve for the equation in terms of x, and then use the equation above. I figured it makes sense to solve for x since we are rotating the curve about the y-axis. I wasn't able to solve for x, so then I tried to use this method in my textbook where you leave x as it is, and then substitute u for whatever is within the square root sign in such a way that you can eliminate x. I tried to do that, but its turning into a mess since you get 1+(e-x-xe-x)2 underneath the square root and I don't really see how substitution could be used here...any ideas?
2. Feb 2, 2010
### Staff: Mentor
All you need to do is set up the integral. Don't worry about trying to evaluate this integral.
3. Feb 2, 2010
### darkblue
So does this mean that the way I have set it up is correct? I had a feeling it wasn't right because I couldn't see what steps I'd take next in the event that I had to solve it.
4. Feb 2, 2010
### Staff: Mentor
Seems to be OK, but I'm a little rusty on these surface area integrals. You have an extra x in what you typed, though, right after b. Did you mean for that to be there?
5. Feb 2, 2010
### darkblue
oops, i meant to put a "*" for multiplication.
Thanks for your help!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9058904051780701, "perplexity": 365.8118238327701}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376829542.89/warc/CC-MAIN-20181218164121-20181218190121-00301.warc.gz"} |
https://www.physicsforums.com/threads/indefinite-and-definite-integral-of-e-sin-x-dx.808702/ | # Indefinite and definite integral of e^sin(x) dx
1. Apr 15, 2015
### Emmanuel_Euler
Look to this indefinite integral →∫e^(sin(x))dx
Antiderivative or integral could not be found.and impossible to solve.
Look to this definite integral
∫e^(sin(x))dx (Upper bound is π and Lower bound is zero)=??
my question is : can we find any solution for this integral (definite integral) ??
Last edited: Apr 15, 2015
2. Apr 15, 2015
### axmls
There is no closed-form solution for the antiderivative, but we can still approximate the definite integral.
3. Apr 15, 2015
### pasmith
It may be possible to use contour integration to find an analytic value for $\int_0^\pi e^{\sin x}\,dx$.
4. Dec 1, 2016
### Cgty
Assume that we have a solution like that int(y dy)=int(e^sinx dx). It is clear we must find y^2/2=int(e^sinx dx). In order to equality, int[ln(y) dy]=int(sinx dx). Due to int(lny dy) is equal to y(lny-1); y(lny-1)=-cosx+c and y=[-cosx+c]/[lny-1]. We need to find y^2/2 therefore, y^2/2=[(cosx+c)/(lny-1)]^2/2. This is the solution of int(e^sinx dx) and we have a non-linear euation.
5. Dec 3, 2016
### lurflurf
lets consider
$$\frac{1}{\pi}\int_0^\pi\!e^{\sin(x)}\,\mathrm{d}x$$
I flipped through some books and did not find much on that, but I did find that
$$\frac{1}{\pi}\int_0^\pi\!e^{\cos(x)}\,\mathrm{d}x=\operatorname{I}_0(1)$$
I is the modified Bessel function of the first kind.
Also we know that
$$\operatorname{I}_0(1)=\frac{1}{\pi}\int_0^\pi\!\cosh(\sin(x))\,\mathrm{d}x\\ \operatorname{I}_0(1)\sim1.26606587775201$$
http://people.math.sfu.ca/~cbm/aands/page_376.htm
and
$$\operatorname{L}_0(1)=\frac{1}{\pi}\int_0^\pi\!\sinh(\sin(x))\,\mathrm{d}x\\ \operatorname{L}_0(1)\sim0.710243185937891$$
L is the Modified Struve Function
http://people.math.sfu.ca/~cbm/aands/page_498.htm
so
$$\frac{1}{\pi}\int_0^\pi\!e^{\sin(x)}\,\mathrm{d}x=\operatorname{I}_0(1)+\operatorname{L}_0(1)\sim1.97630906368990$$
6. Jan 11, 2017
### Emmanuel_Euler
can you give me the name of the books please, because i need them and thank you so much for help
7. Jan 11, 2017
### lurflurf
I found that in the famous Handbook of Mathematical Functions edited by M. Abramowitz and I. A. Stegun a "work for hire performed for the US Government" thus freely available.
For example here
http://people.math.sfu.ca/~cbm/aands/toc.htm
It is also of course available in print if you prefer.
8. Jan 11, 2017
### Emmanuel_Euler
thank you so much for help.....
9. Jun 23, 2018 at 1:21 PM
### alexpeter_pen
10. Jun 23, 2018 at 4:30 PM
### jostpuur
By using the formulas
$$\sin(\alpha)\sin(\beta) = \frac{1}{2}\big(\cos(\alpha - \beta) - \cos(\alpha + \beta)\big)$$
$$\sin(\alpha)\cos(\beta) = \frac{1}{2}\big(\sin(\alpha - \beta) + \sin(\alpha + \beta)\big)$$
it is possible to write the powers $(\sin(x))^n$ in a form where non-trivial powers do not appear. By using this approach we get a series that starts as
$$\int\limits_0^{\pi} e^{\sin(x)}dx = \pi + 2 + \frac{1}{2!}\frac{\pi}{2} + \frac{1}{3!}\frac{4}{3} + \frac{1}{4!}\frac{3\pi}{8} + \frac{1}{5!}\frac{16}{15} + \cdots$$
It is unfortunate of course that it might be impossible to get a nice formula for these terms, but it's not obvious if that's the way it's going to be. It could be that there exists some theory for the coefficients in the formula for $(\sin(x))^n$.
11. Jun 24, 2018 at 12:45 PM
### alexpeter_pen
Just to add to my previous answer the actual formula so one does not have to follow the site
$$\displaystyle \int e^{\sin(x)} dx=I_0(1)x + \frac{\pi}{2}L_0(1) + 2\sum_{n=1}^{+\infty} \frac{I_n(1)}{n} \sin \left ( nx - \frac{n\pi}{2} \right )$$
Another nice way of solving definite integral apart for simply stating its value through Struve and Bessel (which is the shortest possible known expression at the moment) goes like this:
First let us get rid of $\sin(x)$, introducing $u=\sin(x), du=\cos(x)dx$ This leads to
$$\displaystyle \int_{0}^{\pi} e^{\sin(x)} dx=2\int_{0}^{1} \frac{e^u}{\sqrt{1-u^2}} du$$
Notice that we have taken it twice from $0$ to $\frac{\pi}{2}$ as $e^{\sin(x)}$ is symmetrical.
Now we use expansion of $e^u$ reducing it all to the sum of integrals
$$\displaystyle \int_{0}^{\pi} e^{\sin(x)} dx=2 \sum_{k=0}^{\infty} \int_{0}^{1} \frac{u^k}{k!\sqrt{1-u^2}} du$$
Now
$$\displaystyle \int_{0}^{1} \frac{u^k}{k!\sqrt{1-u^2}} du= \frac{1}{k!}\frac{\sqrt{\pi}\Gamma(\frac{k+1}{2})}{2\Gamma(\frac{k}{2}+1)}$$
coming from the connection between Beta and Gamma function, making it all
$$\displaystyle \int_{0}^{\pi} e^{\sin(x)} dx=\sum_{k=0}^{\infty} \frac{\sqrt{\pi}\Gamma(\frac{k+1}{2})}{\Gamma(\frac{k}{2}+1)k!}$$
or in a split form
$$\displaystyle \int_{0}^{\pi} e^{\sin(x)} dx=\sum_{k=0}^{\infty}\frac{\pi}{4^n(n!)^2} + \sum_{k=0}^{\infty} \frac{2^{n+1}n!}{(2n+1)!(2n+1)!!}$$
First 10 terms are giving 20 digit precision already.
I am happy with 4 terms 4 digit precision
$$\displaystyle \frac{328}{147} + \frac{2917 π}{2304} \approx 6.2087$$
Just to make the connection
$$\displaystyle \pi L_0(1)=\sum_{k=0}^{\infty} \frac{2^{n+1}n!}{(2n+1)!(2n+1)!!}$$
$$\displaystyle \pi I_0(1)=\sum_{k=0}^{\infty}\frac{\pi}{4^n(n!)^2}$$
Last edited: Jun 24, 2018 at 1:40 PM
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook
Have something to add?
Draft saved Draft deleted | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9610797762870789, "perplexity": 1479.755459571582}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867364.94/warc/CC-MAIN-20180625014226-20180625034226-00610.warc.gz"} |
http://math.ucr.edu/home/baez/time/time.html | Next: Bibliography
The Physical Basis of the Direction of Time
by H. D. Zeh
Book Review by John C. Baez
April 6, 1993
This appeared in the Mathematical Intelligencer.
Also available in Postscript and LaTeX.
In this book Zeh brings some clarity to a very murky problem: why are the future and the past so different? One need only to read the physics journals to see that this is a multi-faceted and very real issue that vexes the experts even now. To understand its seriousness, it is first necessary to see how similar the future and the past are. They don't seem so in everyday life: we remember the past but not the future, our actions affect the future but not the past, and so on. From this standpoint it is really quite surprising that the dynamical laws of physics - with one small exception - seem to be symmetrical under time reversal.
Before we go any further, it's important to get a clear idea of what time reversal symmetry really means. At the simplest level, we may think of the laws of physics as equations involving a time variable , and say that they are symmetric under time reversal if given any solution, and making the substitution , we obtain another solution. To take an easy example, consider a point particle of mass in three-dimensional space with no forces acting on it. If we write its position as a function of time, , Newton's second law says that
If satisfies this equation then so does .
Typically the laws are more complicated, and one may have to be more careful in defining time reversal symmetry. Different laws of physics involve very different mathematical structures, but they are almost always separated into two components, the "kinematics'' and the "dynamics''. The kinematics consists of the description of the set of states that the system can be in at any given time. For example, in classical mechanics, we can specify the state of a point particle in three-dimensional space by giving its position and its velocity , so . The dynamics tells how states change with time. In a theory where we can predict both the future and past from the present, and where there are no time-dependent external influences, we usually describe dynamics with a family of maps , where . If the state of the system is at some time , the state is at time . The maps should form a "one-parameter group,'' that is,
We say that the physical system given by has "time-reversal symmetry'' if there is a map , called time reversal, such that
For example, our point particle with no forces on it moves with constant velocity, so
It's easy to check that is a one-parameter group, and that the system has time-reversal symmetry, where
Note, by the way, that time-reversal symmetry in the sense described above is different from requiring that a given state be invariant under time-reversal:
Our world is evidently in a state that is not even approximately invariant under time reversal; there are many processes going on whose time-reversed versions never seem to happen. But this is logically independent from the question of whether the dynamical laws of physics admit time reversal symmetry. Keeping this distinction straight is crucial for thinking clearly about the direction of time. Even people who claim to understand the distinction often slip. When reading about time reversal symmetry, I become infuriated when authors confuse symmetry of the laws with symmetry of the state, and I am happy to report that not once did I hurl Zeh's book to the floor in anger.
At this point, we could go through all theories of physics and check to see whether they have time reversal symmetry. But let us simply turn to the most up-to-date and complete laws of physics we know: the standard model and general relativity. The "standard model'' is a complicated theory of quantum fields that describes the most fundamental particles we know (mainly leptons and quarks) and the forces - electromagnetism and the weak and strong nuclear forces - by which they interact. In other words, it treats everything except gravity. The standard model has time-reversal symmetry except for effects involving the weak force. This is the force that permits a proton and electron to turn into a neutron and a neutrino, as happens in some radioactive atoms, or vice-versa, as in some others.
In fact, in quantum field theory, time reversal, or is one of a trio of possible symmetries, the others being charge conjugation or which amounts to interchanging particles with their antiparticles, and parity, or which is related to spatial inversion
in somewhat the same way that time reversal relates to the map
In a quantum field theory states are given by unit vectors in a Hilbert space . The symmetries and are given by unitary operators on - if the theory in question admits these symmetries - while is given by an antiunitary operator, that is, a conjugate-linear one-to-one and onto norm-preserving map from to itself. I will restrain myself from explaining why must be antiunitary rather than unitary, fascinating though this is. The key point here is that in the standard model, the weak force violates , , and symmetry, while electromagnetism and the strong force admit all these symmetries. Moreover, while violation of symmetry is quite common and blatant - particularly for neutrinos - violation of symmetry has so far only been seen in the decays of a single particle, the neutral kaon, and the amount of violation is minute. Most physicists believe that this small symmetry violation is not particularly related to the gross time asymmetry of the state of the universe. But there is something very curious about this, as in the elaborate Islamic designs that are perfectly symmetrical except for one tiny flaw put in to avoid the wrath of Allah. Zeh's book does not treat the asymmetry of the weak interaction very thoroughly, but luckily there is already a good book that does just this [1].
On the other hand, general relativity treats gravity, which is a great puzzle in its own right, since it seems very difficult to unify with the rest of the forces. Unlike all the other forces, it is not at all natural to formulate its dynamics in terms of a one-parameter time evolution group. Essentially, this is because it treats of the geometry of spacetime itself, and how it wiggles around. While the dynamics of general relativity is by now moderately well understood, the modifications required for a quantum theory of gravity are still very poorly understood, and seem to require a radical rethinking of the very notion of time. In his last chapter, "The Quantization of Time,'' Zeh tours this fascinating subject. While a quantum theory of gravity would be likely to have profound implications for the study of time reversal, one can fairly say that so far the dynamics of gravity seems to admit time reversal symmetry.
It's worth noting that there are some cases where at first glance it looks as if the laws of physics are asymmetric under time reversal, but on closer inspection it turns out to be the fault of the particular state of the universe we are in. The two most famous examples are the "time arrow of radiation'' and the "time arrow of thermodynamics.'' Here an "arrow of time'' is used loosely to denote something that is not symmetric under time reversal.
The time arrow of radiation refers simply to the fact that when we shake an electrically charged object, it emits waves of radiation that ripple outwards as time progresses into the future, rather than the past. This is expressed mathematically in terms of what are called Green's functions. To understand these, it's easier to consider the scalar wave equation rather than Maxwell's equations of electromagnetism in their full glory. Thus we have a "field'' being produced by a "source'' , and we assume both are smooth functions and that
where
The source does not uniquely determine the field, but it is possible to write down formulas that give us for any source a field with . In particular, we say that is a Green's function (actually a distribution) for the scalar wave equation if
where is short for , implies that . Two Green's functions are the "advanced'' one,
and the "retarded'' one,
where and is the Dirac delta distribution. In electromagnetism one typically uses the retarded Green's function, so that if is nonzero only for times , then is typically nonzero after , but is zero before .
It may seem odd that while the equation is preserved by the transformation , we are solving it in a way that doesn't respect this symmetry. But there are two things that help resolve this puzzle. First, it is worth noting that working with the retarded rather than the advanced Green's function is, at least for vanishing outside a bounded set, equivalent to an assumption about the nature of the field , namely that it vanish as . In short, we are making a time-asymmetric assumption about the state of the system when we are choosing the retarded Green's function. Why do we make this assumption? For a quite interesting reason: because it's dark at night. In a sense, light radiates out from the sun and from our flashlights, rather than coming into them from the distance, because the universe is a rather dark and cold place. The very fact that space is mostly dark and empty, with a speckling of hot bright stars that radiate outwards, is blatantly time-asymmetric, so the time arrow of radiation appears to be cosmological in origin. This fact about the universe is crucial to life as we know it, since all life on earth is powered by the outgoing radiation of the sun, and the earth in turn dumps its waste heat into the blackness of space.
A second, subtler point is that the equation does not fit into the general framework of one-parameter groups, because the field is subject to an arbitrary time-dependent external influence, the source . Here one wants to imagine oneself, the experimenter, as being able to do whatever one wants with the source , and see what it does to the field . This is related to the notion of free will: we like to think that the laws of physics govern the behavior of everything else, but that we are free to do whatever we want. However, in the most fundamental laws of physics we know - the standard model and general relativity - no "arbitrary external influences'' appear. In these laws, there is no need to choose between a retarded and advanced Green's function (or some other Green's function, for that matter). There is only the need to choose the state that best matches what we observe.
The time arrow of thermodynamics is perhaps the most famous aspect of time reversal symmetry - so I will treat it very briefly here. Why is it so much more likely that a porcelain cup will fall to the floor and smash to smithereens, than it is for a pile of porcelain smithereens to form into a cup and jump into ones hand? Disorder seems to be always on the increase. In fact, in thermodynamics there is a quantity called entropy, , which is a a measure of disorder - although one must be very careful not to fall for the negative connotations of "disorder,'' which here is interpreted in a very precise and sometimes counterintuitive sense. The second law of thermodynamics is that
This law appears utterly time-asymmetric, and reconciling it with the (almost) time-symmetric fundamental laws of physics has exercised the minds of many physicists for many years. But the final resolution seems to be a simple one: this law is not true except for certain states. That is, it expresses a time asymmetry of the state of a system, rather than an asymmetry in the dynamical laws. As long as the dynamical laws admit symmetry under time reversal, for every state with there is a time-reversed state with . It is also worth noting that the vast majority of states typically have quite large and .
As with the time arrow of radiation, in the last analysis it appears to be nothing but a raw experimental fact that the entropy of our universe is increasing. In a sense this is not surprising, because pondering chemistry and biology a bit it becomes apparent that life as we know it requires the entropy to be changing monotonically, rather than staying about the same. One might ask why rather than , but this is essentially a matter of convention. Processes like remembering and planning, which define the psychological notions of future and past, are only able to occur in the direction of increasing entropy. That is, a memory at time can only be of an event at time for which , while a plan at time can only be for an action at time for which . Since we have settled on using calendars for which the number of the years increase in the direction of plans, rather than memories, we have chosen a time coordinate for which implies .
The main remaining mystery, then, is why the state of the universe is grossly asymmetric under time reversal, even though the dynamical laws of physics are almost - but not quite! - symmetric. If the reader wishes to puzzle over this some more, or wants supporting evidence for some of the (perhaps upsetting) claims I've made above, he or she could not do better than to read Zeh's book. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9039524793624878, "perplexity": 311.76900147366194}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257832475.43/warc/CC-MAIN-20160723071032-00306-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/determine-projectile-motion-velocity.91748/ | # Determine Projectile Motion velocity
1. Oct 1, 2005
### dekoi
If the velocity of a projectile is known at t=1second and the gravitational force is unknown, how does one determine the velocity at t=0, 2, and 3 seconds.
The projectile does not start from rest.
If i use the formula: vfy = viy - gt, i can't solve for v, since the value for g is unknown.
Thank you .
2. Oct 1, 2005
### hotvette
Use the information you have. How can knowing the velocity at t=1 sec help? Hint: use your equation of motion.
3. Oct 2, 2005
### dekoi
Which equation for motion?
I still do not understand.
4. Oct 2, 2005
### Grogs
You're sure this is correct and you've listed everything that's been given to you (vy1)? If that's the case, you can't solve for vy0, vy2, etc. numerically. The best you could do is solve for them in terms of vy0 . For example:
$$v_{y1} = v_ {y0} + g't \Longrightarrow g' = \frac{v_{y1}-v_{y0}}{t}$$
You could then put vy2 and vy3 in terms of vy0, but that's as far as you could go. If you had one other piece of information you could solve for g' numerically and use it to solve for the other velocities as well.
5. Oct 2, 2005
### dekoi
The only problem with that is that in the second part of the question, I have to find the value of "g", so there must be a way to do it. Here is the exact question:
A physics student on Planet Exidor throws a ball, and it follows a parabolic trajectory. The ball's position is shown at 1 s intervals until t = 3 s. At t = 1 s, the ball's velocity is v = (2.0i + 2.0j) m/s. (i and j being unit vectors for x and y)
a) Determine the ball's velocity at t = 0s, 2s, and 3s.
b) What is the value of g on Planet Exidor?
c) What was the ball's launch angle?
6. Oct 2, 2005
### Päällikkö
Does the problem have a graph/figure ("The ball's position is shown" implies it does)? Can you see the height from that?
7. Oct 2, 2005
### dekoi
It does not land at 3 seconds; the graph only shows up until 3 seconds.
The graph is attached.
File size:
6.1 KB
Views:
475
8. Oct 2, 2005
### Päällikkö
$$v_y = v_{y0} - gt$$
What do you know at t = 1s and t = 2s?
9. Oct 2, 2005
### dekoi
At t = 1s, vx = 2.0 m/s and vy = 2.0 m/s.
At t = 2s, vx = 2.0 m/s and vy = (2.0 - g) m/s
10. Oct 2, 2005
### Fermat
From your graph, you can see that the total time of flight is 4s.
So the time to reach max height = 2s
You also know the vertical velocity at t = 1.
By symmetry the velocity at t = 3 will be the same as at t= 1, but in the opposite direction.
Now you can use your eqns of motion to solve for g, etc.
11. Oct 2, 2005
### Grogs
{quote snipped for clarity}
This answer is correct; however, by looking at your graph I can tell you the *exact* numerical value of v2y. Why is that?
12. Oct 2, 2005
### dekoi
Is this correct?
v2y= 0 m/s
therefore
0 m/s = 2.0 - g
g = 2.0 m/s2
v0 = (2.0i + 4.0j) m/s
v1 = (2.0i + 2.0j) m/s
v2 = (2.0i) m/s
v3 = (2.0i - 2.0j) m/s
v0 = sqrt(2.0^2 + 4.0^2) = sqrt(4 + 16) = sqrt20 m/s
angle = arccos(v0x / v0) = arccos(2 / sqrt20) = 1.1 degrees
I am not sure about the angle.
13. Oct 2, 2005
### dekoi
Any responses?
14. Oct 2, 2005
### Päällikkö
You have radians on.
Looks good.
15. Oct 2, 2005
### dekoi
Thank you everyone.
Have something to add?
Similar Discussions: Determine Projectile Motion velocity | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8115414977073669, "perplexity": 2522.0600416564484}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719754.86/warc/CC-MAIN-20161020183839-00026-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/260444/localization-at-a-maximal-ideal | # Localization at a Maximal Ideal
While studying, I came across this question:
If $A$ is a ring in which $x^n=x$ for all $x\in A$ (where $n$ is an integer greater than $1$ and may depend on $x$), show all prime ideals are maximal.
I didn't find the solution too hard (if it is correct). We let $P$ be a prime ideal. That $A/P$ is an integral domain; if we can show it is also a field, we'll know that $P$ is maximal. To that end, we know $a^n+P=a+P$, and in particular, $(a+P)(a^{n-1}+P)=(a+P)(1+P)$. Since we're in an integral domain, the cancellation law applies, so we have $(a^{n-1}+P)=(1+P)$. Now we have $(a+P)(a^{n-2}+P)=(1+P)$, hence $(a+P)$ has an inverse, so $A/P$ is a field.
The next question has me confused, though:
Let $\mathbf{m}$ be a prime ideal in a ring in which $x^n=x$ for all $x\in A$ (where $n$ is an integer greater than $1$ and may depend on $x$). Show that $A_\mathbf{m}$ is a field.
It seems like $A_\mathbf{m}$ should not be a field since $\frac{m}{1}$, $m\in\mathbf{m}$ wouldn't have an inverse, thus it would have to be in the equivalence class of $\frac{0}{1}$, but I don't see why that is true (if it indeed is true). Please help clear this up for me!
