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	<html><!-- Created using the cpp_pretty_printer from the dlib C++ library. See http://dlib.net for updates. --><head><title>dlib C++ Library - vector_abstract.h</title></head><body bgcolor='white'><pre>
	<font color='#009900'>// Copyright (C) 2003 Davis E. King ([email protected])
	</font><font color='#009900'>// License: Boost Software License See LICENSE.txt for the full license.
	</font><font color='#0000FF'>#undef</font> DLIB_VECTOR_ABSTRACT_
	<font color='#0000FF'>#ifdef</font> DLIB_VECTOR_ABSTRACT_

	<font color='#0000FF'>#include</font> "<a style='text-decoration:none' href='../serialize.h.html'>../serialize.h</a>"
	<font color='#0000FF'>#include</font> <font color='#5555FF'><</font>functional<font color='#5555FF'>></font>
	<font color='#0000FF'>#include</font> <font color='#5555FF'><</font>iostream<font color='#5555FF'>></font>
	<font color='#0000FF'>#include</font> "<a style='text-decoration:none' href='../matrix/matrix_abstract.h.html'>../matrix/matrix_abstract.h</a>"

	<font color='#0000FF'>namespace</font> dlib
	<b>{</b>
	<font color='#0000FF'>template</font> <font color='#5555FF'><</font>
	<font color='#0000FF'>typename</font> T,
	<font color='#0000FF'><u>long</u></font> NR <font color='#5555FF'>=</font> <font color='#979000'>3</font>
	<font color='#5555FF'>></font>
	<font color='#0000FF'>class</font> <b><a name='vector'></a>vector</b> : <font color='#0000FF'>public</font> matrix<font color='#5555FF'><</font>T,NR,<font color='#979000'>1</font><font color