-
When you say «such a ring» in the second displayed piece of text, what do you mean? – Mariano Suárez-Alvarez Dec 17 '12 at 3:48
A ring $A$ that has the property where $x^n=x$ for some positive integer greater than $1$ and may depend on $x$. The question has two parts; the first one wasn't too challenging, but I don't see why the second question is true. – Clayton Dec 17 '12 at 3:49
If you edit the question to be explicit, we can delete these comments :-) – Mariano Suárez-Alvarez Dec 17 '12 at 3:50
Let $x\in \mathfrak{m}$ and let $n$ be such that $x^n=x$. Then $x^{n-1}-1$ is not in $\mathfrak{m}$ because this would imply that $1\in\mathfrak{m}$ which is imposible since $\mathfrak{m}$ is a prime ideal. But $(x^{n-1}-1)x=0$ and this implies that $x/1=0$ by definition of $A_\mathfrak{m}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9658306241035461, "perplexity": 77.2777123780047}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042986806.32/warc/CC-MAIN-20150728002306-00120-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.arxiv-vanity.com/papers/cond-mat/9904064/ | # Dimer Order With Striped Correlations In the J1-J2 Heisenberg Model
Rajiv R. P. Singh[*], Zheng Weihong[], C.J. Hamer[], and J. Oitmaa[§] Department of Physics, University of California, Davis, CA 95616;
School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia.
February 16, 2021
###### Abstract
Ground state energies for plaquette and dimer order in the square-lattice spin-half Heisenberg model are compared using series expansion methods. We find that these energies are remarkably close to each other at intermediate values of , where the model is believed to have a quantum disordered ground state. They join smoothly with those obtained from the Ising expansions for the 2-sublattice Néel-state at , suggesting a second order transition from a Néel state to a quantum disordered state, whereas they cross the energy for the 4-sublattice ordered state at at a large angle, implying a first order transition to the 4-sublattice magnetic state. The strongest evidence that the plaquette phase is not realized in this model comes from the analysis of the series for the singlet and triplet excitation spectra, which suggest an instability in the plaquette phase. Thus, our study supports the recent work of Kotov et al, which presents a strong picture for columnar dimer order in this model. We also discuss the striped nature of spin correlations in this phase, with substantial resonance all along columns of dimers.
###### pacs:
PACS Indices: 75.10.-b., 75.10J., 75.40.Gb
## I Introduction
There has been considerable study, over the last decade, of the frustrated spin- square lattice Heisenberg antiferromagnet (the “ antiferromagnet”). These studies include exact diagonalizations on small systems[5, 6, 7, 8], spin-wave calculations[9, 10], series expansions[11, 12], and a field-theoretic large- expansions[13].
These studies, and others, have provided a substantial body of evidence that the ground state of this system, in the region , has no long-range magnetic order and has a gap to spin excitations. For the model has conventional antiferromagnetic Néel order whereas for the system orders in a columnar phase. Whether this “intermediate phase” is a spatially homogeneous spin-liquid, or whether it has some type of spontaneously broken symmetry leading to a more subtle type of long-range order, has not been conclusively established.
Zhitomirsky and Ueda[14] have proposed a plaquette resonating valence bond (RVB) phase, which breaks translational symmetry along both and axes, but preserves the symmetry of interchange of the two axes. The horizontal and vertical dimers resonate within a plaquette. An early series study[15] had investigated the relative stability of various spontaneously dimerized states and had concluded that a columnar dimerized phase was the most promising candidate for the intermediate region, in agreement with the large- expansions. Zhitomirsky and Ueda[14] claim their plaquette phase has a lower energy than this columnar dimer phase, but we find this to be incorrect.
Further support for the columnar dimer scenario comes from recent work of Kotov et al.[16], who combine an analytic many body theory with extended series and diagonalization results to study the nature and stability of the excitations in the intermediate region. It is argued that where the Néel phase becomes unstable the system will develop not only a gap for triplet excitations but also a gapped low-energy singlet which reflects the spontaneous symmetry breaking. This is clearly seen in the calculations. At a second order transition occurs, with the energies of Néel phase and dimerized phase joining smoothly, and the energy gap and dimerization vanishing.
It is the aim of this paper to further investigate, using series methods, the competing possibilities of columnar dimerization versus plaquette order in the intermediate region of the antiferromagnet. It is conceivable that both occur, with a transition from one to the other. However, such a transition, reflecting a change of symmetry, is expected to be first-order and not well suited to series methods. If both phases are locally stable the most direct way to compare them is by comparison of the ground state energies. If one is unstable this should show up by the closing of an appropriate gap or by the divergence of an appropriate susceptibility. In this paper we calculate the ground state energy and the singlet and triplet excitation spectra by series expansions about a disconnected plaquette Hamiltonian. We also calculate the susceptibility for the dimer phase to break translational symmetry in the direction perpendicular to the dimers. This susceptibility will be large if there is substantial resonance in the dimer phase and will diverge if there is an instability to the plaquette RVB phase.
Combining the plaquette expansion results with the dimer expansions of Kotov et al.[16], a very interesting picture emerges for the quantum disordered phase. We find that the plaquette phase is unstable and hence is not the ground state for this model. The dimer phase, on the other hand, is stable. However, there is substantial resonance in the dimer phase. The spin-spin correlations are not simply those of isolated dimers. Instead, the nearest neighbor correlations are nearly identical along the rungs and chains of dimer columns. In contrast, the correlations from one dimer column to the next are much weaker. The spin-gap phase appears separated from the Néel phase by a second order transition, whereas it is separated from the columnar phase by a first order transition. These results are in remarkable agreement with the large-N theories [13]. The existence of a quantum critical point separating an antiferromagnetic phase and a quantum disordered phase with striped correlations in a microscopic model makes this critical point a particularly interesting one. The role of doping and its implications for high- materials deserves further attention.
## Ii Series Expansions and Results
We study the Hamiltonian
H=J1∑n.n.Si⋅Sj+J2∑n.n.n.Si⋅Sj (1)
where the first sum runs over the nearest neighbor and the second over the second nearest neighbor spin pairs of the square-lattice. We denote the ratio of couplings as . The linked-cluster expansion method has been previously reviewed in several articles[17, 18, 19], and will not be repeated here. To carry out the series expansion about the disconnected-plaquette state for this system, we take the interactions denoted by the thick solid and dashed bonds in Fig. 1 as the unperturbed Hamiltonian, and the rest of the interactions as a perturbation. That is, we define the following Hamiltonian
H=H0+H1 (2)
where the unperturbed Hamiltonian () and perturbation () are
H0 = J1∑⟨ij⟩∈ASi⋅Sj+J2∑⟨ij⟩∈BSi⋅Sj (3) H1 = λJ1∑⟨ij⟩∈CSi⋅Sj+λJ2∑⟨ij⟩∈DSi⋅Sj
and the summations are over intra-plaquette nearest-neighbor bonds (A), intra-plaquette second nearest-neighbor bonds (B), inter-plaquette nearest-neighbor bonds (C), inter-plaquette second nearest-neighbor bonds (D), shown in Fig. 1. With this Hamiltonian, one can carry out an expansion in powers of , and at one recovers the original Hamiltonian in Eq. (1). Thus, although we expand about a particular state, i.e. a plaquette state, our results at describe the original system without broken symmetries, provided no intervening singularity is present. Such perturbation expansions about an unperturbed plaquette Hamiltonian have been used previously to study Heisenberg models for CaVO[20].
It is instructive to consider the states of an isolated plaquette. There are two singlet states, one with energy and the other with energy . The former is the ground state for and corresponds to pair singlets resonating between the vertical and horizontal bonds of the plaquette. It is even under a rotation. The latter is the ground state for and is odd under a rotation. The wavefunctions for these two singlet states are
ψ1 = 1√12[(++−−)+(+−+−)+(−−++)+(−+−+)−2(+−−+)−2(−++−)] (4) = ψ2 = 12[(++−−)−(+−+−)+(−−++)−(−+−+)] (5) =
where means these two spins form a singlet. There are three triplet states, one with energy and a degenerate pair with energy ; like the singlets, these have a level crossing at . Under a rotation the former is odd, while the latter two are even and odd, respectively. Finally there is a quintuplet state at , which is even under a rotation. For and the first excited state of the plaquette is a triplet, while for it is the other singlet. These states and corresponding energies are shown in Figure 2. The eigenstates of , the unperturbed Hamiltonian, are direct products of these plaquette states.
To derive the plaquette expansions we identify each plaquette as a 16 state quantum object, and these lie at the sites of a square lattice with spacing , where is the original lattice spacing. Interactions between plaquettes connect first and second-neighbor sites on this new lattice. The cluster data is thus identical to that used by us previously[12] to derive Ising expansions for this model. Because there are 16 states at each cluster site, the vector space grows very rapidly with the number of sites and thus limits the maximum attainable order for plaquette expansions to considerably less than can be achieved for dimer or Ising expansions.
We have computed the ground state energy to order , for fixed values of the coupling ratio . The series are analysed using integrated differential approximants[21], evaluated at to give the ground state energy of the original Hamiltonian. The estimates with error bars representing confidence limits, are shown in Figure 3. For comparison we also show previous results obtained from Ising expansions[12] and dimer expansions[16]. We find that, in the intermediate region, the ground state energy for both plaquette and dimer phase are very close to each other and cannot be used to distinguish between them. The dimer expansion yields slightly lower energies near the transition to the Néel phase. We do not draw any conclusions from this.
Zhitomirsky and Ueda[14] have claimed that the ground state energy from a second-order plaquette expansion is -0.63 (at ), much lower that the dimer expansion result -0.492. This result appears incorrect. At the ground state energy is given by
4E0/NJ1 = −7/4−277λ2/1440−0.001357λ3−0.0210609λ4 (6) −0.000319586λ5−0.00580643λ6−0.001822686λ7+O(λ8)
The second order result (at ) is , rather than . We note that if the second order coefficient were 4 times larger then the resulting energy would be .
We have also derived series, to order , for the singlet and triplet excitation energies, , using the method of Gelfand[18], and taking as unperturbed eigenfunctions the corresponding plaquette states. The low order terms for are given by:
Δs(kx,ky)/J1 = 1−301λ2/1440+137λ3/86400+217λ3cos(kx)cos(ky)/172800 (7) +(−5λ2/16−89λ3/9600)[cos(kx)+cos(ky)]/2 Δt(kx,ky)/J1 = 1−3691λ2/30240+(−2λ/3+11λ2/720)[cos(kx)+cos(ky)]/2 (8) −λ2[cos(2kx)+cos(2ky)]/120+(λ/3−5λ2/96)cos(kx)cos(ky) −λ2[cos(2kx)cos(ky)+cos(kx)cos(2ky)]/90+7λ2cos(2kx)cos(2ky)/360
The full series are available on request. We first consider the triplet excitations. Figure 4 shows along high symmetry directions in the Brillouin zone for and various coupling ratios . For the series are well converged and direct summation and integrated differential approximants give essentially identical results. We find that the minimum gap occurs at for and moves to for . Next we seek to locate the critical point where the triplet gap vanishes. This is done using Dlog Padé approximants to the gap series at the appropriate . In practice this works well when the minimum gap lies at . For we find a critical point at . We can compare this result with recent work of Koga et al.[22] who obtain from a modified spin-wave theory and from a 4th order plaquette expansion. The critical point increases with increasing . At , at the approximate centre of the intermediate phase, we find . This result has some uncertainty but, if accurate, means that the plaquette phase becomes unstable before the full Hamiltonian () is reached. The associated critical exponent describing the vanishing of the triplet gap is about 0.7 for , suggesting that the transition lies in the universality class of the classical Heisenberg model. On the other hand, for the exponent is about 0.4. This supports the existence of an intermediate phase lying in a different universality class.
Figure 5 shows the singlet excitation energy along high symmetry directions in the Brillouin zone for and the same coupling ratios as Figure 4. Again the series are well converged and direct summation and integrated differential approximants give essentially identical results. We find that the minimum gap occurs at for all . We have also noted that for , the triplet excitation and the singlet excitation have same gap at , but at , the singlet gap is considerable larger than the triplet gap, this means probably that the triplet gap close before the singlet gap at . The critical point obtained by the Dlog Padé approximant to the singlet gap is also generally slightly larger than that obtained from the triplet gap around (see Fig. 6).
The full phase diagram in the parameter space of and could be very interesting from the point of view of quantum phase transitions, but may not be easy to determine by numerical methods. Some possible scenarios are shown in Fig. 7. One possibility is that the plaquette phase, for all , has an instability to some magnetic phase and the dimerized phase exists only very close to inside the magnetic phases. A second possibility is that the plaquette-Néel critical line meets the Néel-dimer critical line at some multicritical point at a value of around , after which there is a first order transition between the plaquette and the dimer phases. A third possibility is that the plaquette-Néel, Néel-dimer and plaquette-columnar critical lines all meet at some multicritical point. The numerically determined phase diagram is particularly uncertain in the interesting region, ., where incommensurate correlations could also become important.
Lastly we have derived expansions for a number of generalized susceptibilities. These are defined by adding an appropriate field term
ΔH=h∑ijQij (9)
to the Hamiltonian and computing the susceptibility from
χQ=−1Nlimh→0∂2E0(h)∂h2 (10)
A divergence of any susceptibility signals an instability of that phase with respect to the particular type of order incorporated in .
We have computed two different susceptibilities from the plaquette expansion. One is the antiferromagnetic (Néel) susceptibility with the operator
Qi,j=(−1)i+jSzi,j (11)
The other is the dimerization susceptibility with the operator
Qi,j=Si,j⋅Si+1,j−Si,j⋅Si,j+1 (12)
which breaks the symmetry of interchange of and axes. We have computed series to order for the antiferromagnetic susceptibility and to order for the dimerization susceptibility. The series have been analyzed by Dlog Padé approximants. The series for the antiferromagnetic susceptibility shows the same critical points (within error bars) as those obtained from the triplet gap for . The series for the dimerization susceptibility is very irregular, and does not yield useful results. For example, for , the series is:
χd=629/90+101λ/300+2.0097647λ2−0.269629λ3+0.438527λ4+O(λ5) (13)
For completeness, we also compute the susceptibility for the dimer phase to become unstable to the plaquette phase from an expansion about isolated columnar dimers[16], by adding the following field term:
ΔH=h∑i,j(−1)jSi,j⋅Si,j+1 (14)
which breaks the translational symmetry in the direction perpendicular to the dimers. The series has been computed up to order , (note that here is the parameter of dimerization).
An analysis of the series shows that this susceptibility becomes very large as , for all and the critical , where the susceptibility appears to diverge, approaches unity from above as is increased to . This implies that there are staggered bond correlations in the direction perpendicular to the dimers, which extend over a substantial range. An interesting question is, in the absence of the plaquette phase as discussed earlier, what could these correlations represent? At this stage it is useful to recall another calculation by Kotov et al.[16]. Within the dimer expansion, they calculated two different dimer order parameters,
Dx=||, (15)
and,
Dy=||, (16)
where the elementary dimers connect spins at and . They found that for , is nearly zero, whereas only goes to zero at the critical point. These results suggest that the dimer phase consists of strongly correlated two-chain ladders, which are then weakly correlated from one ladder to next. This striped nature of spin correlations in the dimer phase has not been noted before and is clearly a very interesting result. The situation for is again less clear. As discussed before, there are many possibilities for the phase diagram in that region and much longer series are needed to throw more light on the situation. Perhaps there is an interesting multicritical point in that region of the phase diagram.
## Iii Discussion
We have attempted to further elucidate the nature of the intermediate, magnetically disordered, phase of the spin- Heisenberg antiferromagnet on the square lattice. This phase is believed to occur in the range . Our approach has been to derive perturbation expansions (up to order ) for the ground state energy, singlet and triplet excitation energies, and various susceptibilities, starting from a system of decoupled plaquettes () and extrapolating to the homogeneous lattice (). We have also derived expansions about an unperturbed state of isolated columnar dimers (“dimer phase”). Both of these have been proposed as candidates for the intermediate phase.
We find that the ground state energy for both plaquette and dimer phases are very similar, any difference lying within the error bars. From this result alone we cannot favor one phase over the other.
The analysis of the singlet and triplet excitation spectra suggests an instability in the plaquette phase. In particular, in the disconnected plaquette expansions, Dlog Padé analysis indicates that the gaps would vanish for less than unity. The gap appears to close first for the triplets and then for the singlets. This is the strongest evidence that the plaquette phase is not realized in this model. However, we should mention here that the critical exponents associated with the vanishing of the gaps are rather small () and the gap closes not too far from equal to unity. Thus, with a relatively short series, this should be treated with some caution. One could ask why the energy series appear to converge well despite the instability. However, this is a well known feature of series expansions, that quantities having weak singularities may continue to show reasonable values even if extrapolated past the singularity.
A consistent interpretation of these results is that within the parameter space of our non-uniform Hamiltonian, the plaquette phase is first unstable to a magnetic phase, which then must give way to the columnar dimer phase. Similar results for the instability of the staggered dimer phase were suggested before by Gelfand et al[11]. However, the full phase diagram in the and parameter space is difficult to obtain reliably, especially near the transition to the columnar phase. There are possibilities of some novel multicritical points, which deserve further attention.
One of our most interesting results is the finding of striped spin correlations in the dimer phase. In this phase, the nearest neighbor spin correlations are nearly equal along the rungs and along the chains of a two spin column and there are extended bond correlations along the chains. However, spin correlations from one column to the next are much weaker. In other words, the dimers are strongly resonating along vertical columns. The existence of a quantum critical point separating an antiferromagnetic phase with such a quantum disordered phase with striped correlations is a very interesting feature of this model which deserves further attention in the context of high- materials.
###### Acknowledgements.
We would like to thank Subir Sachdev and Oleg Sushkov for many useful discussions. This work has been supported in part by a grant from the National Science Foundation (DMR-9616574) (R.R.P.S.), the Gordon Godfrey Bequest for Theoretical Physics at the University of New South Wales, and by the Australian Research Council (Z.W., C.J.H. and J.O.). The computation has been performed on Silicon Graphics Power Challenge and Convex machines. We thank the New South Wales Centre for Parallel Computing for facilities and assistance with the calculations.
## References
• [5] E. Dagotto and A. Moreo, Phys. Rev. Let. 63, 2148(1989).
• [6] R.R.P. Singh and R. Narayanan, Phys. Rev. Lett. 65, 1072(1990).
• [7] H.J. Schulz and T.A.L. Ziman, Europhys. Lett. 18, 355(1992). T. Einarsson and H.J. Schulz, Phys. Rev. B51, 6151(1995); H.J. Schulz, T.A.L. Ziman, and D. Poiblanc, J. Phys. I France 6, 675(1996).
• [8] J. Richter, Phys. Rev. B47, 5794(1993).
• [9] H.A. Ceccatto, C.J. Gazza and A.E. Trumper, Phys. Rev. B45, 7832(1992).
• [10] A.V. Dotsenko and O.P. Sushkov, Phys. Rev. B50, 13821(1994).
• [11] M.P. Gelfand, R.R.P. Singh and P.A. Huse, Phys. Rev. B 40, 10801(1989).
• [12] J. Oitmaa and W.H. Zheng, Phys. Rev. B 54, 3022 (1996).
• [13] N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773(1991); S. Sachdev and N. Read, Int. J. Mod. Phys. B5, 219 (1997).
• [14] M.E. Zhitomirsky and K. Ueda, Phys. Rev. B 54, 9007 (1996).
• [15] M.P. Gelfand, Phys. Rev. B 42, 8206(1990).
• [16] V.N. Kotov, J. Oitmaa, O. Sushkov and W.H. Zheng, cond-mat/9903154.
• [17] H.X. He, C.J. Hamer and J. Oitmaa, J. Phys. A 23, 1775(1990).
• [18] M.P. Gelfand, R.R.P. Singh and P.A. Huse, J. Stat. Phys. 59, 1093(1990).
• [19] M. P. Gelfand, Solid State Commun. 98, 11 (1996).
• [20] W.H. Zheng, M.P. Gelfand, R.R.P. Singh, J. Oitmaa, and C.J. Hamer, Phys. Rev. B 55, 11377 (1997); M. P. Gelfand, W.H. Zheng, Rajiv R. P. Singh, J. Oitmaa and C. J. Hamer, Phys. Rev. Lett. 77, 2794(1996);
• [21] A.J. Guttmann, in Phase Transitions and Critical Phenomena, edited by C. Domb and M.S. Green (Academic, New York, 1989), Vol. 13.
• [22] A. Koga, S. Kumada and N. Kawakami, preprint cond-mat/9810414.
If you find a rendering bug, file an issue on GitHub. Or, have a go at fixing it yourself – the renderer is open source!
For everything else, email us at [email protected]. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8442201614379883, "perplexity": 1118.2380279255187}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487586239.2/warc/CC-MAIN-20210612162957-20210612192957-00104.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-12th-edition/chapter-9-section-9-5-common-and-natural-logarithms-9-5-exercises-page-620/2 | Intermediate Algebra (12th Edition)
A. $e$
We know that logarithms with base $e$ are called natural logarithms and the base $e$ logarithm of $x$ is written as $ln(x)$. Therefore, the base of $ln(x)$ is $e$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.924212634563446, "perplexity": 210.15001101054196}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670731.88/warc/CC-MAIN-20191121050543-20191121074543-00135.warc.gz"} |
http://indico.gsi.de/event/7878/ | AP-Seminare
Laboratory X-ray Astrophysics with Highly Charged Ions
by Sven Bernitt (IOQ, FSU Jena)
Wednesday, November 14, 2018 from to (Europe/Berlin)
at GSI ( KBW Lecture Hall - Sider Room )
Description Space observatories, like the satellites XMM-Newton and Chandra, observe the x-ray spectra of hot astrophysical plasmas. Such are present in stellar atmospheres, the accretion discs around black holes, intracluster media, and many other environments. The comparison of observed x-ray spectra with plasma models can reveal the state and dynamics of different components of those hot objects. The models heavily depend on the accurate knowledge of the underlying atomic and molecular processes. Recent observations of the Perseus galaxy cluster with the Hitomi Soft X-ray Spectrometer microcalorimeter provided a high-resolution spectrum in the photon energy range from 0.1 to 12 keV, with many well-resolved line features originating from highly charged ions of most astrophysically relevant elements, from silicon to nickel. However, its analysis has uncovered significant shortcomings of commonly used spectral modelling software packages. These include inaccurate transition energies, but also atomic-scale processes completely missing from the models. One component not included in spectral models was emission following charge exchange between bare sulfur ions and atomic hydrogen. We have studied this process with an electron beam ion trap (EBIT) and found it to be a likely explanation of a weak line feature around 3.5 keV found in galaxy cluster spectra. This feature had previously sparked enormous interest in the scientific community, when it was attributed to a possible dark matter decay process. This illustrates how incomplete knowledge of atomic-scale processes limits the amount of information that can be extracted from astrophysical x-ray spectra. We have combined EBITs with ultrabrilliant synchrotron and free-electron laser x-ray light sources to resonantly excite electronic transitions in trapped highly charged ions. These experiments have provided valuable atomic data and help to benchmark atomic structure theory. Furthermore, we have used a newly developed compact EBIT to provide an accurate calibration of molecular photoabsorption spectra relevant for current and future x-ray satellite missions. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8324682116508484, "perplexity": 2631.676885545665}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039744381.73/warc/CC-MAIN-20181118135147-20181118161147-00434.warc.gz"} |
https://proofwiki.org/wiki/Group_of_Reflection_Matrices_Order_4 | # Group of Reflection Matrices Order 4
It has been suggested that this page or section be merged into Klein Four-Group as Order 2 Matrices. (Discuss)
## Definition
Consider the algebraic structure $S$ of reflection matrices:
$R_4 = \set {\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} }$
under the operation of (conventional) matrix multiplication.
$R_4$ is the group of reflection matrices of order $4$.
### Cayley Table
$\begin{array}{r|rrrr} \times & r_0 & r_1 & r_2 & r_3 \\ \hline r_0 & r_0 & r_1 & r_2 & r_3 \\ r_1 & r_1 & r_0 & r_3 & r_2 \\ r_2 & r_2 & r_3 & r_0 & r_1 \\ r_3 & r_3 & r_2 & r_1 & r_0 \\ \end{array}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8679466843605042, "perplexity": 1575.2789762363786}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195529276.65/warc/CC-MAIN-20190723105707-20190723131707-00420.warc.gz"} |
https://tex.stackexchange.com/questions/26114/how-to-insert-a-dot-after-the-theorem-environment | # How to insert a dot after the theorem environment?
How to insert a dot after the theorem environment? I use \newtheorem{exerc}{}.
\documentclass[a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{theorem}
\newtheorem{exerc}{}
\begin{document}
\begin{exerc}
Exercise
\end{exerc}
\end{document}
I need.
1. Exercise
2. Exercise
\renewcommand{\theexerc}{\arabic{exerc}.}% #.
just after you declare your new theorem \newtheorem{exerc}{}.
• Generally, it's a bad idea to touch \theexerc for that sort of thing as it will also affects the \label/\ref mechanism. Using \theoremseparator{.} from the ntheorem package (which is an expansion of the theorem package) will avoid any side effect. – Philippe Goutet Aug 19 '11 at 20:48
I usually use amsthm instead of theorem. It provides \newtheoremstyle in which you can define your own theorem styles:
\documentclass[a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsthm}
% the following style is based on the example
% found in the amsthm manual
\newtheoremstyle{mystyle}% name
{3pt}% space above
{3pt}% space below
{}% body font
{}% indent amount
\theoremstyle{mystyle}
\newtheorem{exerc}{}
\begin{document}
\begin{exerc}
Exercise
\end{exerc}
\end{document}
Hope it helps. =)
Below is a solution using ntheorem. As a side note, if you're making Exercises/Problems with associated solutions, you might like to look up the answers package.
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{ntheorem}
\theoremstyle{plain}
\theorembodyfont{}
\theoremsymbol{}
\theoremprework{}
\theorempostwork{}
\theoremseparator{.}
\newtheorem{exerc}{}
\begin{document}
\begin{exerc}
Exercise
\end{exerc}
\end{document} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8419496417045593, "perplexity": 3581.0564286919125}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526408.59/warc/CC-MAIN-20190720024812-20190720050812-00285.warc.gz"} |
http://link.springer.com/article/10.1007%2Fs10712-012-9184-0 | Surveys in Geophysics
, Volume 33, Issue 3, pp 779–816
# Observing and Modeling Earth’s Energy Flows
## Authors
• Max-Planck-Institüt for Meteorology
• Stephen E. Schwartz
• Atmospheric Sciences DivisionBrookhaven National Laboratory
DOI: 10.1007/s10712-012-9184-0
## Abstract
### Keywords
Climate change Cloud radiative effects Aerosol Energy budget Climate sensitivity Radiative forcing
## 1 Introduction
Modern climate science, wherein descriptive pictures of the climate system began to be complemented by quantitative theory, is only about a hundred years old. In the late nineteenth Century understanding of radiative transfer, particularly at the infrared wavelengths associated with terrestrial radiation, was developing rapidly, and it became possible to formulate quantitative descriptions of the relationship between the flux of energy through the Earth system and quantities like the average surface temperature. Exemplary in this respect are Arrhenius’ 1896 calculations suggesting that changes in carbon dioxide would induce changes in the surface temperature. Arrhenius’ study was essentially an exercise in radiative transfer in which he quantified the flow of solar and terrestrial radiation through the Earth system and the roles of the various processes that influenced these transports. Greenhouse gases such as carbon dioxide and water vapor played a key role in Arrhenius’ calculations, and other factors influencing solar radiation such as clouds and surface properties were also accounted for. Although prescient in many respects, for instance with respect to the role of the carbon cycle and feedbacks associated with water vapor and surface albedo, Arrhenius did not touch on a number of issues that have come to dominate the discourse with respect to climate change. Among these are rate of increase in ocean enthalpy and changes in the ocean circulation; changes in patterns of precipitation; the role of aerosols, both in influencing clouds and in the energy budget as a whole; and also the possibility that changes in cloudiness may enhance or offset other changes in the climate system.
An increased emphasis on precisely those issues largely left out of the early studies has marked a fundamental shift in climate science over the last forty or so years. Through the latter part of the 1960s, climate science was preoccupied with resolving controversies related to the radiative transfer that was the basis for the analysis of Arrhenius and his followers. Confusion about the nature and importance of the details of the spectroscopy of CO2 and H2O in the thermal infrared (e.g., Plass 1956; Kaplan 1960) was resolved only by the calculations of Manabe and Wetherald (1967), which showed that earlier controversies lost relevance when the vertical structure of the atmosphere is properly accounted for (cf., Pierrehumbert 2011). Their research propelled the field into the current era, wherein qualitatively new questions, in particular the role of clouds and aerosol particles, demanded a more detailed understanding and accounting of the energy flows through the climate system.
Arrhenius was perhaps the first to appreciate how small changes in flows of energy through the system can have a large impact on Earth’s climate. A doubling of the mixing ratio (commonly “concentration”) of atmospheric CO2, which is the paradigmatic example of a forcing of climate change, gives rise to a radiative perturbation of 3–4 W m−2, which is about 1 % of the solar radiation incident at the top-of-atmosphere;1 the total radiative forcing attributed to long-lived greenhouse gases introduced through human activities over the industrial era thus far is about 3 W m−2. By way of comparison, in the tropics, the diurnal variation of incident solar radiation is more than 1,000 W m−2. The presence of a high cloud can change the outgoing long-wave radiation by 100–200 W m−2, comparable in magnitude to seasonal changes in radiative fluxes, albeit more short lived. From the perspective of climate system response, the central value of current estimates of the increase in global mean surface temperature that would result from a doubling of CO2, 3 K, is about 1 % of the global mean surface temperature, 288 K, and again, much less than geographical and temporal variability. The rather large consequences of such small changes in Earth’s energy flows, and the complexity of the system that mediates these flows, make determining the effects of changing atmospheric composition on Earth’s energy flows, through measurement or modeling, a challenging scientific problem.
In the following, we reflect on this challenge as framed by discussions at a recent ISSI workshop titled “Observing and Modelling Earth’s Energy Flows”. Our presentation is organized around three basic issues: (1) what is the status of present understanding of Earth’s energy budget; (2) how does the composition of the atmosphere, particularly clouds and aerosols, influence this budget; and (3) how can modeling help constrain a description of processes regulating the flow of energy through the climate system. These issues, particularly those aspects central to advancing understanding of the climate system, are discussed in turn below. Although the discussion presented herein benefitted greatly from presentations and discussions at the Workshop, responsibility for the material presented rests with the present authors and should not be taken as representing the views of participants in the workshop or as a workshop consensus.
## 2 Present Understanding of Earth’s Energy Budget
### 2.1 Global Energy Balance
Measurements have been indispensable to the advancing understanding of energy flows through the climate system. Current understanding of these flows is summarized in Fig. 1, which we have constructed based on the available literature (especially Trenberth et al. 2009; Kato et al. 2012; Stephens et al. 2012). The energy flows are more certain at the top of the atmosphere than at the surface, as measurements at the top of the atmosphere have benefitted greatly from advances in satellite remote sensing. Measurements from the Earth Radiation Budget Experiment (ERBE) (Ramanathan 1987) and now those from the Clouds and Earth’s Radiant Energy System (CERES) mission (Wielicki et al. 1996) have convincingly shown that Earth reflects less short-wave radiation and emits more long-wave radiation than previously thought (cf., Table 1) and have contributed to closure of the top-of-atmosphere energy budget to within a few watts per square meter.
Table 1
Estimates of Earth’s energy budget, subjectively determined based on a review of the existing literature and the best estimates of the net imbalance at the surface and top-of-atmosphere
Source
H-1954
L-1957
R-1987
K-1997
Z-2005
T-2009
S-2012
This study
Top of atmosphere
Incident SW
338
349
343
342
342
341
340
340
Reflected SW
115
122
106
107
106
102
98
100
Outgoing LW
223
227
237
235
233
239
239
239
Surface
Absorbed SW
160
167
169
168
165
161
168
162
Downward LW
352
334
327
324
345
333
347
342
Upward LW
398
395
390
390
396
396
398
397
Latent heat flux
82
68
90
78
n/a
80
88
86
Sensible heat flux
30
33
16
24
n/a
17
24
20
References to prior estimates are: Houghton (1954), London (1957), Ramanathan (1987), Kiehl and Trenberth (1997), Trenberth et al (2009), Stephens et al. (2012). Because both the H-1954 and L-1957 estimates were for the northern hemisphere only, these estimates have been rescaled using the ratio of the global versus the northern hemisphere average from a high-resolution AMIP simulation using the ECHAM6 model. For basis of present estimates see Appendix
Also contributing to the reduction of uncertainty in the balance of energy flows at the top of the atmosphere are new measurements of total solar irradiance and ocean heat uptake. The total solar irradiance determined by the Solar Radiance and Climate Experiment, 1,360.8 ± 0.5 W m−2 at solar minimum (Kopp and Lean 2011), is well below the range of previous estimates as summarized in Table 1. Confidence in these lower estimates and their associated assessment of uncertainty is gained through the identification of artifacts in the older measurements that lends enhanced credence to the newer lower estimates. Because the atmosphere has a relatively small heat capacity, an imbalance of energy flows at the TOA can be sustained only an increase in ocean enthalpy, augmented to lesser extent by melting of the cryosphere and warming of the land surface and the atmosphere. As discussed by Lyman (2011), the rate of heating of the top 700 m of the world ocean over the period from 1993 to 2008, as inferred from temperature measurements and expressed per the area of the entire planet, is 0.64 ± 0.11 W m−2 (90 % confidence interval). Sparser measurements extending to ocean depths of 3 km reported by Levitus et al. (2005) suggest that the upper ocean takes up about three-quarters of the ocean heating. Levitus et al. (2005) also estimate that the contributions of other enthalpy sinks, including the atmosphere, the land, and the melting of ice, can account for an additional 0.04] W m−2; based on these estimates the flux imbalance at the TOA is estimated to be 0.9 ± 0.3 W m−2. The uncertainty is based on the 90 % confidence interval given by Lyman and the assumption that the relative uncertainty in the deep ocean enthalpy uptake estimates and in the estimates of heating by other components of the Earth system are about 50 %. 2 For reference, in constructing the energy-balanced version of the CERES data, Loeb et al. (2009) estimated the surface enthalpy uptake to be 0.85 W m−2 similar to the 0.9 W m−2 estimated here and employed by Trenberth et al. (2009) based on a somewhat different line of reasoning. An analysis based on more recent ARGO data and a revised analysis of the CERES measurements Loeb et al. (2012) suggest a somewhat lower central value of 0.50 ± 0.43 W m−2 for the rate of increase in Earth system enthalpy, per unit area of Earth’s surface. In summary, although there is a rather larger, 6.5 W m−2, inherent uncertainty in measurements of the reflected short-wave and emitted long-wave radiation at the TOA, associated principally with uncertainty in the absolute calibration of the CERES instruments (Loeb et al. 2009), the net irradiance at the TOA is constrained to within 1 W m−2 by improved measurements of increases in ocean enthalpy.
The surface energy budget is distributed over several terms, each of which exhibits uncertainty that is several-fold greater than the uncertainty in the net budget at TOA. As pointed out by Trenberth et al. (2009), if each of the terms in the surface energy budget is estimated individually, in isolation of the others, an imbalance can arise in the net surface flux that is as much as 20 W m−2; this is more than an order of magnitude greater than current measurement-based estimates of the rate of increase in the enthalpy of the ocean, cryosphere, and land. In absolute terms, the uncertainty is largest for the long-wave irradiance downwelling at the surface and for the latent heat flux (precipitation). Here too, new measurements and improved modeling of energy flows are beginning to improve understanding. Active remote sensors such as the cloud profiling radar flown as part of the CloudSat mission (Stephens et al. 2008) and the Cloud-Aerosol Lidar with Orthogonal Polarization flown as part of the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation mission (Winker et al. 2010) are providing improved estimates of the vertical distribution of clouds, particularly cloud base. These measurements are crucial for estimates of the downward long-wave irradiance, which are based on radiative transfer modeling given a realistic atmospheric state. Using these measurements, Kato et al. (2012) estimate a downward long-wave irradiance at the surface of 345.4 ± 6.9 W m−2, which is considerably larger than the value derived by Trenberth et al (2009) from the residual of the surface energy balance given existing precipitation climatologies. CloudSat and surface-based measurements also suggest that existing precipitation climatologies underestimate light rain from warm clouds, which is common over the tropical ocean (Nuijens et al. 2009; Stephens et al. 2012), and which may help reconcile the differences in the energy-based versus precipitation-based estimates of the strength of the hydrological cycle. Although still large, overall the uncertainty in the surface energy budget, as presented in Fig. 1 is somewhat smaller than that presented by Stephens et al. (2012), as global modeling (including reanalysis) estimates constrained by observations are given somewhat more weight in the present analysis, particularly for the upward turbulent fluxes and the long-wave irradiance.
Advances in understanding of Earth’s energy flows become most apparent when looking beyond the globally and annually averaged estimates of these flows. Measurements by CERES instruments track day-to-day variations in Earth’s energy flows on regional scales. Measurements extending over more than a decade have made it possible to explore how these energy flows fluctuate on intradecadal time-scales. As an example, Fig. 2 shows the existing record of monthly and globally averaged radiation anomalies. The radiation anomalies are calculated using the SSF1deg (single scanner footprint at one degree) data set, which includes information only from the twice daily measurements of the CERES platform on the polar orbiting Terra (ascending node equator crossing time 10:30 am) and Aqua (ascending node equator crossing time 1:30 pm). Unlike the CERES SYN or EBAF products, the SSF1deg data set does not incorporate measurements from a shifting complement of geostationary satellites, and thus more fully represents the inherent stability of the CERES measurements themselves. Monthly anomalies were constructed by subtracting the monthly climatology of the same data set from the global and monthly averaged values. The remarkable agreement between the Aqua and Terra measurements, over the period of the record where they coincide, suggests that the observed variability is an intrinsic property of Earth’s climate system, rather than an artifact due to instrument precision or insufficient sampling. This inherent variability limits quantification of secular trends in the flow of energy through the TOA.
This variability in Earth’s energy flows, manifest as fluctuations in the anomaly of net irradiance as large as 3 W m−2 over annual to inter-annual timescales, e.g., Fig. 2, is expected to be mirrored in other components of the climate system. By combining these trends with concurrent measurements of ocean enthalpy, Wong et al. (2006) and Trenberth and Fasullo (2010b) have begun to explore this question. By drawing attention to the what ended up being insignificant (and spurious) trends over the length of the record, Trenberth and Fasullo (2010b) have given what turns out to be a false impression of irreconcilable trends in the relation between ocean enthalpy increases and the TOA energy imbalance.3 These results unfortunately overshadowed the real revolution in such approaches, namely the recognition that the present observing system shows evidence of large inter-annual fluctuations in the flow of energy through the Earth system and that these might be trackable as they work their way through the system— for instance from anomalies in the TOA net radiation, to changes in ocean heat uptake.
We disagree with the assertion of Stephens et al. (2012) that the satellite measurements such as CERES have “not significantly changed our understanding of the TOA energy balance” relative to what was known based on the ERBE data. Although we acknowledge that estimates of the surface radiation budget do not significantly differ from those of Houghton (1954) more than a half century ago, we would argue in both cases that, when viewed from the perspective of initial attempts to infer the planetary energy balance from measurements looking down and up from Mt Wilson (Abbot and Fowle 1908), or on the basis of a network of pyrheliometers over North America (e.g., Houghton 1954), progress has been tremendous, even if the small magnitude of changes in estimates of the globally and annually averaged budgets (Table 1) would seem to belie this progress. The present era of global sensing is unprecedented: high-precision instruments are tracking Earth’s energy flows on decadal timescales and regional spatial scales; an absolute accuracy of better than one part per thousand has been achieved in measurements of solar irradiance and is helping to close the TOA energy budget; networks of ocean-going floats are quantifying ocean heat uptake; and surface-based networks and profiling instruments are beginning to advance understanding of the surface energy budget. As should become increasingly clear through the course of this article, sustaining and expanding such measurements will be key to further advances in understanding of Earth’s climate and its susceptibility to change.
### 2.2 Radiative Forcing, Response and Climate Sensitivity
In addition to providing understanding of energy flows in the current climate, consideration of the energy budget also provides a framework for understanding climate change. This framework, which has developed over the last 30 years, rests on the assumption that changes in the globally averaged surface temperature can be linearly related to a radiative forcing, F. The constant of proportionality between the forcing and the response is called the equilibrium climate sensitivity, $$S_{\rm{eq}},$$ which can be formally defined as the steady-state change in $$T_{\rm s}$$ (the globally averaged near-surface air temperature) that would result from a sustained change in a radiative flux component of the Earth energy budget at the TOA (forcing), normalized to that flux change, with unit: K (W m−2)−1. That is, the equilibrium sensitivity is the proportionality constant between the steady-state change in surface temperature and the applied forcing,
$$\Updelta T_{\rm s} =S_{\rm{eq}} F.$$
(1)
Because it codifies the sensitivity of such an essential feature of the climate state, the globally averaged surface temperature, Eq. 1 provides a powerful description of past and prospective future climate change. To the extent that other properties of the climate system scale with its value, $$S_{\rm{eq}}$$ assumes an even broader significance. For these reasons, determination of $$S_{\rm{eq}}$$ has evolved into a central focus of climate science (e.g., Knutti and Hegerl 2008).
Equation 1 is well suited to the interpretation of models, for which the changes in the equilibrium state can be determined on the basis of numerical simulations. However, by extending the framework that leads to Eq. 1, a related framework has been developed which allows for the determination of $$S_{\rm{eq}}$$ from observations. Assuming that the forcing, F were known, it could be related to the rate of change in the enthalpy of the Earth system, $$\dot{H}$$ , and the temperature change between that at t = 0, defined to correspond to a time when the system is in stationarity and the forcing is applied, and that at some later time, t
$$\dot{H} =F - S_{\rm{eq}}^{-1} (T_{\rm s}(t)-T_{\rm s}(0)).$$
(2)
If both the forcing and the rate of change of the planetary enthalpy are known, the equilibrium sensitivity might be determined (Gregory et al. 2002) as
$$S_{\rm{eq}} = {\frac{T_{\rm s}(t)-T_{\rm s}(0)} {F - \dot{H}}}$$
(3)
where the forcing and the change in global temperature are relative to a prior unperturbed state (at time, t = 0) and the rate of heating of the planet $$\dot{H}$$ can be inferred either from satellite measurements at the top of the atmosphere or from the change in enthalpy of the planet inferred from ocean calorimetry as discussed by Lyman (2011). However, application of Eq. 3 is limited by the fact that changes in the enthalpy and surface temperature must be measured relative to an equilibrium state that is not generally known. To sidestep this issue Forster and Gregory (2006) and Murphy et al. (2009) proposed to determine $$S_{\rm{eq}}$$ over shorter time-periods, not starting with the unperturbed state, as
$$S_{\rm{eq}} = {\frac{T_{\rm s}^{\prime}}{F^{\prime}- \dot{H}^{\prime}}}$$
(4)
where primes denote changes in the quantities over a given time-period. The utility of any such determination of sensitivity rests on the uncertainties in the several quantities on the right-hand side of the equation.
Central to the conceptual framework surrounding the use of Eq. (1), or its time-dependent variants, are the assumptions: (1) that $$S_{\rm{eq}}$$ does not depend on the nature or magnitude of the forcing; and (2) that the forcing resulting from a given change in atmospheric composition or surface properties can be determined unambiguously. Both assumptions become increasingly problematic if the forcing-sensitivity framework is pushed too far, for instance by demanding ever more certain estimates of $$S_{\rm{eq}}$$. Models have long suggested that $$S_{\rm{eq}}$$ depends on both the temperature and the nature of the forcing (e.g., Hansen et al. 1997; Colman and McAvaney 2009). For instance, Hansen et al. (1997) showed that if the TOA radiative forcing is equated with the initial radiative perturbation associated with a change in CO2, the change in $$T_{\rm s}$$ will differ by as much as 25 % as compared to what arises for an equal radiative perturbation arising from a change in the solar irradiance. To get around this issue, the concept of an effective, or adjusted, forcing has been developed. The adjusted radiative forcing attempts to account for a rapid, and perturbation-dependent, adjustment of the climate system to a compositional change. The familiar example, and the one that motivated the original concept of an adjusted radiative forcing, is the adjustment of the stratosphere which proceeds differently depending on whether the initial radiative perturbation is caused by changes in well-mixed greenhouse gases as compared to changes in the solar irradiance, or stratospheric aerosols (e.g., Hansen et al. 1997). However, several studies have begun to show that, in addition to the stratosphere, changes in other components of the atmosphere, for instance clouds, also adjust rapidly in ways that depend on the nature of the external perturbation and thus must be factored into the calculation of an adjusted radiative forcing. An elegant framework for dealing with these adjustments has been proposed by Gregory et al. (2004). In their approach, the adjusted radiative forcing is defined as the y-intercept of the regression of the TOA radiative imbalance against $$T_{\rm s}$$. This definition formalizes the idea of a rapid adjustment of the system to a perturbation which determines the effective, or adjusted radiative forcing, and which is temporally well separated from the slow response of the surface temperature to this adjusted radiative forcing. Of course this definition means that the forcing is no longer independent, or external to the system and thus cannot be determined a priori, for instance, through complete knowledge of radiative transfer and compositional changes alone. And as a result, internal physical processes, such as the nature of cloud adjustment, contribute to uncertainty in estimates of the effective radiative forcing driving changes in Earth’s climate.
The dependence of the forcing on the system to which the perturbation is applied makes the framework surrounding Eq. (1) less useful, as it implies that both F and $$S_{\rm{eq}}$$ depend on the system. The extra uncertainty, which the fast response of the system introduces in the determination of the forcing, poses a particular challenge to observationally based attempts to estimate $$S_{\rm{eq}},$$ for example through the use of either Eqs. (3) or (4). By focusing on periods over which the observational record is better constrained the numerator and denominator of Eqs. (3) and (4) become small; but in this case, because the denominator is the difference of two rather large terms, the answer becomes proportionally more sensitive to errors in either $$F, \dot{H}$$ (or their associated changes in time). So, although in the past, estimates of the forcing F have generally been thought to be limited by understanding to changes in the aerosol over the period of industrialization (e.g., Gregory et al. 2002), recent work suggests that factors such as clouds, which contribute to the fast response of the system, (Hansen et al. 1997; Gregory and Webb 2008; Andrews et al. 2009 2011), may place a more fundamental limit on the quantification of the effective radiative forcing that arises as a result of compositional changes to the atmosphere.
## 3 Clouds
### 3.1 Radiative Effects, Feedbacks, and Forcing
The definition of the cloud radiative effect can be made precise as follows. Let Q denote the rate of absorption of solar energy (short-wave irradiance) and E denote the rate of emission of infrared energy (long-wave irradiance), both at TOA. If an inward directed flux is defined to be positive, E at the TOA must be negative. With this sign convention and letting H denote the enthalpy of the Earth system, including components such as oceans, land and surface ice, then the rate of change of H with time,4 $$\dot{H}$$ is given by $$\dot{H} = Q + E.$$ Under the assumption that three-dimensional radiative effects are negligible the two radiative flux terms may be conceptually distinguished into components pertaining to contributions from cloudy and cloud-free regions of the planet:
$$Q = Q_{\star}\left[ 1 - \alpha_0(1-A_{\rm c}) - \alpha_{\rm c}A_{\rm c}\right]$$
(5)
$$E = -E_0(1-A_{\rm c}) - E_{\rm c} A_{\rm c}$$
(6)
Here $$A_{\rm c}$$ denotes the fraction of the area of the planet that is, on average, cloudy; $$Q_{\star}$$ denotes the average solar irradiance incident at the TOA ($$Q_{\star} \approx 340$$ W m−2, e.g., Fig. 1); α0 and $$\alpha_{\rm c}$$ denote the effective TOA albedo of the cloud-free and cloudy scenes, respectively; and E 0, and $$E_{\rm c}$$ denote the emitted long-wave irradiance obtained by compositing over the cloud-free and cloudy scenes, respectively. The albedos are designated as “effective” quantities because Eq. (5) depends non-linearly on cloudiness and insolation and, thus, the effective albedos must account for the co-variability between these two quantities. So for instance, $$\alpha_{\rm c}$$ is not the average cloud albedo, but rather the cloud albedo that the average cloudiness requires so that the planetary albedo, $$\alpha = \alpha_0 (1-A_{\rm c}) - \alpha_{\rm c} A_{\rm c}$$ matches that observed, likewise for the α0, the effective albedo of the cloud-free scenes. In general each of the terms on the right-hand side of Eqs. (5) and (6) except for $$Q_{\star},$$ is a function of the state of the system, importantly the vertical distribution of temperature, T 0, the composition of the atmosphere and the surface properties. $$E_{\rm c}$$ is the effective emitted irradiance of cloudy scenes that is calculated by compositing over all columns not identified as being cloud-free. It depends on the distribution of clouds, but because clouds have some transparency and the atmosphere has some opacity, it also depends on the composition and temperature of the atmosphere, and the co-variability among the two. The short- and long-wave components of the CRE, denoted by superscripts (Q) and (E) respectively, follow naturally as the difference between the all-sky radiative flux and the fluxes which would be manifest in the absence of clouds, i.e.,
$$F^{(Q)}_{{\rm c}} = -Q_{\star}(\alpha_{\rm c} - \alpha_0) A_{\rm c}$$
(7)
$$F^{(E)}_{{\rm c}} = -(E_{\rm c} - E_0)A_{\rm c}$$
(8)
Given our sign convention, and because $$\alpha_{\rm c}$$ is generally greater than α0 whereas $$E_{\rm c}$$ is generally less than E 0, the short-wave CRE is negative and the long-wave CRE is positive. Both quantities increase in magnitude with cloud amount, $$A_{\rm c}.$$ The net CRE, F c , is given by F c (Q) + F c (E) . A secular increase in the magnitude of F c (Q) would exert a cooling influence on the Earth system, whereas an increase in the magnitude of F c (E) would exert a warming influence. It should be emphasized that the CRE depends not just on the properties of the cloudy fraction of the planet but on the differences between the cloudy and cloud-free portions of the planet. As reviewed by Loeb et al. (2009) the application of this concept to various data sets shows the short-wave CRE to range from −45.4 to −53.3 W m−2 and the long-wave CRE to range from 26.5 to 30.6 W m−2; the associated net CRE from these prior estimates ranges from −16.7 to −24.5 W m−2. The CERES EBAF data, upon which the TOA estimates in Fig. 1 are largely based, give a short-wave cloud radiative effect of −47.1 W m−2 and a long-wave CRE of +26.5 W m−2; overall clouds, more precisely cloudy scenes, exert a net cooling influence on the Earth system of about 20 W m−2.
The CRE concept has seen considerable use in the interpretation of feedbacks in the climate system. From the perspective of Eq. (1), a feedback is a change in a radiative flux that results from a change in global temperature; such a further change in radiative flux in addition to that caused by an initial forcing imposed on the climate system can enhance or diminish the temperature change induced by a given forcing (positive or negative feedback, respectively). The feedback concept has been quite useful in interpreting the contributions of different components of the climate system to $$S_{\rm{eq}}.$$ It is quite straightforward to demonstrate that changes in cloud radiative effects are not the same as cloud feedbacks. To appreciate this point note that, in the limit of small changes, the strength of the short-wave cloud feedback can be derived formally from (5) and (6) as (e.g., Soden et al. 2008; Schwartz 2011)
$$\lambda_{{\rm c}}^{(Q)} = {\frac{\partial Q}{\partial A_{\rm c}}} \; {\frac{\partial A_{\rm c}}{\partial T_{\rm s}}} + {\frac{\partial Q}{\partial \alpha_{\rm c}}} \; {\frac{\partial \alpha_{\rm c}}{\partial T_{\rm s}}}$$
(9)
$$= - Q_{\star} \left[ (\alpha_{\rm c} - \alpha_0) {\frac{\delta A_{\rm c}}{\delta T_{\rm s}}} + A_{\rm c} {\frac{\delta \alpha_{\rm c}}{\delta T_{\rm s}}} \right].$$
(10)
A change in the CRE from surface temperature induced changes in cloud amount is, however, not simply equal to the cloud feedback times the change in surface temperature; but rather includes an additional term that accounts for the change in the albedo of the cloud-free scenes with the change in surface temperature:
$$\delta F_{{\rm c}}^{(Q)} = \lambda_{{\rm c}}^{(Q)} \delta T_{\rm s} + Q_{\star} A_c {\frac{\partial \alpha_0}{\partial T_s}} \delta T_s.$$
(11)
The relationship between the cloud feedback and the change in the CRE follows similarly for the long-wave part of the spectrum. Equation (11) explicitly includes the dependence of the CRE on factors other than cloudiness, in the present example also the cloud-free sky albedo and long-wave emission from cloud-free scenes. Thus if, for example, in a changing climate, the surface albedo or cloud-free-sky aerosol changed, the CRE, as conventionally defined, would change for reasons that have nothing to do with changes in cloud properties. If the cloud properties remained fixed, the actual cloud feedback would be zero. This situation is described by Soden et al (2008) as a masking effect. This example is readily extended to the long-wave, where the change in the cloud-free-sky emission, E 0 can cause a change in the CRE. Because δE 0 and δa 0 are in principle observable, the net effect of changing clouds on the response of the system to an external perturbation can, in principle, be determined from measurements. Soden et al. (2008), illustrate how radiative kernels can be used to diagnose cloud feedbacks from changes in CRE, although the kernel methods conflate cloud feedbacks with cloud mediated CO2 indirect forcing (adjustment), as discussed below. Irrespective of how a feedback is calculated, care must be taken in its interpretation, as the definition of a feedback is dependent on how one defines their system. Choosing relative humidity instead of absolute humidity as a thermodynamic coordinate, or potential temperature instead of height as a vertical coordinate, can give very different pictures of the feedbacks in the system for reasons that have nothing to do with the processes taking place in the atmosphere.
In Sect. 2.2, it was pointed out that rapid adjustments by clouds can contribute to the adjusted radiative forcing associated with an external perturbation to the atmospheric composition. It proves instructive to illustrate this idea using the conceptual framework developed above. To do so, consider the simplified case in which only the long-wave emission of the atmosphere depends on the greenhouse gas concentration, so that $$E = E(\chi,T_{\rm s})$$ where χ denotes the greenhouse gas concentration. In this case, the net radiative forcing that results from a perturbation in a greenhouse gas concentration, δχ, can be expressed as follows
$$F_{\chi} = F^{(E)}_{\chi} = {\frac{\partial E}{\partial \chi}} \delta \chi.$$
(12)
To the extent that the perturbation in greenhouse gas concentrations also influences cloud amount (for reasons that will become clear shortly), there is a resulting further contribution to the radiative forcing of the greenhouse gas perturbation, so that the effective radiative forcing due to the perturbation in concentration becomes
$$F_{\chi} = \left[ \left( {\frac{\partial Q}{\partial A_{\rm c}}} + {\frac{\partial E}{\partial A_{\rm c}}} \right) {\frac{\partial A_{\rm c}}{\partial \chi}} + {\frac{\partial E}{\partial \chi}}\right] \delta \chi.$$
(13)
The first two terms on the right-hand side introduce the idea of an indirect forcing of greenhouse gases that is mediated by clouds. The use of the word “indirect" signifies that the change in the TOA irradiance is not a direct consequence of the greenhouse gas concentration on the clear-sky emissivity, but rather results from the sensitivity of cloud amount to the long-wave emissivity of the atmosphere, i.e., a cloud adjustment. Given our description of the system through Eqs. (5) and (6), an indirect CO2 forcing follows as soon as one admits that cloudiness may depend on the concentration of atmospheric CO2 (Forster and Gregory 2006; Gregory and Webb 2008; Andrews et al. 2009). The idea that clouds may be sensitive to the concentration of greenhouse gases and thus may rapidly adjust in ways that change the initial forcing, actually predates the idea that clouds may depend on surface temperature and hence act as a feedback (Plass 1956). This is not just a mathematical abstraction. Because the radiative cooling at the top of stratiform cloud layers, which is important to their sustenance, is sensitive to the downwelling long-wave radiative flux, which in turn depends on the long-wave opacity of the overlying atmosphere (Caldwell and Bretherton 2009; Stevens et al. 2003) it is likely that clouds respond rapidly (adjust) to perturbations in atmospheric CO2, thereby providing at least one justification for the ansatz that $$A_{\rm c} = A_{\rm c}(\chi).$$
### 3.2 Cloud Amount
A determination of whether changes in cloudiness are causing changes in Earth’s energy flows depends on the ability to identify clouds, and cloud changes, unambiguously. This proves to be a challenge, which stems in part from the very nature of clouds. Cloud are, in a word, nebulous. A cloud, like an aerosol more generally, is a dispersion of particulate matter in an often turbulent flow. But, additionally and in contrast to clear-air aerosols, clouds are inherently ephemeral, as they contain a substantial amount of condensed (liquid or solid) water, the presence and amount of which are maintained by local supersaturation, and which can quickly dissipate by evaporation, converting a cloudy scene to a cloud-free scene, making it difficult to determine the boundaries or even the presence of a cloud. This situation leads inevitably to a certain arbitrariness in whether a cloud is present at a given location. And this arbitrariness can lead to large differences in quantities, such as cloud fraction, which are central to interpreting Earth’s energy flows (Stephens 1988).
It might be argued that a clear basis for defining a cloud is provided by Köhler theory, namely as the set of particles that exist in an environment that is supersaturated relative to the equilibrium supersaturation over the particle surface and for which the equilibrium state is unstable. However, such a definition is not very useful in practice as the theory applies at best only to liquid clouds, and deliquesced aerosol in humid environments, or evaporating hydrometers in subsaturated environments are often optically indistinguishable from clouds defined on the basis of Köhler theory, the transition region extending over distances up to kilometers or more (Koren et al. 2007; Tackett and Di Girolamo 2009; Twohy et al. 2009; Bar-Or et al. 2010).
As a consequence of such concerns, the presence of clouds is often determined based on their radiative properties. This approach is advantageous from a practical perspective, as satellite borne instruments can be built to be sensitive to such properties, and thus afford the opportunity for reproducible measurements with global coverage and high spatial and temporal resolution. But it also introduces the possibility that quantities such as cloud fraction will be determined, in part, by the characteristics of the instruments with which they are measured. The seriousness of this issue is illustrated with the help of Figs. 3, 4 and 5, which show that co-located measurements by multiple approaches yield results that can differ profoundly at a single time, in monthly averages, and in the seasonal pattern. These issues can be ameliorated by focusing on anomalies over longer time-periods, which as show in Fig. 5 are less instrument dependent. When a comparison to models is of interest, instrument simulators can also help address these issues. Even so, differences among measurement techniques make the unambiguous determination of long-term trends more difficult, and potentially more sensitive to inhomogeneities in the observational record.
Natural variability in cloudiness compounds these issues. CERES SSF1deg short-wave CRE data analyzed and presented in Fig. 6 show that the CERES instruments on the different satellites are in relatively good agreement in their quantification of monthly anomalies during their period of overlap. Given that these estimates are based on similar instruments whose measurements are processed in a similar way, albeit for different samples of the planetary cloudiness, it is not surprising that the correlation between the two time-series is markedly stronger than what one deduces with different instruments, even if they sample the same cloud field, as in Fig. 5. The agreement between the CERES measurements aboard the TERRA and AQUA satellites suggests that natural variability is the main cause of month-to-month fluctuations, which can be as large 1–2 W m−2. These fluctuations are responsible for the stated uncertainty of about 0.3 W m−2 dec−1 in the 95 % significance ranges in monthly global anomalies of Q c, an uncertainty range which is just large enough to explain a nearly 0.6 W m−2 difference in the decadal trends estimated from the two instruments.
Such a large uncertainty, associated with natural variability alone, suggests that even very high quality, dedicated measurements, such as those provided by CERES will not be able to document potentially substantial changes in global cloud radiative effects on timescales shorter than half a century. From the perspective of climate change, a change in the irradiance associated with cloud-covered scenes that is comparable to a given radiative forcing over the time-period of interest can be considered substantial. Over the period 1960–2005, CO2 increased at an average rate of 1.4 ppm yr−1 (Forster et al. 2007). A 3.7 W m−2 forcing associated with a doubling of CO2 (Forster et al. 2007) would result in a trend in the energy budget of about 0.25 W m−2 dec−1, in the absence of any feedbacks (and assuming a linear increase in the forcing with time. In this context, a change in cloud properties in response to this forcing that resulted in a further change in the radiation budget $$|\dot{Q}| > 0.05$$ W m−2 dec−1 would constitute an appreciable change. Based on this we argue that observationally constraining the response of the climate system to such a perturbation, and to some meaningful degree, requires an ability to detect changes in the radiation budget $$|\dot{Q}|$$ of 0.05 W m−2 dec−1. The analysis of Loeb et al. (2007) suggests detection of such a trend with 90 % confidence would require 50 years of data, and this analysis is optimistic as it assumes a perfect instrument and insignificant decadal variability. A similar conclusion has been reached by Dessler (2010). Moreover, as the number of years of data required to establish a trend is proportional to $$|\dot{Q}|^{2/3},$$ where $$|\dot{Q}|$$ is the magnitude of the trend (Weatherhead et al. 2000), establishing a trend even twice as large, which becomes interesting as the rate of forcing increases super linearly with time, would still require 30 years—a timeframe which, if anything is a lower bound as it does not factor in the effects of instrumental limitations, or issues related to the ambiguity of cloud identification, as discussed above.
## 4 Aerosol Radiative Effects and Forcing
The complexity of aerosols is one reason it has proven difficult to understand how human activity has contributed to a changing aerosol burden. This complexity is expressed in terms of the heterogeneous chemical and microphysical properties and also the highly variable spatial and temporal distribution of the aerosol. The heterogeneous composition, which contrasts with the well defined molecular properties of the greenhouse gases, is a consequence of the numerous contributions to atmospheric aerosols: Primary emissions from natural and anthropogenic sources and gas-to-particle conversion resulting from atmospheric reactions of precursor gases, importantly sulfur and nitrogen oxides from combustion sources, ammonia from agriculture and animal husbandry, organics from anthropogenic sources and vegetation, and numerous other sources. Gas-to-particle conversion processes lead both to new particle formation and to growth of pre-existing particles. The resulting aerosols undergo further evolution in the atmosphere through condensation and coagulation and in cloud processing. Ultimately, the aerosol particles are removed from the atmosphere, importantly by precipitation. The optical and cloud-nucleating properties of aerosols, and thus their influences on climate and climate change, are strongly dependent on the size and chemical composition of the particles comprising the aerosol. For example, growth of particles with increasing relative humidity, which greatly increases their ability to scatter visible light, is dependent on composition.
The complexity in the spatio-temporal distribution of aerosols is hinted at even upon inspection of their long-term average global distribution undifferentiated by aerosol type as seen in the large spatial variability of aerosol optical depth (AOD) at 550 nm averaged over the available 11 years of multi-angle imaging spectroRadiometer (MISR) data, Fig. 7. Major contributions arise from windblown dust (e.g., Northern Africa, Arabian peninsula, western China) and biomass burning (e.g., central Africa, Amazonia, Indonesia). Substantial contributions from human activity, mainly combustion related, can be inferred over southeast and eastern Asia and extending into the western North Pacific. The highly industrialized regions of Europe and North America (extending to the North Atlantic) also exhibit noticeable enhancement of AOD relative to pristine continental regions and major portions of the Southern Hemisphere Ocean. The spatial heterogeneity of the distribution of these aerosols is a consequence of the heterogeneous distribution of sources together with the short atmospheric residence times of these aerosols, about a week, together with the intermittent removal by precipitation. Because aerosol sources have pronounced seasonality, and because sink and transport processes of all aerosols are heavily dependent on variable meteorological conditions, the distribution of aerosols shown in Fig. 7, being a long-term average, considerably understates the complexity of the spatial distribution of atmospheric aerosols. This points out that the complexity of aerosols is manifested not only by their varied chemical and microphysical properties, but also by their heterogeneous spatial distribution; for example an aerosol particle above a bright surface has a different radiative effect compared to even the same particle over a darker surface. All of these considerations make it much more difficult to quantify aerosol forcing than is the case with the incremental greenhouse gases.
The short residence time of aerosol particles in the troposphere not only complicates characterization of their radiative influences, but also has implications on climate change that would result from future changes in emissions, especially as most of the incremental aerosol arises from emissions associated with fossil fuel combustion. If at some point in the future, emissions of CO2 from combustion are substantially reduced and if this were accompanied by reduction of associated emissions of sulfur and nitrogen oxides, major precursors of light-scattering tropospheric aerosols, the result would likely be for temperatures to initially increase because of the reduction of aerosol forcing. An initially abrupt increase in temperature following an abrupt cessation of aerosol forcing has been shown in climate model studies (e.g., Brasseur and Roeckner 2005; Matthews and Caldeira 2007).
As a consequence of all these considerations, understanding of energy flows in the Earth system and changes in these flows over the industrial era is challenged by poor understanding of the effect of aerosols on cloud-free skies where one can speak of the direct aerosol radiative forcing. This challenge is even greater with respect to aerosol effects on clouds (Lohmann and Feichter 2005; Stevens and Feingold 2009) where it has become common to speak of the indirect aerosol radiative forcing, which results from modification of the radiative influences of clouds that result from changes in the aerosol environment in which they form. The discrimination of aerosol forcing into direct and indirect components structures thinking about aerosol influences on climate and is used to structure the discussion below.
### 4.2 Aerosol Direct Forcing
Analogous to the way in which a CRE is calculated, the direct aerosol radiative effect, DARE, is the change in the irradiance at the top of the atmosphere that results from the total aerosol present. This can be contrasted with the direct aerosol radiative forcing, DARF, which is the quantity pertinent to forcing of climate change. As is the case with greenhouse gas forcing, DARF describes only the aerosol radiative effect associated with the secular (mainly anthropogenic) change in the DARE due to incremental aerosols, as that is the externally forced contribution. A negative forcing denotes a decrease in absorbed short-wave irradiance; opposite in sign to the positive greenhouse gas forcing, and, within the framework of the forcing-response paradigm, would offset some fraction of the greenhouse gas forcing.
Importantly, in determining the DARE, and how it has changed, it is necessary to account for the effects of clouds. The effect of cloud contamination of pixels used to determine aerosol optical depth as already been noted; such contamination would result in a gross overestimation of aerosol optical depth, and it is thus necessary to apply stringent cloud screening (Mishchenko et al. 1999). However, being overly stringent runs the risk that high-humidity regions will be misclassified as being cloudy, with the resultant effect that the swelling of particles at high relative humidity and the attendant increase in light-scattering cross-section, optical depth, and forcing is excluded from the measurements. These effects can be substantial; for sulfate aerosols, the scattering cross-section increases fourfold between 90 and 97 % relative humidity (Nemesure et al. 1995). Surface-based measurements are also subject to a similar concern as the technique requires a direct path to the Sun. More intrinsic to the radiative forcing issue, the direct radiative effect of light scattering by aerosols is greatly diminished in the presence of clouds, which prevents solar irradiance from reaching the aerosol if clouds are above the aerosol, or by providing a bright underlying albedo, minimizing the effect of aerosol scattering if clouds are below the aerosol. Clouds beneath an absorbing aerosol greatly increase the amount of solar absorbed irradiance relative to the cloud-free situation. Because it is difficult to retrieve aerosol amounts in the presence of clouds, it is has been common to estimate the DARE on a global basis simply by multiplying the cloud-free DARE, determined by measurement or by modeling of the amount and optical properties of the aerosol, by the cloud-free sky fraction. Such an approach has been shown in model calculations to yield a value of DARF whose magnitude is erroneously large (Bellouin et al. 2008) by a factor of two.
Increases in the amount of aerosol loading that can confidently be ascribed to anthropogenic emissions have been thought to give rise to changes in global average atmospheric radiative fluxes (aerosol forcing) that are a substantial fraction of the greenhouse gas forcing over the industrial era. As part of the fourth assessment report of the IPCC the DARF was estimated to be −0.5 ± 0.4 W m−2 (Forster et al. 2007, 90 % confidence limits). Although considerably reduced in magnitude as compared to earlier estimates, such as the value of −1.3 W m−2 initially estimated for the sulfate aerosol alone (Charlson et al. 1992), a value of DARF at the high magnitude end of the range cited by IPCC would still offset a substantial fraction of greenhouse gas warming over the industrial era (about 3 W m−2), and thereby imply a much larger climate sensitivity as inferred on the basis of Eq. (3) with F denoting the sum of greenhouse gas and aerosol forcing. To the extent that the DARF is large, advances in satellite remote sensing make it conceivable to look for a direct signal, by looking for signatures of aerosol trends in radiative fluxes measured by passive satellite borne instrumentation over the past decade of intensive Earth observations. Eleven years of MISR measurements (upper panel of Fig. 8) of AOD show large-scale shifts in specific regions. Large increases in AOD over northern India southeast and eastern Asia and around the Persian Gulf likely reflects the economic development of these regions. Over the region ranging from southwest of North America, across the Atlantic and into North Africa, the Mediterranean and central Europe, and over the maritime continent, AODs have decreased sharply, by as much as 0.1. A modest decrease is evident across the southern ocean, and there is an apparent increase over western Canada. The uncertainties associated with the retrievals of AOD from space (Kahn et al. 2010), and the susceptibility of decadal trends to the effects of inter-annual modes of variability, such as El Nino, preclude drawing confident conclusions from Fig. 8; however, the broad conclusions drawn by the figure are also supported by a more systematic analysis based on multiple platforms (Zhang and Reid 2010).
A concern over the interpretation of these measurements is that the marked changes in aerosol optical depth inferred from the MISR and multi-instrument aerosol measurements are not mirrored in trends in the outgoing short-wave irradiance in cloud-free scenes over the same time-period as measured by the highly calibrated CERES radiometer on the same platform, lower panel of Fig. 8. Although the irradiance in cloud-free circumstances shows trends in some regions where aerosol optical depth retrieved by MISR has been increasing, for instance over south Asia, particularly the middle east and southeast China, and to some extent over western Canada, the relationship between changes in AOD and changes in clear-sky short-wave radiative fluxes is not striking. The largest changes (irrespective of sign) in cloud-free-sky radiation appear over land in regions where there are no discernible trends in AOD. In almost every case statistically significant trends in CERES absorbed short-wave radiative irradiance over land are also evident in changes in land-surface properties measured over effectively the same time-period: the brightening, at TOA, of Australia; the brightening in southeastern Asia; the brightening/dimming patterns over the southwestern United States and northern Mexico as well as the brightening/dimming patterns over Argentina and South America, and the dimming of the very northern tip of Africa are all associated with consistent changes in soil moisture, whereby an increase in the outgoing short-wave irradiance for scenes identified as being cloud-free corresponds to a decrease in soil moisture and evapotranspiration (Jung et al. 2010). Notwithstanding the merit of exploring why such large apparent trends in AOD over the ocean are not seen in the CERES clear-sky short-wave irradiance, this analysis provides little support for a strong DARF.
In summary, the evolving understanding of radiative forcing, one in which the process of adjustment plays an important role; the complexity of the aerosol and its co-variability with clouds and surface features; and the role of even small amounts of absorption of aerosols over bright surfaces, suggests that back-of-the-envelope estimates of the DARF can easily be misleading. Based on this reasoning, and in consideration of more detailed calculations that suggest previous work underestimates the uncertainty in aerosol forcing (Loeb and Su 2010), it would not be surprising if the sum of the various contributions to the DARF is much closer to zero than previously thought.
### 4.3 Aerosol Indirect Forcing (Cloud Adjustments)
It has long been appreciated that clouds adjust to changes in tropospheric aerosols and this adjustment affects their albedo and precipitation development. Based on this understanding, it has been hypothesized that changes in the loading and properties of tropospheric aerosols may indirectly affect the radiative influences of clouds (i.e., alter the CRE) by modifying cloud properties and/or amount. These changes are referred to as aerosol indirect effects, or the indirect radiative forcing resulting from anthropogenic (or secular) changes to tropospheric aerosol loading and properties. They are analogous to the CO2 indirect forcing discussed in the context of Eq. (13) and, like the CO2 indirect forcing, are more usefully thought of as an adjustment to a compositional change of the atmosphere. Because such adjustments convolve spatially heterogeneous changes in the aerosol with changes in cloudiness, they tend to be complex and uncertain, and are only briefly touched upon them here.
To be sure, a wealth of observational support exists for Twomey’s hypothesis that the cloud drop number concentration increases, cloud drop radius decreases, and cloud albedo increases with increasing aerosol particle concentration. However, quantification of the net radiative forcing that can be attributable to the indirect effects of aerosols on clouds has proven elusive. That cloud drop radii are reduced and cloud albedo is increased by aerosols is clearly shown in ship tracks (e.g., Segrin et al. 2007), but the net effect of such changes are partially offset by changes in cloud water content. Regionally, reduction in cloud drop effective radius is associated with enhanced concentration of anthropogenic aerosol (e.g., Schwartz et al. 2002), but the expected increase in cloud albedo is often absent. Such lack of enhancement of cloud albedo is also likely due to a decrease in cloud liquid water path with increasing aerosol concentration, at variance with Twomey’s ansatz of other things, especially cloud water content, remaining equal. A global survey using satellite observations showed roughly equal likelihood of negative, near-zero, or positive correlation of column liquid water and column drop concentrations in liquid water clouds (Han et al. 2002). Although several studies show strong correlations between cloud amount and aerosol optical depth (e.g., Nakajima et al. 2001; Koren et al. 2010), the interpretation of such correlations is difficult, as a variety of processes (both physical and retrieval artifacts) can be expected to produce such correlations, quite independently of whether or not the aerosol is interacting with the cloud (e.g., Loeb and Schuster 2008). For example, both aerosol optical depth and cloudiness increase with humidity and, thus, it is not surprising that modeling studies might overestimate the tendency of clouds to adjust to aerosol perturbations. By regressing the logarithm of the retrieved aerosol optical depth against the logarithm of the retrieved cloud droplet concentrations over a number of geographic regions, Quaas et al. (2005) found statistically significant slopes that range from 0.1 to 0.3 depending on location, with values over the ocean three times greater than those over land and with a global mean value of just under 0.2 (see also Quaas et al. 2009). Based on this analysis, Quaas et al. (2009) estimate quite a low aerosol indirect forcing, −0.2 ± 0.1 W m−2. As noted by Quaas et al. (2005), this uncertainty is parametric, and contributions to the uncertainty from structural effects can be expected to be substantial; hence, observational estimates cannot, on their own, establish with confidence even the sign of the effect hypothesized by Twomey, despite arguments based on simple physical considerations that it is negative. The structural uncertainty that frustrates attempts to quantify the Twomey, or Twomey-like, effects also makes it more difficult to test cloud lifetime hypotheses and all the more to quantify the resultant forcing. Moreover, to the extent that precipitation processes become involved the difficulties are compounded, in no small part because of the sensitivity of aerosol amount to wet scavenging by precipitation.
In summary, although there is little doubt of the importance of aerosol-cloud interactions in influencing the amount of atmospheric aerosol, as well as cloud properties, the variety of ways in which clouds adjust to aerosol perturbations (Stevens and Feingold. 2009), many of which are not possible to account for given the relatively crude description of cloud processes in climate models, lends weight to the argument that, after a full accounting, the radiative forcing attributable to cloud adjustments to aerosol perturbations is likely to be small, at least on a global scale.
## 5 Modeling Earth’s Energy Flows
The preceding discussion demonstrates that, irrespective of the sophistication of observing systems, models are indispensable for estimates of fundamental properties of the climate system. The forcing-feedback-response framework is useful only in so far as compositional perturbations can be associated with a radiative forcing. And if it is to have any general meaning, this radiative forcing must incorporate the perturbation-dependent fast response of the system, the adjustment. Complications posed by adjustment have long been appreciated for inhomogeneous perturbations, such as those due to aerosols, and in special cases for homogeneous forcings such as those due to greenhouse gases, i.e., stratospheric adjustment. Adjustments that involve changes in cloudiness, whether due to aerosol or greenhouse gas forcing, can be large, are often model dependent, and add significant uncertainty to ultimate estimates of the response of the climate system to compositional changes.
Uncertainties in the modeling arise because models are imperfect. And, given the singular nature of the task to which climate models are to be applied, these imperfections are difficult to quantify. Independent realizations of perturbations to the energy flows in Earth-like planets, which could be used to evaluate the reliability of climate models, do not exist. So the empiricism through which the adequacy of models could be assessed does not exist, and critical tests are invariably indirect. As an example, Hall and Qu (2006) showed that in models the surface albedo feedback, which is not observable, correlates with the relationship between seasonal variations of surface temperature and surface albedo, which is observable. The implication is that models which capture the observed relationship between seasonal changes in surface temperature and surface albedo more reliably represent the surface albedo feedback. Of course, the relationship that models show between their seasonal cycle and their response to a secular perturbation might simply be an artifact of how the models are constructed, and indeed there is evidence that relationships between quantities one desires from a model, and quantities that one can measure, may say more about the models from which such relationships are derived than they do about the physical system (e.g., Klocke et al. 2011). Nonetheless, the Hall and Qu (2006) example shows how focusing on perturbations to Earth’s energy flows might lead to the development of a critical framework for assessing the reliability of models. This idea hinges on advancements in the ability to measure Earth’s energy flows, as developed further below.
One of the key energy flows illustrated by Fig. 1, is that of short-wave radiation, which powers the climate system. The near constancy of the total solar irradiance is thought to be a prerequisite for the development of life, but this would come to nought were it not for a commensurate constancy in the planetary albedo. The constancy of the planetary albedo is evident not only on globally and annually averaged scales, but also within latitude zones. This point is illustrated by the latitudinal dependence of zonal averages from a decade of CERES measurements of the reflected short-wave radiation at the TOA, Fig. 9. Also shown are the range of annual averages of the reflected short-wave radiation, evaluated as
$$\left\langle Q^{\uparrow}_n \right\rangle = {\frac{1}{2}} \int Q^{\uparrow}_n \cos(\varphi) {\rm d}\varphi$$
(14)
where $$Q^{\uparrow}_n$$ denotes the reflected short-wave radiation as a function of latitude $$\varphi$$ and year n; and the values averaged over the northern and southern hemisphere separately. The surprising feature of Earth’s climate system that is revealed in these measurements is the small inter-annual variability (the range in the yearly averages is 1.16 W m−2 and the standard deviation is 0.36 W m−2), despite the zonal average spanning more than 40 W m−2, with a root-mean squared variability (weighted by area) of nearly 9 W m−2. Also the difference between the two hemispherically averaged values is very small, only 0.35 W m−2.
Because the globally and annually averaged albedo can be, and is, readily tuned by adjusting global parameters in models, the fidelity with which it is represented is not a critical test of models. However, the ability of models to represent natural fluctuations in the energy flows about the parameter constrained global values begins to provide a much more critical test. The simplest example of such a fluctuation is that embodied by the climatology of zonally averaged anomalies, Fig. 10. These zonal patterns help to regulate the meridional heat transport and are not directly specified through the adjustment of global parameters. Broadly speaking, climate models, here represented by a subjective selection of five climate models, are skillful in representing zonally averaged anomalies; correlations between the observed and simulated latitudinal anomaly range between 0.60 and 0.94 for the models shown. However, the departures from the observations, which are due mainly to treatment of clouds in the models, are substantial; particularly when viewed in the context of radiative forcing over the industrial era. The root-mean square error in the zonal residuals ranges from 4 to 8 W m−2, comparable to the departure of the zonal mean from the global mean and considerably greater than a short-wave cloud feedback that would be important in the context of understanding climate feedbacks, 1–2 W m−2. The ways in which the models are wrong also varies, although some patterns emerge. In a comparison of zonal monthly mean albedo calculated with twenty climate models and ERBE observations over 1985–1990 Bender et al. (2006) found substantial positive and negative departures, not infrequently as great as 0.1, that were reproducible from year-to-year in a given model but differed substantially from model to model in space and time. Trenberth and Fasullo (2010a) show that most models reflect too little solar radiation in the southern storm tracks and misrepresent in one fashion or another the structure of the tropical convergence zones (cf., Lin. 2007). Most models reasonably represent the poleward increase of reflected short-wave radiation in the mid-latitudes of the northern hemisphere, presumably because part of this is carried by the influence of specified surface features such as the Saharan desert and Tibetan Plateau. The models also tend to accurately represent the remarkable constancy in the globally averaged values, and even the year-to-year variability within latitude bands, but almost all fail to properly capture the near equality of the hemispherically averaged values.
Given a globally and annually averaged flow of energy into the Earth system, models would ideally also produce a model state that is consistent with what is observed. In this respect, there is also room for improvement. The pre-industrial control climate can differ substantially among models, even if the flow of energy into the system is prescribed, or tuned to match the best estimate of the observations. This is evident in Fig. 11, which shows simulated global mean temperature over the twentieth century taken from all of the relevant simulations, 58 in total, in the CMIP3 archive. Simulated temperatures at the end of the twentieth century exhibit a range of nearly 3 °C (from 12.8 to 15.5 °C). Most models are biased cold, despite being forced with a total solar irradiance that is now thought to be too large. The multi-model mean temperature is more than 0.5 °C lower than measured, an offset that is comparable to the temperature change observed over the twentieth century. From a certain perspective, the agreement is excellent; errors in temperature of 1 K out of 288 K corresponds to an error of only 0.35 %, albeit somewhat larger (5.5 W m−2 or nearly 1.4 % when translated into an energy flux). However, even such a small temperature error can alter the modeled climate in ways that are as great as the climate change that has occurred over the twentieth century or are projected for the twenty-first century. Such an error would seem to have implications for model projections of climate change; so it is surprising that, despite these differences the models, individually and collectively, still represent the trend in twentieth century temperatures as accurately as they do. This surprise is tempered by a realization that the agreement in the twentieth century temperature trend may also be a reflection of the model development process and the considerable latitude that uncertainty in the aerosol forcing gives model developers in matching the observed temperature trend (Kiehl 2007).
Although tests like the ones just discussed are neither exhaustive, nor absolute, it seems reasonable to assume that the ability of a model to represent perturbations to Earth’s energy flows that arise from natural forcings should be indicative of their ability to represent perturbations stemming from human sources (cf., Lucarini and Ragone 2011). Viewed comprehensively, an improvement in an ability to model these natural perturbations in Earth’s energy flows, could reasonably be associated with more reliable projections of perturbations in Earth’s energy flows stemming from prospective future changes in atmospheric composition. Viewed over many model generations, substantial improvements in these respects, should correlate with a reduction of model spread in the representation of perturbations to Earth’s energy flows that are otherwise not observable. To test the conjecture that models which better represent natural perturbations in Earth’s energy flows (for instance associated with seasons, decadal variations in solar forcings, volcanos, or different surface boundary conditions) are also better at representing forced perturbations requires both sustaining and advancing observations of Earth’s energy flows, but also standardized experiments within evolving intercomparison protocols, such as the coupled model intercomparison project (CMIP).
## 6 Concluding Remarks
Tremendous advances in observing systems now make it possible to track energy flows through the Earth system with high precision, and on temporal and spatial scales that would have been difficult to imagine just a few decades ago. More accurate measurements of the total solar irradiance and the increase in ocean enthalpy are helping to constrain the balance between net solar irradiance and outgoing long-wave irradiance to less than ±2 W m−2 at the top of the atmosphere. Precise measurements of Earth’s energy flows, as part of NASA’ s CERES program are providing new insights into how the energy flows at the top of the atmosphere vary spatially on timescales ranging from days to now more than a decade. And although, with the exception of the planetary albedo, which the satellite record has convincingly shown to be much smaller than pre-satellite estimates, the several terms in Earth’s energy budget do not appear to have changed markedly as a result of the satellite record, the confidence in this budget has been greatly enhanced by the accuracy and precision of the satellite measurements. Measurements have advanced to the point where estimates of long-term trends in the top-of-atmosphere energy budget are now limited principally by natural variability, rather than instrument precision. Active remote sensing, from both surface- and space-based platforms is helping to constrain the surface energy budget. And seasonally resolved climatologies of aerosol optical depth are available both from surface- and space-based networks.
These measurement advances are not without their attendant controversies, nor do they yet provide a complete picture of Earth’s energy flows. In particular, satellite-based estimates of precipitation remain difficult to reconcile with estimates of downwelling surface irradiance in the long-wave, given uncertainties in other terms in the energy budget. Also, the absolute uncertainty of global climatologies of energy related quantities remains comparable to, or larger than, perturbations expected from changes in atmospheric composition resulting from human activities, precluding measurement-based quantification of changes in flux terms resulting from the anthropogenic perturbation. Furthermore, distinguishing aerosols from clouds, or complex surfaces, and quantifying aerosol forcings remain challenging, even for a well characterized aerosol perturbation. These challenges limit confidence in estimates of total secular forcing. Nonetheless, the comprehensiveness of the measurements is making it possible to track energy flows through the Earth system on space and time scales that are unprecedented, thereby offering the possibility to pose critical tests of both models and understanding of how energy flows through the Earth system.
Climate change, as quantified, for instance, by changes to the globally averaged surface temperature, can be expected to result from compositional changes to Earth’s atmosphere which perturb the flow of energy through the Earth system. The forcing-response-feedback framework, which has developed to understand such changes, posits that compositional changes in the Earth system can be associated with an effective radiative forcing, so that distinct perturbations that result in quantitatively the same effective radiative forcing will elicit the same response, as measured by globally averaged surface temperature. The magnitude of the response to a sustained forcing (per unit forcing), the climate sensitivity, is thought to be an intrinsic property of the Earth system. Notwithstanding objections that climate change can be manifest in changes that do not scale with changes in the globally averaged surface temperature, this framework relies on an ability to determine this effective radiative forcing, given some externally imposed change to the system. However, because this effective forcing depends on the fast response (or adjustments) of the system, to a given perturbation, it cannot be determined from first principles. And so climate models, which previously were thought to be indispensable to an assessment of the climate sensitivity, have become equally indispensable to the estimation of the effective radiative forcing. Consequently those poorly modeled elements of the climate system that affect the flow of energy through the system, for instance clouds and convection, have become the principal limit in quantification of not only the climate sensitivity, but also the effective forcing – especially for complex perturbations such as those associated with aerosols.
Although climate models have become indispensable, they also lack critical tests. Independent realizations of perturbations to Earth-like planets do not exist and, hence, model-based estimates of the effective radiative forcing that accompanies a compositional perturbation, or the climate sensitivity that emerges as a result of feedbacks in the Earth system, can not be independently assessed. However, by focusing on the ability of models to represent the ways in which energy is observed to flow through the Earth system, particularly for repeatable events such as seasons, ENSOs, solar cycles, and perhaps volcanic perturbations, and as a function of surface state or latitude, it might be possible to develop a measure of reliability of climate models, somewhat analogous to what is done for weather forecasting. So doing would complement the heuristic use of models that currently proves essential to development of understanding. However, this will required sustained and expanded efforts to measure the flows of energy through the Earth system and international programs dedicated to documenting the evolving skill of models to represent these flows.
Footnotes
1
Unless otherwise noted the radiative forcing estimates we cite are for the radiative perturbation that arises at the TOA after the stratosphere has adjusted to the presence of compositional change, but with the tropospheric temperatures held constant.
2
Other sources of energy—geothermal, combustion of fossil fuel and nuclear production—are yet an order of magnitude smaller (P. Pilewskie, ISSI Workshop on Observing and modeling Earth’s energy flows, Bern, 10–14 Jan., 2011).
3
The analysis by Trenberth and Fasullo (2010b) suggested that the radiative imbalance at the top of the atmosphere measured by CERES has been increasing by as much as 1 W m−2 dec −1. Because of the limited heat capacity of the atmosphere such an imbalance would imply a large change in ocean enthalpy and/or surface ice amount, neither of which is observed. However, Trenberth and Fasullo (2010b) made use of preliminary CERES data for the time-period 2005–2010, over which the striking trend in radiative imbalance was noted, and failed to account for uncertainty in their estimated trend.
4
By conservation of energy, and as other sources of energy in the Earth system are negligible (P. Pilewskie ISSI Workshop on Observing and modeling Earth’s energy flows, Bern, 10–14 Jan., 2011) this time-rate of change in Earth’s enthalpy is equal to the net flux at the top of the atmosphere, N. Because the atmospheric enthalpy changes so little, a consequence of the low heat capacity of the atmosphere, N at the top of the atmosphere is usually taken as identical with N at the surface. The enthalpy notation is adopted because it emphasizes what is being measured, namely the change in the heat content, or enthalpy of the Earth system.
5
It might appear that there are exceptions. Intrieri et al. (2002) and Mauritsen et al (2011) calculate the clear-sky radiative fluxes using measurements of the atmosphere in cloudy conditions, but do not include the clouds in the radiative transfer calculations. However, this is not an actual measurement of radiative effects, but rather a calculation of radiative effects given measured atmospheric properties.
## Acknowledgments
We thank the organizers of the workshop for giving us the opportunity to present our views. In addition through the development of this manuscript exchanges with Andrea Brose, Robert Cahalan, Robert Cess, Ralph Kahn, Seiji Kato, Norman Loeb, John Lyman, Jochem Marotzke, David Randall, and Graeme Stephens, were a great resource in the development of our ideas and analysis methods. MISR and CERES data were obtained from the NASA Langley Research Center Atmospheric Sciences Data Center. We thank Wei Wu (BNL) for providing cloud cover data for north Central Oklahoma as well as Stefan Kinne and Thorsten Mauritsen for critical and substantive comments on an initial draft version of this manuscript. Several critical reviews of the manuscript, two of which were anonymous, also resulted in major changes to the manuscript that we believe improved the presentation of our ideas, and these reviewers are thanked for their frankness, and their attention to our ideas. We acknowledge the modeling groups, the Program for Climate Model Diagnosis and Intercomparison (PCMDI) and the WCRP’s Working Group on Coupled Modeling (WGCM) for their roles in making available the WCRP CMIP3 multi-model dataset. Support of this dataset is provided by the Office of Science, U.S. Department of Energy. Work by S. Schwartz was supported by the United States Department of Energy (Office of Science, OBER) under Contract No. DE-AC02-09CH10886. B. Stevens was supported by the Max Planck Society for the Advancement of Science.
## Appendix: A Energy Budget Justification
This appendix provides a brief, term-by-term, justification, based on observations and simulations with the ECHAM6 climate model, for energy flows at the top of the atmosphere and at the surface. The best estimate for each term is provided in Table 1, and the uncertainty is provided as a “likely” range (which we interpret as the central 68 %, or two-sigma, of the distribution) in Fig. 1. Estimates were constrained by measurements of the increase in ocean enthalpy, from which the energy flux imbalance at the TOA and surface, both taken as about 1 W m−2, can be inferred. ECHAM6 simulations were performed using prescribed sea-surface temperatures for the period 1977–1986 with T255 spectral resolution, or roughly 0.5°; these simulations benefitted from modifications to ECHAM6, relative to ECHAM5, designed to include more advanced treatments of clear-sky radiative transfer and surface albedo, as well as tuning that better reflects present understanding of the radiative imbalance at TOA. The syntheses of the energy budgets presented here were greatly informed by those of Trenberth et al (2009) and Stephens et al. (2012), denoted below as T-2009 and S-2012 respectively.
Kopp and Lean (2011) estimate a total solar irradiance of 1360.8 ± 0.5 W m−2 at the last solar minimum. This estimate gains credibility because of laboratory measurements that show that the measurement geometry of earlier generations of satellites that had reported higher values of solar irradiance were biased high as a result of scattered light. Accounting for the total solar maximum being roughly 1.6 W m−2 greater (Kopp and Lean 2011) yields an average over the solar cycle of 1361.6 W m−2. For the average ratio of Earth’s surface area to its projected area, 4.0034, the average TOA irradiance is 340.1 ± 0.1 W m−2 (Table 2).
Table 2
Model-based estimates of non-radiative heat fluxes as constrained by observations for the reanalysis products, and sea-surface temperature and top-of-atmosphere balance in the case of ECHAM6
Source
NRA1
NRA2
JRA1
JRA2
ERA
ERA-interim
ECHAM6 (1980–1990)
Latent heat flux
80.2
83.1
85.1
90.2
82.3
81.6
87.2
Sensible heat flux
15.3
15.6
18.8
19.4
15.3
17.7
19.0
NRA, JRA and ERA refer to the reanalysis products of the National (US) Center for Environmental Prediction, the Japanese Meteorological Agency, and the European Center Reanalysis Products, ERA40 and ERA-Interim. The ERA-Interim data are for the period from January 1989 to December 2007. The different eras, e.g., NRA1 and NRA2 refer to estimates over the ERBE epoch (lasting from February 1985 to March 1989) versus the CERES period which T-2009 define as lasting from March 2000 to April 2004 (cf., Trenberth et al. 2009)
Based largely on the CERES data S-2012 estimate this term to be 97.7 ± 2 W m−2. We maintain their estimate of the uncertainty at ±2 W m−2, in recognition of constraints posed by estimates of the rate of increase in ocean enthalpy and associated uncertainty in the outgoing long-wave irradiance at TOA. However, we center our estimate at a slightly greater value (99 W m−2) and give a best estimate of 100 W m−2, in the light of the estimates of T-2009 and previous assessments of the ERB data which gave much higher values of reflected short-wave irradiance at TOA.
Both S-2012 and T-2009 estimate a value of 239 W m−2, which we adopt as a central value. S-2012 estimate the uncertainty in measurements of this term to be ±3 W m−2. We adopt a somewhat smaller estimate of the uncertainty, as we believe this better represents the constraint posed by measurements of the other terms at TOA, together with the rate of change of ocean enthalpy.
In the absence of notable advancements in measurements of this quantity we let the values cited by T-2009, 161 W m−2, and S-2012, 168 W m−2, span the range of our estimate. The resultant range encompasses nearly all past estimates, as well as global climate modeling results using ECHAM6. Larger values, as suggested by S-2012 were not adopted because we could not reconcile them with other measurements and the need to maintain surface energy balance. For reference, Stackhouse et al. (2011) and Kato et al (2012) estimate this term to be 167 and 169 W m−2, respectively: these values are at the upper end, or even just beyond, the range of values we present as most likely.
In estimating this term, we discount the much lower estimates by T-2009 and adopt values closer to the central estimates of S-2012, which largely summarizes results from earlier studies by Kato et al (2012) and Stackhouse et al. (2011). However, because the central estimate of S-2012 is at the upper end of the range of earlier studies upon which it is based, we do not extend our “likely” range to include values at the upper end of their uncertainty range.
The two-sigma range presented in Fig. 1 is slightly larger than the range spanned by the estimates of S-2012 and T-2009. Our ECHAM6 estimates, constrained by observed sea-surface temperatures yield a value of 397.7 W m−2. Although one would expect this quantity to be well constrained by surface temperature, uncertainties in the surface emissivity, as reviewed by T-2009 and references therein, contribute to uncertainty. We note that if the upwelling surface infrared flux is evaluated according to the Stefan-Boltzmann radiation law with the emissivity of the surface taken as unity. Under this assumption, the upwelling thermal infrared flux given in Fig. 1 as 395–399 W m−2 corresponds to Ts of 288.9–289.6 K. Likewise the downwelling flux from the atmosphere to the surface is evaluated under the assumption of unity absorptivity. A departure from unity in emissivity and absorptivity would result in substantial reduction in both fluxes. Measurement-based emissivities in the thermal infrared range from 0.89 to 0.99 for terrestrial surfaces (Sutherland 1986); and 0.96–0.99 for the ocean surface (Hanafin and Minnett 2005). Other things being equal, for an upwelling flux of 400 W m−2, a decrease in emissivity by 0.1 would result in a reduction in flux of 40 W m−2. However, as a reduction in emissivity is accompanied by equal reduction in absorptivity, reduction in both fluxes from those calculated for unity emissivity and absorptivity would be locally compensated to great extent. Zhou et al. (2002) present calculations with a radiation transfer model coupled to a land-surface model that shows that a reduction in emissivity/absorptivity over terrestrial surfaces results in an increase in surface temperature and near-surface air temperature; a decrease in the emissivity by 0.1 was found to result in a decrease in net and upward long-wave radiation by about 6.6 and 8.1 W m−2, respectively. The decrease in radiative emission was found to be further offset by an increase in sensible heat flux. In view of these considerations, it would seem that the uncertainty associated with the rates of bidirectional radiative transfer at the surface may be somewhat greater than has previously been appreciated; however, the net effect is likely smaller than might be assumed based on emissivity considerations alone because of compensating effects. In addition to considerations of emissivity/absorptivity, insufficient resolution of the spatio-temporal variability tends to bias estimates low. For model-based estimates the latter are somewhat ameliorated by the ECHAM6 resolution in which radiative fluxes are calculated at 20 min intervals on a T255 grid; however, the ECHAM6 calculations are based on a constant surface emissivity of 0.996 which is likely too large; hence we take the ECHAM6 estimates as an upper bound.
### Latent Heat Flux
Our “likely” range of 78–88 W m−2 attempts to bridge the gap between estimates derived from the climatology of precipitation versus those inferred from a consideration of the surface energy budget. The GEWEX radiation panel specifies a best estimate of globally and annually averaged precipitation of 2.61 ± 0.233 mm d−1 (Gruber and Levizzani 2008). Assuming an effective evaporation/precipitation temperature of 23 °C yields a conversion factor of 28.25 W m−2(mm d−1)−1 so the global precipitation estimate become 73.7 ± 6.6 W m−2 . Assuming a density of water of 1 g cm−3 and an enthalpy of evaporation of 2.501 kJ kg−1, which would correspond to an effective evaporation/precipitation temperature of 4 °C, leads to a conversion factor of 28.83 W m−2 (mm d−1)−1, which is about 2 % or 1.5 W m−2 larger than the value at 23 °C. Kiehl and Trenberth (1997) appear to use this larger factor, whereas T-2009 assume a conversion factor of 80.0/2.76 = 29.0 W m−2 (mm d−1)−1, which is yet larger. T-2009 cite a value of 2.63 mm d−1 for the global and annual averaged precipitation rate from the global precipitation climatology project; thus their best estimate (assuming an effective precipitation/evaporation temperature of 23 °C) is nearly 8 %, not 5 %, larger than that measured by GPCP. These calculations suggest that because the precipitation does not fall at the same temperature, or in the same phase, as it evaporated, differences in the contribution of precipitation, versus evaporation/sublimation, to the surface energy budget may be as large as 1 W m−2; this difference is not accounted for in Fig. 1. S-2012 estimate a yet larger, 88 ± 10 W m−2, value for globally and annually averaged precipitation, justified in part by a consideration of larger estimates of downwelling long-wave irradiance at the surface and the constraints of surface energy imbalance. Our estimated likely range overlaps with the upper part of the likely range given by Gruber and Levizzani (2008) and incorporates the central value of S-2012, but gives less weight to the upper range of estimates by S-2012. Values larger than 88 W m−2 are necessary only if values in the upper end of the range for the downwelling irradiance estimated by S-2012 are adopted but would require a bias in the precipitation climatologies more than twice as large as the stated (1-σ) uncertainty.
### Surface Sensible Heat Flux
Our “likely” range of 17–21 W m−2 attempts to bridge the gap between the estimate by T-2009 and the much larger central value estimated by S-2012. We adopt somewhat smaller values than estimated by the latter investigators because the larger values are difficult to reconcile with the reanalysis of meteorological observations and because high-resolution climate models constrained by sea-surface temperatures robustly give estimates toward the lower end of our stated range. Here we note that estimates of surface fluxes from the ERA-Interim reanalysis are nearly 2 W m−2 larger than those by the ERA40. The ERA40 reanalysis helped motivate the lower estimate of surface sensible heat flux by T-2009, but the physics of the model used for the ERA40 is generally assumed to be inferior to what is used in the later (ERA-Interim) reanalysis product. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8945810198783875, "perplexity": 1287.7668537488607}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049276780.5/warc/CC-MAIN-20160524002116-00205-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://www.cut-the-knot.org/m/Geometry/BottemasPointSibling.shtml | # Bottema's Point Sibling
### Solution
1. We prove that triangles $BCB_c\,$ and $A_cCA\,$ are equal. (Thus $AA_c=BB_c,$ as a consequence.) Indeed, $CB_c=CA,\,$ $CA_c=CB,\,$ and $\angle BCB_c=90^{\circ}+\angle ACB=\angle ACA_c.\,$ One is obtained from the other with a rotation through $90^{\circ}\,$ around point $C,\,$ making corresponding sides perpendicular. In particular, $AA_c\perp BB_c.$ Let $N\,$ be the intersection of the two.
2. Let $M\,$ be Bottema's point. Among the many properties it has, $M\,$ lies on the circle with diameter $AB.\,$ Since $\angle ANB=90^{\circ},\,$ $N\,$ lies on the same circle. $M\,$ is known to divide the arc $\overset{\frown}{AB}\,$ to which it belongs in half. This means that the arc $overset{\frown}{BM}\,$ is one quarter of the circle, making inscribed $\angle BNM=45{\circ}.$
$N\,$ also belongs to the circle $(CBA_bA_c)\,$ because $\angle BNA_c=90^{\circ}\,$ and $BA_c\,$ is a diameter of that circle. Inscribed $\angle BNA_b=45^{\circ}\,$ for it too is subtended by a quarter of a circle. Thus $M\,$ lies on $NA_b.$
Similarly, $N\in (ACB_cB_a)\,$ and, for that reason, $\angle ANB_a=45^{\circ},\,$ implying that A_bNB_a is a straight line. Thus the three lines $AA_c,BB_c,A_bB_a\,$ are indeed concurrent and
3. $A_bB_a\,$ is the bisector of angles $ANB_c\,$ and $BNA_c,$
4. Inscribed $angle CNA_b=90^{\circ},\,$ as inscribed and subtended by the diameter $CA_b.\,$ If $M'\,$ is the second intersection of $CN\,$ with $(ANB),\,$ $\angle MNM'=90^{\circ},\,$ making $MM'\,$ a diameter of $(ANB).$
### Acknowledgment
The problem has been tweeted by Antonio Gutierrez and introduced with a GeoGebra applet by Tim Brzezinski.
The configuration is a part of Vecten's construction involving squares on each side of a triangle that is known as Bride's chair. Here we relate it to the famous Bottema's problem. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9769241213798523, "perplexity": 280.4516571692855}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488528979.69/warc/CC-MAIN-20210623011557-20210623041557-00033.warc.gz"} |
http://psychclassics.yorku.ca/Fisher/Methods/chap5.htm | # Classics in the History of Psychology
An internet resource developed by
Christopher D. Green
York University, Toronto, Ontario
ISSN 1492-3173
STATISTICAL METHODS FOR RESEARCH WORKERS
By Ronald A. Fisher (1925)
Posted April 2000
V
TESTS OF SIGNIFICANCE OF MEANS, DIFFERENCES OF MEANS, AND REGRESSION COEFFICIENTS
23. The Standard Error of the Mean
The fundamental proposition upon which the statistical treatment of mean values is based is that -- If a quantity be normally distributed with standard deviation s, then the mean of a random sample of n such quantities is normally distributed with standard deviation s/[sqrt]n.
The utility of this proposition is somewhat increased by the fact that even if the original distribution were not exactly normal, that of the mean usually tends to normality, as the size of the sample is increased; the method is therefore applied widely and legitimately to cases in which we have not sufficient evidence to assert that the original distribution was normal, but in which we have reason to think that it does not belong to the exceptional class of distributions for which the distribution of the mean does not tend to normality.
If, therefore, we know the standard deviation of a population, we can calculate the standard deviation of [p. 102] the mean of a random sample of any size, and so test whether or not it differs significantly from any fixed value. If the difference is many times greater than the standard error, it is certainly significant, and it is a convenient convention to take twice the standard error as the limit of significance ; this is roughly equivalent to the corresponding limit P=.05, already used for the c2 distribution. The deviations in the normal distribution corresponding to a number of values of P are given in the lowest line of the table of t at the end of this chapter (p. 137)· More detailed information has been given in Table I.
Ex. 16. Significance of mean of a large sample. -- We may consider from this point of view Weldon's die-casting experiment (Ex. 5, p. 66). The variable quantity is the number of dice scoring "5" or "6" in a throw of 12 dice. In the experiment this number varies from zero to eleven, with an observed mean of 4.0524; the expected mean, on the hypothesis that the dice were true, is 4, so that the deviation observed is .0524· If now we estimate the variance of the whole sample of 26,306 values as explained on p. 50, but without using Sheppard's correction (for the data are not grouped), we find
s2 = 2.69825,
whence s2/n = .0001025,
and s/[sqrt]n = .01013.
The standard error of the mean is therefore about .01, and the observed deviation is nearly 5.2 times as great; thus by a slightly different path we arrive [p. 103] at the same conclusion as that of p. 68. The difference between the two methods is that our treatment of the mean does not depend upon the hypothesis that the distribution is of the binomial form, but on the other hand we do assume the correctness of the value of s derived from the observations. This assumption breaks down for small samples, and the principal purpose of this chapter is to show how accurate allowance can be made in these tests of significance for the errors in our estimates of the standard deviation.
To return to the cruder theory, we may often, as in the above example, wish to compare the observed mean with the value appropriate to a hypothesis which we wish to test; but equally or more often we wish to compare two experimental values and to test their agreement. In such cases we require the standard error of the difference between two quantities whose standard errors are known; to find this we make use of the proposition that the variance of the difference of two independent variates is equal to the sum of their variances. Thus, if the standard deviations are s1, s2, the variances are s12, s22; consequently the variance of the difference is s12+s22, and the standard error of the difference is [sqrt]s12+s22.
Ex. 17· Standard error of difference of means from large samples. -- In Table 2 is given the distribution in stature of a group of men, and also of a group of women; the means are 68.64 and 63.85 inches, giving a difference of 4.79 inches. The variance obtained for the men was 7.2964 square inches; this is the value obtained by dividing the sum of the squares of [p. 104] the deviations by 1164 ; if we had divided by 1163, to make the method comparable to that appropriate to small samples, we should have found 7.3027. Dividing this by 1164, we find the variance of the mean is .006274. Similarly the variance for the women is .63125, which divided by 1456 gives the variance of the mean of the women as .004335. To find the variance of the difference between the means, we must add together these two contributions, and find in all .010609; the standard error of the difference between the means is therefore .1030 inches. The sex difference in stature may therefore be expressed as
4.79 [plus or minus] .103 inches.
It is manifest that this difference is significant, the value found being over 46 times its standard error. In this case we can not only assert a significant difference, but place its value with some confidence at between 4½ and 5 inches. It should be noted that we have treated the two samples as independent, as though they had been given by different authorities; as a matter of fact, in many cases brothers and sisters appeared in the two groups; since brothers and sisters tend to be alike in stature, we have overestimated the probable error of our estimate of the sex difference. Whenever possible, advantage should be taken of such facts in designing experiments. In the common phrase, sisters provide a better "control" for their brothers than do unrelated women. The sex difference could therefore be more accurately estimated from the comparison of each brother with his own sister. In [p. 105] the following example (Pearson and Lee's data), taken from a correlation table of stature of brothers and sisters, the material is nearly of this form; it differs from it in that in some instances the same individual has been compared with more than one sister, or brother.
Ex. I8. Standard error of mean of differences. -- The following table gives the distribution of the excess in stature of a brother over his sister in 1401 pairs
Treating this distribution as before we obtain: mean=4.895, variance=6.4074, variance of mean=.004573, standard error of mean =.0676 ; showing that we may estimate the mean sex difference as 4¾ to to 5 inches.
In the above examples, which are typical of the use of the standard error applied to mean values, we have assumed that the variance of the population is known with exactitude. It was pointed out by "Student" in 1908, that with small samples, such as are of necessity usual in field and laboratory experiments, the variance of the population can only be roughly estimated from the sample, and that the errors of estimation seriously affect the use of the standard error. [p. 106]
If x (for example the mean of a sample) is a value with normal distribution and s is its true standard error, then the probability that x/s exceeds any specified value may be obtained from the appropriate table of the normal distribution; but if we do not know s, but in its place have s, an estimate of the value of s,the distribution required will be that of x/s, and this is not normal. The true value has been divided by a factor, s/s, which introduces an error. We have seen in the last chapter that the distribution in random samples of s2/s2 is that of c2/n, when n is equal to the number of degrees of freedom, of the group (or groups) of which s2 is the mean square deviation. Consequently the distribution of s/s calculable, and if its variation is completely independent of that of x/s (as in the cases to which this method is applicable), then the true distribution of x/s can be calculated, and accurate allowance made for its departure from normality. The only modification required in these cases depends solely on the number n, representing the number of degrees of freedom available for the estimation of s. The necessary distributions were given by "Student" in 1908; fuller tables have since been given by the same author, and at the end of this chapter (p. 137) we give the distributions in a similar form to that used for our table of c2.
24. The Significance of the Mean of a Unique Sample
If x1, x2, ..., xn, is a sample of n' values of a variate, x, and if this sample constitutes the whole of [p. 107] the information available on the point in question, then we may test whether the mean of x differs significantly from zero, by calculating the statistics
The distribution of t for random samples of a normal population distributed about zero as mean, is given in the table of t for each value of n. The successive columns show, for each value of n, the values of t for which P, the probability of falling outside the range [plus or minus]t, takes the values .9,...,.01, at the head of the columns. Thus the last column shows that, when n=10, just I per cent of such random samples will give values of t exceeding +3.169, or less than -3.169. If it is proposed to consider the chance of exceeding the given values of t, in a positive (or negative) direction only, then the values of P should be halved. It will be seen from the table that for any degree of certainty we require higher values of t, the smaller the value of n. The bottom line of the table, corresponding to infinite values of n, gives the values of a normally distributed variate, in terms of its standard deviation, for the same values of P.
Ex. 19. Significance of mean of a small sample. -- The following figures (Cushny and Peebles' data) [p. 108] which I quote from Student's paper show the result of an experiment with ten patients, on the effect of the optical isomers of hyoscyamine hydrobromide in producing sleep.
The last column gives a controlled comparison of the efficacy of the two drugs as soporifics, for the same patients were used to test each; from the series of differences we find
For n=9, only one value in a hundred will exceed 3250 by chance, so that the difference between the results is clearly significant. By the methods of the [p. 109] previous chapters we should, in this case, have been led to the same conclusion with almost equal certainty; for if the two drugs had been equally effective, positive and negative signs would occur in the last column with equal frequency. Of the 9 values other than zero, however, all are positive, and it appears from the binomial distribution,
(½+½)9,
that all will be of the same sign, by chance, only twice in 512 trials. The method of the present chapter differs from that in taking account of the actual values and not merely of their signs, and is consequently the more reliable method when the actual values are available.
To test whether two samples belong to the same population, or differ significantly in their means. If x'1, x'2,…,x'n1+1, and If x1, x2,…,xn2+1 be two samples, the significance of the difference between their means may be tested by calculating the following statistics.
The means are calculated as usual; the standard [p. 110] deviation is estimated by pooling the sums of squares from the two samples and dividing by the total number of the degrees of freedom contributed by them; if a were the true standard deviation, the variance of the first mean would be s2/(n1+1), of the second mean s2/(n2+1), and therefore of the difference s2{1/(n1+1)+1/(n2+1)}; t is therefore found by dividing x[bar]-x'[bar] by its standard error as estimated, and the error of the estimation is allowed for by entering the table with n equal to the number of degrees of freedom available for estimating s; that is n=n1+n2. It is thus possible to extend Student's treatment of the error of a -an to the comparison of the means of two samples.
Ex. 20. Significance of difference of means of small samples. -- Let us suppose that the above figures (Table 27) had been obtained using different patients for the two drugs; the experiment would have been less well controlled, and we should expect to obtain less certain results from the same number of observations, for it is a priori probable, and the above figures suggest, that personal variations in response to the drugs will be to some extent correlated.
Taking, then, the figures to represent two different sets of patients, we have
The value of P is, therefore, between .1 and .05, and [p. 111] cannot be regarded as significant. This example shows clearly the value of design in small scale experiments, and that the efficacy of such design is capable of statistical measurement.
The use of Student's distribution enables us to appreciate the value of observing a sufficient number of parallel cases; their value lies, not only in the fact that the probable error of a mean decreases inversely as the square root of the number of parallels, but in the fact that the accuracy of our estimate of the probable error increases simultaneously. The need for duplicate experiments is sufficiently widely realised; it is not so widely understood that in some cases, when it is desired to place a high degree of confidence (say P =.01) on the results, triplicate experiments will enable us to detect with confidence differences as small as one-seventh of those which, with a duplicate experiment, would justify the same degree of confidence.
The confidence to be placed in a result depends not only on the actual value of the mean value obtained, but equally on the agreement between parallel experiments. Thus, if in an agricultural experiment a first trial shows an apparent advantage of 8 tons to the acre, and a duplicate experiment shows an advantage of 9 tons, we have n=1, t=17, and the results would justify some confidence that a real effect had been observed; but if the second experiment had shown an apparent advantage of 18 tons, although the mean is now higher, we should place not more but less confidence in the conclusion that the treatment was [p. 112] beneficial, for t has fallen to 2.6, a value which for n=1 is often exceeded by chance. The apparent paradox may be explained by pointing out that the difference of 10 tons between the experiments indicates the existence of uncontrolled circumstances so influential that in both cases the apparent benefit may be due to chance, whereas in the former case the relatively close agreement of the results suggests that the uncontrolled factors are not so very influential. Much of the advantage of further replication lies in the fact that with duplicates our estimate of the importance of the uncontrolled factors is so extremely hazardous.
In cases in which each observation of one series corresponds in some respects to a particular observation of the second series, it is always legitimate to take the differences and test them as in Ex. 18, 19 as a single sample; but it is not always desirable to do so. A more precise comparison is obtainable by this method only if the corresponding values of the two series are positively correlated, and only if they are correlated to a sufficient extent to counterbalance the loss of precision due to basing our estimate of variance upon fewer degrees of freedom. An example will make this plain.
Ex. 21. Significance of change in bacterial numbers. -- The following table shows the mean number of bacterial colonies per plate obtained by four slightly different methods from soil samples taken at 4 P.M. and 8 P.M. respectively (H. G. Thornton's data): [p. 113]
From the series of differences we have x[bar]=+10.775, ¼s2=3.756, t=5.560, n=3, whence the table shows that P is between .01 and .02. If, On the contrary, we use the method of Ex. 20, and treat the two separate series, we find x[bar]-x'[bar]=+10.775, ½s2=2.188, t =7.285, n=6; this is not only a larger value of n but a larger value of t, which is now far beyond the range of the table, showing that P is extremely small. In this case the differential effects of the different methods are either negligible, or have acted quite differently in the two series, so that precision was lost in comparing each value with its counterpart in the other series. In cases like this it sometimes occurs that one method shows no significant difference, while the other brings it out; if either method indicates a definitely significant difference, its testimony cannot be ignored, even if the other method fails to show the effect. When no correspondence exists between the members of one series and those of the other, the second method only is available. [p. 114]
25. Regression Coefficients
The methods of this chapter are applicable not only to mean values, in the strict sense of the word, but to the very wide class of statistics known as regression coefficients. The idea of regression is usually introduced in connection with the theory of correlation, but it is in reality a more general, and, in some respects, a simpler idea, and the regression co-efficients are of interest and scientific importance in many classes of data where the correlation coefficient, if used at all, is an artificial concept of no real utility. The following qualitative examples are intended to familiarise the student with the concept of regression, and to prepare the way for the accurate treatment of numerical examples.
It is a commonplace that the height of a child depends on his age, although knowing his age, we cannot accurately calculate his height. At each age the heights are scattered over a considerable range in a frequency distribution characteristic of that age; any feature of this distribution, such as the mean, will be a continuous function of age. The function which represents the mean height at any age is termed the regression function of height on age; it is represented graphically by a regression curve, or regression line. In relation to such a regression line age is termed the independent variate, and height the dependent variate.
The two variates bear very different relations to the regression line. If errors occur in the heights, this [p. 115] will not influence the regression of height on age, provided that at all ages positive and negative errors are equally frequent, so that they balance in the averages. On the contrary, errors in age will in general alter the regression of height on age, so that from a record with ages subject to error, or classified in broad age-groups, we should not obtain the true physical relationship between mean height and age. A second difference should be noted: the regression function does not depend on the frequency distribution of the independent variate, so that a true regression line may be obtained even when the age groups are arbitrarily selected, as when an investigation deals with children of "school age." On the other hand a selection of the dependent variate will change the regression line altogether.
It is clear from the above instances that the regression of height on age is quite different from the regression of age on height; and that one may have a definite physical meaning in cases in which the other has only the conventional meaning given to it by mathematical definition. In certain cases both regressions are of equal standing; thus, if we express in terms of the height of the father the average adult height of sons of fathers of a given height, observation shows that each additional inch of the fathers' height corresponds to about half an inch in the mean height of the sons. Equally, if we take the mean height of the fathers of sons of a given height, we find that each additional inch of the sons' height corresponds to half an inch in the mean height of the fathers. No selection [p. 116] has been exercised in the heights either of fathers or of sons; each variate is distributed normally, and the aggregate of pairs of values forms a normal correlation surface. Both regression lines are straight, and it is consequently possible to express the facts of regression in the simple rules stated above.
When the regression line with which we are concerned is straight, or, in other words, when the regression function is linear, the specification of regression is much simplified, for in addition to the general means we have only to state the ratio which the increment of the mean of the dependent variate bears to the corresponding increment of the independent variate. Such ratios are termed regression coefficients. The regression function takes the form
Y = a+b(x-x[bar]),
where b is the regression coefficient of y on x, and Y is the mean value of y for each value of x. The physical dimensions of the regression coefficient depend on those of the variates; thus, over an age range in which growth is uniform we might express the regression of height on age in inches per annum, in fact as an average growth rate, while the regression of father's height on son's height is half an inch per inch, or simply ½. Regression coefficients may, of course, be positive or negative.
Curved regression lines are of common occurrence ; in such cases we may have to use such a regression function as
Y = a+bx+cx2+dx3, [p. 117]
in which all four coefficients of the regression function may, by an extended use of the term, be called regression coefficients. More elaborate functions of x may be used, but their practical employment offers difficulties in cases where we lack theoretical guidance in choosing the form of the regression function, and at present the simple power series (or, polynomial in x) is alone in frequent use. By far the most important case in statistical practice is the straight regression line.
26. Sampling Errors of Regression Coefficients
The straight regression line with formula
Y = a+b(x-x[bar])
is fitted by calculating from the data, the two statistics
these are estimates, derived from the data, of the two constants necessary to specify the straight line; the true regression formula, which we should obtain from an infinity of observations, may be represented by
a+b(x-x[bar]),
and the differences a-a, b-b, are the errors of random sampling of our statistics. If s2 represent the variance of y for any value of x about a mean given by the above formula, then the variance of a, the mean of n' observations, will be s2/n', while that of b, which is [p. 118] merely a weighted mean of the values of y observed, will be
In order to test the significance of the difference between b, and any hypothetical value, b, to which it is to be compared, we must estimate the value of s2; the best estimate for the purpose is
found by summing the squares of the deviations of y from its calculated value Y, and dividing by (n'-2). The reason that the divisor is (n'-2) is that from the n' values of y two statistics have already been calculated which enter into the formula for Y, consequently the group of differences, y-Y, represent in reality only n'-2 degrees of freedom.
When n' is small, the estimate of s2 obtained above is somewhat uncertain, and in comparing the difference b-b with its standard error, in order to test its significance we shall have to use Student's method, with n=n'-2. When n' is large this distribution tends to the normal distribution. The value of t with which the table must be entered is
Similarly, to test the significance of the difference between a and any hypothetical value a, the table is entered with [p. 119]
this test for the significance of a will be more sensitive than the method previously explained, if the variation in y is to any considerable extent expressible in terms of that of x, for the value of s obtained from the regression line will then be smaller than that obtained from the original group of observations. On the other hand, one degree of freedom is always lost, so that if b is small, no greater precision is obtained.
Ex. 22. Effect of nitrogenous fertilisers in maintaining yield. -- The yields of dressed grain in bushels per acre shown in Table 29 were obtained from two plots on Broadbalk wheat field during thirty years; the only difference in manurial treatment was that "9a" received nitrate of soda, while "7b" received an equivalent quantity of nitrogen as sulphate of ammonia. In the course of the experiment plot "9a" appears to be gaining in yield on plot "7b." Is this apparent gain significant?
A great part of the variation in yield from year to year is evidently similar in the two plots; in consequence, the series of differences will give the clearer result. In one respect the above data are especially simple, for the thirty values of the independent variate form a series with equal intervals between the successive values, with only one value of the dependent variate corresponding to each. In such cases the work is simplified by using the formula
S(x-x[bar])2 = 1/12 n' (n'2-1), [p. 120]
where n' is the number of terms, or 30 in this case. To evaluate 6 it is necessary to calculate
S{y(x-x[bar])};
this may be done in several ways. We may multiply [p. 121] the successive values of y by -29, -27,… +27, +29, add, and divide by 2. This is the direct method suggested by the formula. The same result is obtained by multiplying by 1, 2, ..., 30 and subtracting 15½ times the sum of values of y; the latter method may be conveniently carried out by successive addition. Starting from the bottom of the column, the successive sums 2.69, 9.76, 6.82, ... are written down, each being found by adding a new value of y to the total already accumulated; the sum of the new column, less 15½ times the sum of the previous column, will be the value required. In this case we find the value 599.615, and dividing by 2247.5, the value of b is found to be .2668. The yield of plot "9a" thus appears to have gained on that of "7b" at a rate somewhat over a quarter of a bushel per annum.
To estimate the standard error of 6, we require the value of
S(y-Y)2;
knowing the value of b, it is easy to calculate the thirty values of Y from the formula
Y =y[bar]+(x-x[bar])b;
for the first value, x-x[bar]=-14.5 and the remaining values may be found in succession by adding b each time. By subtracting each value of Y from the corresponding y, squaring, and adding, the required quantity may be calculated directly. This method is laborious, and it is preferable in practice to utilise the algebraical fact that [p. 122]
The work then consists in squaring the values of y and adding, then subtracting the two quantities which can be directly calculated from the mean value of y and the value of b. In using this shortened method it should be noted that small errors in y[bar] and b may introduce considerable errors in the result, so that it is necessary to be sure that these are calculated accurately to as many significant figures as are needed in the quantities to be subtracted. Errors of arithmetic which would have little effect in the first method, may altogether vitiate the results if the second method is used. The subsequent work in calculating the standard error of b may best be followed in the scheme given beside the table of data ; the estimated standard error is .1169, so that in testing the hypothesis that b=0 that is that plot "9a" has not been gaining on plot "7b," we divide b by this quantity and find t=2.282. Since s was found from 28 degrees of freedom n=28, and the table of t shows that P is between .02 and .05.·
The result must be judged significant, though barely so; in view of the data we cannot ignore the possibility that on this field, and in conjunction with the other manures used, nitrate of soda has conserved the fertility better than sulphate of ammonia ; these data do not, however, demonstrate the point beyond possibility of doubt.
The standard error of y[bar], calculated from the above data, is 1.012, so that there can be no doubt that the [p. 123] difference in mean yields is significant; if we had tested the significance of the mean, without regard to the order of the values, that is calculating s2 by dividing 1020.56 by 29, the standard error would have been 1.083. The value of b was therefore high enough to have reduced the standard error. This suggests the possibility that if we had fitted a more complex regression line to the data the probable errors would be further reduced to an extent which would put the significance of b beyond doubt. We shall deal later with the fitting of curved regression lines to this type of data.
Just as the method of comparison of means is applicable when the samples are of different sizes, by obtaining an estimate of the error by combining the sums of squares obtained from the two different samples, so we may compare regression coefficients when the series of values of the independent variate are not identical; or if they are identical we can ignore the fact in comparing the regression coefficients.
Ex. 23. Comparison of relative growth rate of two cultures of an alga. -- Table 30 shows the logarithm (to the base 10) of the volumes occupied by algal cells on successive days, in parallel cultures, each taken over a period during which the relative growth rate was approximately constant. In culture A nine values are available, and in culture B eight (Dr. M. Bristol-Roach's data).
The method of finding Sy(x-x[bar]) by summation is shown in the second pair of columns: the original values are added up from the bottom, giving successive [p. 124] totals from 6.087 to 43.426; the final value should, of course, tally with the total below the original values. From the sum of the column of totals is subtracted the sum of the original values multiplied by 5 for A and by 4½ for B. The differences are Sy(x-x[bar]); these must be divided by the respective values of S(x-xbar])2,
namely, 60 and 42, to give the values of b, measuring the relative growth rates of the two cultures. To test if the difference is significant we calculate in the two cases S(y2), and subtract successively the product of the mean with the total, and the product of b with Sy(x-x[bar]); this process leaves the two values of S(y-Y)2, which are added as shown in the table, and the sum divided by n, to give s2. The value of n is found by adding the 7 degrees of freedom from series A to the 6 degrees from series B, and is therefore 13. [p. 125] Estimates of the variance of the two regression coefficients are obtained by dividing s2 by 60 and 42, and that of the variance of their difference is the sum of these. Taking the square root we find the standard error to be .01985, and t=1.844· The difference between the regression coefficients, though relatively large, cannot be regarded as significant. There is not sufficient evidence to assert culture B was growing more rapidly than culture A.
27. The Fitting of Curved Regression Lines
Little progress has been made with the theory of the fitting of curved regression lines, save in the limited but most important case when the variability of the independent variate is the same for all values of the dependent variate, and is normal for each such value. When this is the case a technique has been fully worked out for fitting by successive stages any line of the form
Y = a+bx+cx2+dx3+.. ;
we shall give details of the case where the successive values of x are at equal intervals.
As it stands the above form would be inconvenient in practice, in that the fitting could not be carried through in successive stages. What is required is to obtain successively the mean of y, an equation linear in x, an equation quadratic in x, and so on, each equation being obtained from the last by adding, a new term being calculated by carrying a single process of [p. 126] computation through a new stage. In order to do this we take
Y = A + Bx1 + Cx2 + Dx3 + …,
where x1, x2, x3, shall be functions of x of the 1st, 2nd, and 3rd degrees, out of which the regression formula may be built. It may be shown that the functions required for this purpose may be expressed in terms of the moments of the x distribution, as follows:
where the values of the moment functions have been expressed in terms of n', the number of observations, as far as is needed for fitting curves up to the 5th degree. The values of x are taken to increase by unity.
Algebraically the process of fitting may now be represented by the equations
[p. 127]
and, in general, the coefficient of the term of the rth degree is
As each term is fitted the regression line approaches more nearly to the observed values, and the sum of the squares of the deviation
S(y-Y)2
is diminished. It is desirable to be able to calculate this quantity, without evaluating the actual values of Y at each point of the series; this can be done by subtracting from S(y2) the successive quantities
and so on. These quantities represent the reduction which the sum of the squares of the residuals suffers each time the regression curve is fitted to a higher degree; and enable its value to be calculated at any stage by a mere extension of the process already used in the preceding examples. To obtain an estimate, s2, of the residual variance, we divide by n, the number of degrees of freedom left after fitting, which is found from n' by subtracting from it the number of constants in the regression formula. Thus, if a straight line has been fitted, n=n'-2; while if a curve of the fifth degree has been fitted, n=n'-6. [p. 128]
28. The Arithmetical Procedure of Fitting
The main arithmetical labour of fitting curved regression lines to data of this type may be reduced to a repetition of the process of summation illustrated in Ex. 23. We shall assume that the values of y are written down in a column in order of increasing values of x, and that at each stage the summation is commenced at the top of the column (not at the bottom, as in that example). The sums of the successive columns will be denoted by S1, S2, ... When these values have been obtained, each is divided by an appropriate divisor, which depends only on n', giving us a new series of quantities a, b, c,... according to the following equations
and so on.
From these a new series of quantities a', b', c',… are obtained by equations independent n', of which we give below the first six, which are enough to carry the process of fitting up to the 5th degree:
[p. 129]
These new quantities are proportional to the required coefficients of the regression equation, and need only be divided by a second group of divisors to give the actual values. The equations are
the numerical part of the factor being
for the term of degree r.
If an equation of degree r has been fitted, the estimate of the standard errors of the coefficients are all based upon the same value of s2, i.e.
from which the estimated standard error of any coefficient, such as that of xp, is obtained by dividing by
and taking out the square root. The number of degrees of freedom upon which the estimate is based is (n'-r-1), and this must be equated to n in using the table of t.
A suitable example of use of this method may be obtained by fitting the values of Ex. 22 (p. 120) with a curve of the second or third degree. [p. 130]
29. Regression with several Independent Variates
It frequently happens that the data enable us to express the average value of the dependent variate y, in terms of a number of different independent variates x1, x2, … xp. For example, the rainfall at any point within a district may be recorded at a number of stations for which the longitude, latitude, and altitude are all known. If all of these three variates influence the rainfall, it may be required to ascertain the average effect of each separately. In speaking of longitude, latitude, and altitude as independent variates, all that is implied is that it is in terms of them that the average rainfall is to be expressed; it is not implied that these variates vary independently, in the sense that they are uncorrelated. On the contrary, it may well happen that the more southerly stations lie on the whole more to the west than do the more northerly stations, so that for the stations available longitude measured to the west may be negatively correlated with latitude measure to the north. If, then, rainfall increased to the west but was independent of latitude, we should obtain merely, by comparing the rainfall recorded at different latitudes, a fictitious regression indicating a falling off of rain with increasing latitude. What we require is an equation taking account of all three variates at each station, and agreeing as nearly as possible with the values recorded; this is called a partial regression equation, and its coefficients are known as partial regression coefficients. [p. 131]
To simplify the algebra we shall suppose that y, x1, x2, x3, are all measured from their mean values, and that we are seeking a formula of the form
Y = b1,x1+b2x2+b3x3.
If S stands for summation over all the sets of observations we construct the three equations
of which the nine coefficients are obtained from the data either by direct multiplication and addition, or, if the data are numerous, by constructing correlation tables for each of the six pairs of variates. The three simultaneous equations for b1, b2, and b3, are solved in the ordinary way; first b3 is eliminated from the first and third, and from the second and third equations, leaving two equations for b1 and b2; eliminating b2 from these, b1 is found, and thence by substitution, b2 and b3.
It frequently happens that, for the same set of values of the independent variates, it is desired to examine the regressions for more than one set of values of the dependent variates; for example, if for the same set of rainfall stations we had data for several different months or years. In such cases it is preferable to avoid solving the simultaneous equations afresh on each occasion, but to obtain a simpler formula which may be applied to each new case.
This may be done by solving once and for all the [p. 132] three sets, each consisting of three simultaneous equations:
the three solutions of these three sets of equations may be written
Once the six values of c are known, then the partial regression coefficients may be obtained in any particular case merely by calculating S(x1y), S(x2y), S(x3y) and substituting in the formulæ,
The method of partial regression is of very wide application. It is worth noting that the different independent variates may be related in any way; for example, if we desired to express the rainfall as a linear function of the latitude and longitude, and as a quadratic function of the altitude, the square of the altitude would be introduced as a fourth independent variate, without in any way disturbing the process outlined above, save that S(x3x4), S(x33) would be calculated directly from the distribution of altitude.
In estimating the sampling errors of partial [p. 133] regression coefficients we require to know how nearly our calculated value, Y, has reproduced the observed values of y; as in previous cases, the sum of the squares of (y-Y) may be calculated by differences, for, with three variates,
S(y-Y)2 = S(y2) - b1S(x1y) - b2S(x2y) - b3S(x3y).·
If we had n' observations, and p independent variates, we should therefore find
and to test if b1, differed significantly from any hypothetical value, b1, we should calculate
entering the table of t with n=n'-p-1.
In the practical use of a number of variates it is convenient to use cards, on each of which is entered the values of the several variates which may be required. By sorting these cards in suitable grouping units with respect to any two variates the corresponding correlation table may be constructed with little risk of error, and thence the necessary sums of squares and products obtained.
Ex. 24. Dependence of rainfall on position and altitude. -- The situations of 57 rainfall stations in Hertfordshire have a mean longitude 12'.4 W., a mean latitude 51° 48'.5 N., and a mean altitude 302 feet. Taking as units 2 minutes of longitude, one [p. 134] minute of latitude, and 20 feet of altitude, the following values of the sums of squares and products of deviations from the mean were obtained:
To find the multipliers suitable for any particular set of weather data from these stations, first solve the equations
1934.1 c11 - 772.2 c12 + 924.1 c13 = 1
-772.2 c11 + 2889.5 c12 + 119.6 c13 = 0
+924.1 c11 + 119.6 c13[sic] + 1750.8 c13 = 0;
using the last equation to eliminate c13 from the first two, we have
2532.3 c11 - 1462.5 c12 = 1.7508
=1462.5 c11 + 5044.6 c12 = 0;
from these eliminate c12, obtaining
10,635.5 c11 = 8.8321;
whence
c11 = .00083043, c12 = .00024075, c13 = -.00045476
the last two being obtained successively by substitution.
Since the corresponding equations for c12, c22, c23 differ only in changes in the right-hand number, we can at once write down
-1462.5 c12 + 5044.6 c22 = 1.7508;
whence, substituting for c12 the value already obtained,
c22 = .00041686, c23 = -.00015554; [p. 135]
finally, to obtain c33 we have only to substitute in the equation
924.1c13 + 119.6c23 + 1750.8c33 = 1,
giving
c33 =.00082182.
It is usually worth while, to facilitate the detection of small errors by checking, to retain as above one more decimal place than the data warrant.
The partial regression of any particular weather data on these three variates can now be found with little labour. In January 1922 the mean rainfall recorded at these stations was 3.87 inches, and the sums of products of deviations with those of the three independent variates were (taking 0.1 inch as the unit for rain)
S(x1y) = +1137.4, S(x2y) = -592.9, S(x3y) = +891.8;
multiplying these first by c11, c12, c13 and adding, we have for the partial regression on longitude
b2 = .39624;
similarly using the multipliers c12, c22, c23 we obtain for the partial regression on latitude
b2 = -11204;
and finally, by using c13, c23, c33,
b3 = .30787
gives the partial regression on altitude.
Remembering now the units employed, it appears that in the month in question rainfall increased by .0198 of an inch for each minute of longitude westwards, [p. 136] is decreased by .0112 of an inch for each minute of latitude northwards, and increased by .00154 of an inch for each foot of altitude.
Let us calculate to what extent the regression on altitude is affected by sampling errors. For the 57 recorded deviations of the rainfall from its mean value, in the units previously used
S(y2) = 1786.6;
whence, knowing the values of b1, b2, and b3, we obtain by differences
S(y-Y)2 = 994.9.
To find s2, we must divide this by the number of degrees of freedom remaining after fitting a formula involving three variates -- that is, by 53 -- so that
s2 = 8.772;
multiplying this by c33, and taking the square root,
s[sqrt]c33 = .12421.
Since n is as high as 53 we shall not be far wrong in taking the regression of rainfall on altitude to be in working units .308, with a standard error .124; or in inches of rain per 100 feet as .154, with a standard error .062. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9008148312568665, "perplexity": 551.5162424239479}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400380358.68/warc/CC-MAIN-20141119123300-00103-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/compare-charge-and-current.828770/ | # Compare charge and current
Tags:
1. Aug 21, 2015
### plasma tail
1. The problem statement, all variables and given/known data
We have seen that a coulomb is an enormous amount of
charge; it is virtually impossible to place a charge of 1 C on an
object. Yet, a current of 10 A, is quite reasonable. Explain
this apparent discrepancy.
2. Relevant equations
I = nqvA
3. The attempt at a solution
i think it is because current is the amount of charge flow per second, and the amount of charge depends on the volume of the wire, hence current can be larger than the placed charges if volume of the wire is pretty big. is this a good answer? is there anything to add? thx
2. Aug 22, 2015
### Noctisdark
I agree with: As you've already said, current is defined as the amount of charge flow per second, there could be many charges little charges but summing them up will result in 1 C, bigger volume mean less resistance so more current flow this can be explained (the bad way) because charges have more space to avoid hitting static atoms in the wire (also the reason you've mentionned) and I should mention all the charges flow on the surface,any way if we get into the math, $I = \frac{\delta Q}{\delta t}$, if we take for example $\delta t = 0.01 s$ mesure that $\delta Q = 0.1C$ passed, then we conclude that I = 10A, Cheers :D
Draft saved Draft deleted
Similar Discussions: Compare charge and current | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9168058037757874, "perplexity": 767.8683930787322}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187826114.69/warc/CC-MAIN-20171023145244-20171023165244-00473.warc.gz"} |
https://www.springermedizin.de/quantitative-summaries-of-treatment-effect-estimates-obtained-wi/9433096?fulltextView=true | main-content
01.12.2013 | Research article | Ausgabe 1/2013 Open Access
# Quantitative summaries of treatment effect estimates obtained with network meta-analysis of survival curves to inform decision-making
Zeitschrift:
BMC Medical Research Methodology > Ausgabe 1/2013
Autoren:
Shannon Cope, Jeroen P Jansen
Wichtige Hinweise
## Electronic supplementary material
The online version of this article (doi:10.1186/1471-2288-13-147) contains supplementary material, which is available to authorized users.
## Competing interests
The authors declare that they have no competing interests.
## Authors’ contributions
SC and JJ conceived of the study. SC performed the statistical analysis and drafted the manuscript. JJ participated in the study design and helped to draft the manuscript. All authors read and approved the final manuscript.
## Background
Randomized controlled trials (RCTs) are often used to inform healthcare decisions [14]. In the absence of a head-to-head or direct comparison, indirect treatment comparisons provide a useful alternative [2, 59]. An evidence base that consist of multiple RCTs where each trial has at least one intervention in common with another can be synthesized by means of a network meta-analysis (NMA). This method provides pooled estimates of available direct comparisons, indirect comparisons of pairwise contrasts for which no head-to-head RCT is available, and a synthesis of consistent direct and indirect evidence, resulting in more precise treatment effect estimates [3, 10]. NMAs provide a comprehensive synthesis of the evidence from RCTs for multiple treatments useful for decision-makers to assess whether a new treatment should be adopted or whether additional evidence is required in the presence of uncertainty [11].
Meta-analyses or NMAs can be performed in a frequentist or a Bayesian framework. The result of a frequentist meta-analysis comparing treatments A and B is an estimate of the treatment effect (i.e. difference between A and B) as well as an associated p-value. The p-value indicates whether the results are statistically ‘significant’ or ‘non-significant’. If results are significant, the probability of erroneously rejecting the null hypothesis is judged to be small enough given the observed data. For example, if treatment A is considered significantly better than treatment B then the difference between the treatments is considered extreme enough to suggest that there is only a small probability (<5%) of incorrectly rejecting the null hypothesis. Therefore the p value reflects the probability of observing such a treatment difference assuming the null hypothesis is true. However, decision-makers are interested in minimizing the risk of an unsupported positive interpretation as well as the risk of overlooking a true difference. In other words, probabilities associated with the alternative hypothesis (i.e. A is better than B) are of interest but cannot be deduced from a frequentist analysis. Moreover, for an analysis of more than two treatments, p values resulting from a frequentist analysis associated with each pairwise comparison do not provide a straightforward interpretation of the relative efficacy or safety of the alternative interventions for decision-makers.
By using a Bayesian NMA it is possible to calculate the probability of being the best treatment out of all those treatments assessed with respect to the outcome of interest. This approach combines a prior probability distribution (representing a prior belief of the possible values for parameter) with a likelihood distribution of the observed effect, resulting in a posterior probability distribution [12]. With Monte Carlo simulations the probability that a treatment is best is calculated based on the proportion of cycles during the sampling process where a treatment ranks first of out all the treatments included in terms of the treatment effect size [13]. Similarly, it is possible to calculate the probability of being the second best treatment, third best treatment, etc., up until the probability of being worst treatment out of those assessed. These probability statements offer an intuitive summary of the joint posterior distribution of the effect sizes for all the included treatments, which naturally facilitates decision-making [13].
Salanti et al. have proposed several methods to present rank probabilities of treatments. Given the challenge of efficiently summarizing results from an analysis involving multiple pairwise comparisons, probabilities are positioned as a useful alternative to ‘p values’ resulting from a frequentist analysis. The importance of presenting a complete overview of the probabilities associated with each ranking is emphasized to avoid the over-interpretation of the probabilities associated with being the ‘best’ treatment, which necessitates a more comprehensive approach to present the information. Therefore, several different approaches are proposed by Salanti et al. to summarize the probabilities in a clear a concise manner. However, all of the methods implicitly assume that the treatment effects are constant over time [13].
In many RCTs the endpoint of interest is the time to the occurrence of a certain event, such as time to progression, progression-free survival, or overall survival. The synthesis of published results across different studies by means of an NMA is typically based on the constant hazard ratio (HR). However, it has been recognized that an NMA that relies on the proportional hazards assumption is biased if the survival curves or hazard functions of competing interventions cross [1417]. Recently NMA models for survival data have been presented that do not assume a constant HR but allow the relative treatment effects to vary over time [1416]. Such analyses can result in time-varying HRs, survival proportions over time, and expected survival by treatment. In order to apply the methods proposed by Salanti et al. to these analyses it is important to acknowledge that treatment effects may vary over time.
In this paper we discuss alternative approaches to present rank probabilities in the context of a Bayesian NMA of parametric survival curves.
## Methods
### Motivating example
#### Evidence base
As an illustrative example the efficacy of systemic chemotherapy for advanced unresectable melanoma was assessed in terms of overall survival. Ten RCTs were included in the network of evidence (Figure 1) [1827], which were identified with a systematic review of the literature. The treatments were categorized as dacarbazine monotherapy (DTIC), DTIC + Interferon (DTIC + IFN), DTIC + non-IFN, and Non-DTIC. Although the most recent treatments are not included, the analysis provides a useful example in oncology, where parametric survival analyses are often utilized.
### Network meta-analysis
The available survival data of the different studies was combined by means of a Bayesian NMA of parametric survival curves with models proposed by Jansen 2011 [15]. With this approach the survival of patients in a trial for the interventions being compared are modeled over time with parametric survival functions and the difference in the shape and scale parameters of these functions between interventions are synthesized and indirectly compared across trials. Within the Bayesian framework, analyses consist of data, likelihood, parameters, and a model. The data was extracted from the included RCTs, where for each arm the reported Kaplan Meier curves were digitized (DigitizeIt v1.6.1). A binomial likelihood distribution was used for the incident number of deaths for every two month interval, which was calculated based on the survival percentages from the Kaplan-Meier curves and the number of patients at risk at the beginning of the interval in each arm of each study, assuming a constant hazard rate within each interval (see Jansen and Cope [16] for more details). A two parameter Weibull NMA model was used with a random effect on the scale parameter [1417] (See Additional file 1 for model details). Non-informative prior distributions were used for the model parameters to avoid influencing the results of the analysis based on prior beliefs. The parameters were estimated using a Markov Chain Monte Carlo within WinBUGS software [28], where inferences were based on 30,000 iterations from two chains and the first 30,000 iterations were discarded as ‘burn-in’.
## Results
### Treatment effects and functional estimates
Relative treatment effects of each intervention versus DTIC were expressed as HRs over time defined as:
$H R Ak t = exp d 0 Ak + d 1 Ak ln t$
(1)
where HR Ak (t) is the HR of intervention k relative to A (i.e. DTIC), and d 0Ak and d 1Ak are the differences in scale and shape of treatment k relative to A as obtained with the NMA.
In order to estimate the hazard and cumulative hazard function by treatment, the pooled differences in scale and shape were added to an average scale and shape for DTIC (obtained from the DTIC studies included in the NMA). These scale and shape estimates describe the hazard over time, as presented in Figure 2A. The ratio of hazard curves over time reflects the HRs, which are illustrated in Figure 2B, including the 95% credible intervals (dotted lines). The hazards (or HRs) were fairly constant over time, with some variation in the early months. The hazard over time for each treatment was transformed into survival functions as presented in Figure 3. Based on the survival functions, the median survival (i.e. time point where 50% of patients are still alive) as well as the expected survival (i.e. mean survival based on the area under the curve up to the time-point when all of the patients have died) were estimated. The area under the survival curve at the left of each time point represents the mean survival up until the corresponding follow-up time. This represents a summary measure of survival which does not require the curves to be fully extrapolated (i.e. up until when all patients have died). In the current evidence network, the mean survival at 22 months was assessed, which reflected the shortest follow-up across the studies, i.e. the DTIC arm in the study by Middleton et al. 2000. The median survival, mean survival at 22 months, and the expected survival are presented in Table 1.
Table 1
Overview of time-independent summary measures
Outcome
DTIC
DTIC + IFN
DTIC + non-IFN
Non-DTIC
Median survival
7.85
7.87
9.88
10.19
Expected survival (after all patients died) and 95% credible interval
12.61 (11.31, 14.13)
11.41 (8.44, 15.48)
16.11 (11.21, 23.14)
15.31 (9.17, 24.34)
Mean survival at 22 months and 95% credible interval
9.84 (9.13, 10.60)
9.61 (7.66, 11.72)
11.15 (8.88, 13.33)
11.23 (8.05, 13.99)
### Graphical and numerical summaries of rank probabilities
The ranking for all four treatments according to the probability of being the 1st, 2nd, 3rd, and 4th best treatments was assessed on the basis of each of the aforementioned effect measures.
Rankograms, presenting the probability per rank for each treatment, are illustrated in Figure 4 for the time independent treatment effects, including median survival, expected survival, and mean survival at 22 months of follow-up. Rankograms for the time-varying treatment effects are presented in Figure 5, corresponding to the hazard (ratio), survival proportions, and mean survival.
The rankograms for the time-independent measures were fairly similar with some minor differences, suggesting DTIC + non-IFN and non-DTIC tended to have the highest probabilities of being the best and second best treatments. DTIC generally had the highest probability of being third best, and DTIC + IFN usually had the highest probability of being the worst. The rankograms for median survival (Figure 4A) and expected survival (Figure 4B) were mostly comparable (with a slight tradeoff between treatment ranks 3 and 4 for DTIC and DTIC + IFN), whereas the rankogram based on the mean survival at 22 months (Figure 4C) differed because the survival curves crossed at about 17 months for non-DTIC and DTIC + non-IFN.
The time-dependent measures were generally comparable and similar to the time-independent measures, indicating that DTIC + non-IFN and non-DTIC were the best treatments, followed by DTIC and DTIC + IFN. Variation in the hazard in the initial period is most obvious in Figure 5A, where the probability of being the best treatment is based on the HR. This indicates that non-DTIC is the best treatment for the first 5 months, after which time DTIC + non-IFN is the best treatment. The rankograms based on the survival proportions (Figure 5B) were similar to those based on the hazards, although the decrease in the probability of DTIC + IFN being the best (and second best) was less dramatic with the former. Rankograms based on the mean survival over time were also similar to those based on HRs and survival proportions, where results for DTIC differed the most which remained more consistent over time with respect to the probability of best and second best. Similarly rankograms for non-DTIC based on the mean survival were less sensitive to differences over time.
Another measure to summarize probabilities proposed by Salanti et al. [13] is the surface under the cumulative ranking curve (SUCRA), which provides a summary statistic for the cumulative ranking. SUCRA ranges from 0 to 1, where 1 reflects the best treatment with no uncertainty and 0 reflects the worst treatment with no uncertainty. SUCRA for treatment k out of competing interventions a can be expressed as follows based on a vector of cumulative probability cum k,b to be among b best treatments:
$SUCR A k = ∑ b = 1 a − 1 cu m k , b a − 1$
(2)
SUCRA was assessed for all effect measures for each treatment. In order to emphasize the importance of assessing SUCRA, probabilities of being the best treatment are compared to the SUCRA scores. Figures 6 and 7 present the probability that each treatment is best as well as SUCRA for the time-independent and time-dependent measures.
Figure 6 illustrates that the pattern associated with the probability of being the best treatment is fairly consistent with the results for SUCRA for the time independent measures. However, for non-DTIC, the probability of being best is lower than SUCRA for the mean at 22 months relative to the median and expected mean because SUCRA accounted for the higher probability of non-DTIC being the second best treatment. Generally, the difference between DTIC and DTIC + IFN was less pronounced for the probability of being the best treatment than for SUCRA, whereas the difference between DTIC + non-IFN and non-DTIC was more pronounced for the probability of being the best than for SUCRA.
The overall pattern for the probability of being the best treatment and SUCRA are similar for the time-independent and time-varying outcomes (Figure 7), although there were some differences depending on the specific time-varying measure. For example, differences between the probability of being the best treatment and SUCRA for DTIC + non-IFN and non-DTIC were greatest when based on the HRs and smallest when based on the mean survival. Additionally, the initial period where non-DTIC is expected to be the best treatment is shortest when based on the HR, longer when based on the survival function, and longest when based on the mean survival (up until almost 30 months). A unique feature of the rankograms based on survival is that the point at which the probability of being the best treatment switches from non-DTIC to DTIC + non-IFN is also the time point when the survival curves cross. As with the time-independent measures, for all three time-dependent measures the probability of being the best treatment suggests that DTIC and DTIC + IFN are comparable and reflect the two worst treatments, whereas DTIC + non-IFN appear to be better than non-DTIC. By evaluating SUCRA it is possible to differentiate DTIC and DTIC + IFN, where a majority of the time points suggest DTIC had a higher proportion than DTIC + IFN. Moreover, differences between DTIC + non-IFN and non-DTIC are less dramatic for SUCRA (as opposed to the probability of being the best treatment), particularly for mean survival. Overall, SUCRA results may raise questions about the additional efficacy of IFN in combination with DTIC as opposed to DITC alone.
## Discussion
### Advantages and disadvantages of different effect measures in relation to treatment ranking
Table 2 outlines the previously described alternative effect measures resulting from a NMA of survival data involving a multi-dimensional treatment effect. While these measures are all related and based on the same analysis, each measure involves a different interpretation and slightly different rank probabilities. Therefore, it is necessary to consider the advantages and disadvantages associated with each measure to calculate the rank probabilities.
Table 2
Summary of alternative methods for calculating rank probabilities
Measure
Probability that a treatment is associated with:
Explicitly reflects time effect
Reflects cumulative effect over time
Requires baseline risk
Median survival
The greatest survival time when 50% patients are alive
No
Yes
Yes
Commonly used and clinically relevant; Easily summarized as statistic; May limit need for extrapolation;
Ignores what happens after 50% of subjects have experienced the event;
Expected survival
The greatest expected survival
No
Yes
Yes
Directly relevant for cost-effectiveness; Easily summarized as statistic;
Sensitive to tail of distribution (may involve extrapolation); Does not illustrate time-varying results or time of greatest treatment effect; May not be as clinically relevant;
Mean survival at time t
Greatest mean survival (area under the curve) up until time t
No
Yes
Yes
Limits need for extrapolation if time t corresponds to follow-up time of trial with shortest duration; Easily summarized as statistic
May be difficult to interpret; Requires subjective selection of time t; Ignores tails of distribution and does not illustrate time-varying results;
Hazard (ratio) over time
The smallest hazard (ratio versus reference treatment) over time
Yes
No
Yes for hazard,
Directly relates to model and may help emphasize changes in treatment effect over time;
Does not capture cumulative effect of treatment over time; May lead to over interpretation near tail of distribution; Cannot be summarized as statistic (requires graphical illustration); May be more difficult to understand;
No for hazard ratio
Survival proportion over time (Cumulative hazard over time)
The greatest survival (proportion) over time
Yes
Yes
Yes
Highly intuitive and clinically relevant; Can be easily compared to data;
Cannot be summarized as statistic (requires graphical illustration);
Mean survival over time
Greatest mean survival (area under the curve) over time
Yes
Yes
Yes
Reflects a cumulative summary of survival proportions up until that time point, thereby de-emphasizing tail of distribution;
Cannot be summarized as statistic (requires graphical illustration); May be more difficult to understand;
### Time independent measures: median survival, expected survival, or mean survival at a specific time point?
Median survival provides an intuitive outcome for clinicians, which can easily be compared across treatments and is not very sensitive to parametric modeling assumptions. However, the median survival does not capture survival information beyond the time point at which 50% of patients have died, thereby providing a limited effect measure to rank treatments.
Expected survival is the primary measure of interest for cost-effectiveness evaluations involving survival, although this measure may require extrapolation of survival proportions beyond the follow-up of the included studies. Consequently, rank probabilities base on the expected survival may often rely on extrapolation, possibly to a different extent for each treatment. Also, althoug the estimates from a parametric model will reflect this uncertainty, the structural uncertainty regarding the choice of the underlying distribution is not captured, which is an important consideration [29].
Rank probabilities based on the mean survival at a specific time point, such as the duration of the trial with the shortest follow-up, may avoid extrapolation. Therefore this summary measure may be less sensitive to the assumptions of extrapolation, although a subjective choice regarding the time point for analysis is required, which leaves it open to criticism.
Overall, a summary measure such as the median survival, expected survival, or mean survival at a specific follow-up time has the advantage of providing a simple statistic that does not require graphical presentation over time. Consequently, rank probabilities based on these effect measures are therefore easier to interpret and compare across different analyses than time-varying effect measures. However, the rank probabilities associated with the one dimensional effect measures do not capture the possible time-varying nature of the underlying hazard of dying and are sensitive to the choice of effect measure.
### Time-varying measures: hazard, survival, or mean survival over time?
Rank probabilities based on the hazard or HR at each time point reflect the treatment ranking at each time point independent of previous time points. Presenting the hazard or HR over time illustrates how treatment effects may vary over time, which may not be easily detected based on the corresponding survival curves. However, HR curves may not be straightforward to interpret and later time points may be less relevant due to the small proportion of subjects that remain at risk of dying.
Rank probabilities based on the survival proportions reflect the cumulative treatment effect of the hazard up to that point in time. Survival curves can be considered the most complete and intuitive representation of treatment effects over time. Presenting survival proportions also allows the results of the meta-analysis to be compared to the observed survival curves reported for the individual studies. Therefore, rank probabilities presented over time based on survival curves may provide the simplest interpretation and the most ‘face validity’ as compared to those based on the hazards or HRs over time, especially considering that decision-makers are likely to be mostly concerned with actual survival over time, as opposed to the risk of dying at each time point.
Mean survival at subsequent time points may provide another measure with value as well for treatment rankings. By evaluating the area under the curve up to each time point, as opposed to the actual survival percentages, more weight is attributed to earlier treatment effects when a greater proportion of patients are still alive. If two treatments cross, the treatment with the more favorable survival in the beginning will result in a longer period of being the best treatment if the mean survival is used in comparison to the survival proportions or the HRs. Emphasizing the early treatment effects (and de-emphasizing later ones) may be considered useful given the increasing uncertainty in treatment effects over time due to reduced population at risk and the possible extrapolation of survival curves.
Overall, rank probabilities of treatment effects over time may provide a more transparent and informative approach to help guide decision-making in comparison to single rank probabilities based on collapsed measures, such as median survival, or expected survival. Rank probabilities based on survival proportions may be the most intuitive and straightforward to communicate, but alternatives based on the hazard function or mean survival over time may be useful as well.
### Probability of being best treatment, rankograms, or SUCRA?
Summarizing treatment effects and their associated uncertainty in terms of the probability that each treatment is best is often presented, although when there is considerable variation in the uncertainty regarding the relative treatment effects, this approach may lead to false conclusions. Rankograms provide the most informative and balanced approach to translate treatment effects and their associated uncertainty into probability statements for decision-making by presenting the probability that each treatment is best, 2nd best, 3rd best, etc. However, rankograms become more difficult to interpret for time-varying treatment effects as compared to one-dimensional effect measures. In the context of time-varying treatment effects, graphing SUCRA may provide a more concise summary measure than presenting all rank probabilities.
## Conclusion
In this paper we present different alternatives for quantitative summaries of treatment effect estimates obtained with NMA of survival data to help inform decision-making. Rank probabilities based on one-dimensional measures such as median survival, expected survival, or mean survival at one follow-up time are relatively easy to understand, but do not provide the wealth of information captured by rank probabilities over time. Survival proportions reflect the cumulative effect of treatments over time and provide the most intuitive basis for rank probabilities and SUCRA. Rank probabilities based on the hazard (ratio) function over time provide information on the treatment effect at each time point, ignoring effects at previous time points. Rank probabilities based on the mean survival at each time point give more weight to the treatment effects when a greater proportion of patients are alive. Rankograms of time-varying treatment effects can be presented efficiently with SUCRA.
## Acknowledgements
The research was performed without specific funding.
## Competing interests
The authors declare that they have no competing interests.
## Authors’ contributions
SC and JJ conceived of the study. SC performed the statistical analysis and drafted the manuscript. JJ participated in the study design and helped to draft the manuscript. All authors read and approved the final manuscript.
Zusatzmaterial
Additional file 1: Appendix. Random effects first order fractional polynomial network meta-analysis model for survival curves. (DOCX 38 KB)
12874_2013_1024_MOESM1_ESM.docx
Authors’ original file for figure 1
12874_2013_1024_MOESM2_ESM.pdf
Authors’ original file for figure 2
12874_2013_1024_MOESM3_ESM.pdf
Authors’ original file for figure 3
12874_2013_1024_MOESM4_ESM.pdf
Authors’ original file for figure 4
12874_2013_1024_MOESM5_ESM.pdf
Authors’ original file for figure 5
12874_2013_1024_MOESM6_ESM.pdf
Authors’ original file for figure 6
12874_2013_1024_MOESM7_ESM.pdf
Authors’ original file for figure 7
12874_2013_1024_MOESM8_ESM.pdf
Literatur
Über diesen Artikel
Zur Ausgabe
## Meistgelesene Bücher aus dem Fachgebiet AINS
• 2014 | Buch
### Komplikationen in der Anästhesie
Fallbeispiele Analyse Prävention
Aus Fehlern lernen und dadurch Zwischenfälle vermeiden! Komplikationen oder Zwischenfälle in der Anästhesie können für Patienten schwerwiegende Folgen haben. Häufig sind sie eine Kombination menschlicher, organisatorischer und technischer Fehler.
Herausgeber:
Matthias Hübler, Thea Koch
• 2013 | Buch
### Anästhesie Fragen und Antworten
1655 Fakten für die Facharztprüfung und das Europäische Diplom für Anästhesiologie und Intensivmedizin (DESA)
Mit Sicherheit erfolgreich in Prüfung und Praxis! Effektiv wiederholen und im entscheidenden Moment die richtigen Antworten parat haben - dafür ist dieses beliebte Prüfungsbuch garantiert hilfreich. Anhand der Multiple-Choice-Fragen ist die optimale Vorbereitung auf das Prüfungsprinzip der D.E.A.A. gewährleistet.
Autoren:
Prof. Dr. Franz Kehl, Dr. Hans-Joachim Wilke
• 2011 | Buch
### Pharmakotherapie in der Anästhesie und Intensivmedizin
Wie und wieso wirken vasoaktive Substanzen und wie werden sie wirksam eingesetzt Welche Substanzen eignen sich zur perioperativen Myokardprojektion?
Kenntnisse zur Pharmakologie und deren Anwendung sind das notwendige Rüstzeug für den Anästhesisten und Intensivmediziner. Lernen Sie von erfahrenen Anästhesisten und Pharmakologen.
Herausgeber:
Prof. Dr. Peter H. Tonner, Prof. Dr. Lutz Hein
• 2013 | Buch
### Anästhesie und Intensivmedizin – Prüfungswissen
für die Fachpflege
Fit in Theorie, Praxis und Prüfung! In diesem Arbeitsbuch werden alle Fakten der Fachweiterbildung abgebildet. So können Fachweiterbildungsteilnehmer wie auch langjährige Mitarbeiter in der Anästhesie und Intensivmedizin ihr Wissen gezielt überprüfen, vertiefen und festigen.
Autor:
Prof. Dr. Reinhard Larsen | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8083931803703308, "perplexity": 1936.4744277263705}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318952.90/warc/CC-MAIN-20190823172507-20190823194507-00536.warc.gz"} |
https://www.physicsforums.com/threads/minimum-variance-unbiased-estimator.625071/ | # Minimum variance unbiased estimator
1. Aug 2, 2012
### dvvv
1. The problem statement, all variables and given/known data
Let $\bar{X}$1 and $\bar{X}$2 be the means of two independent samples of sizes n and 2n from an infinite population that has mean μ and variance σ^2 > 0. For what value of w is w$\bar{X}$1 + (1 - w)$\bar{X}$2 the minimum variance unbiased estimator of μ?
(a) 0
(b) 1/3
(c) 1/2
(d) 2/3
(e) 1
2. Relevant equations
If θ~ is unbiased for θ and
Var(θ~)= 1/E[(d loge f (x)/dθ)^2] = 1/E[(dl(θ)/dθ)^2]
then θ~ is a minimum variance unbiased estimator of θ.
3. The attempt at a solution
E[w$\bar{X}$1 + (1 - w)$\bar{X}$2] = wμ + (1-w)μ = μ
So it's an unbiased estimator of μ.
I tried calculated the variance but I guess it's wrong.
Var[w$\bar{X}$1 + $\bar{X}$2 - w$\bar{X}$2] = w^2.σ^2/n + σ^2/n + w^2.σ^2/n = σ^2/n(2w^2 +1)
I think I have to use the formula above but I don't know how.
Thanks.
2. Aug 2, 2012
### Ray Vickson
Note: use brackets, since otherwise your expressions are ambiguous. Better yet, use LaTeX, as you did in the first part of your post.
Your variance formula is incorrect. Since $\bar{X}_1$ and $\bar{X}_2$ are independent we have
$$\text{Var}(a \bar{X}_1 + b \bar{X}_2) = a^2 \text{Var}(\bar{X}_1) + b^2 \text{Var}(\bar{X}_2)$$
for any constants $a, \: b.$ Use $a = w$ and $b = 1-w.$
I don't know why you wanted to write (1-w)X as X - wX and then apply the variance formula, but you did it incorrectly. Using V(.) for the variance of a random varable, we have (using the fact that X and X are dependent): $$V(X - wX) = 1^2 V(X) + w^2 V(X) - 2 \cdot 1\cdot w\: \text{Cov}(X,X),$$ and, of course, $\text{Cov}(X,X) = V(X).$
RGV
Last edited: Aug 2, 2012
3. Aug 2, 2012
### dvvv
I don't know why I did that either.
So is it right to say:
$$\text{Var}( \bar{X}_1) = \text{Var}(\bar{X}_2) = σ^2/n$$ ?
How do I work out what w is?
4. Aug 2, 2012
### Ray Vickson
In terms of sigma, what would be the variance of a sample mean of size 10 (that is, n=10)? What about for a sample of size 20? Now go back and read (carefully) the original question. Do you honestly still need me to answer?
RGV
5. Aug 2, 2012
### dvvv
I guess it would be (σ^2)/10 and (σ^2)/20, so it's (σ^2)/n and (σ^2)/2n for $\bar{X}_1$ and $\bar{X}_2$, respectively.
I subbed that into
$$\text{Var}(a \bar{X}_1 + b \bar{X}_2) = a^2 \text{Var}(\bar{X}_1) + b^2 \text{Var}(\bar{X}_2)$$
and subbed in a and b, and I got
$$(σ^2(w^2+1))/2n$$
I still don't know how to get w, sorry...
6. Aug 2, 2012
### Ray Vickson
No wonder: you have made a serious algebraic blunder, so you end up with the wrong expression.
RGV
7. Aug 3, 2012
8. Aug 3, 2012
### Ray Vickson
You should not need a powerful tool to do such simple, high-school algebra, but since you have already done it, OK. The question asked you to find the value of w that minimizes the variance. So, that is what you need to do. At this point I am signing off this thread.
RGV
9. Aug 3, 2012
### dvvv
I just used it to confirm I was right. Thanks for your help. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9315566420555115, "perplexity": 1269.6847431349781}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886108709.89/warc/CC-MAIN-20170821133645-20170821153645-00257.warc.gz"} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.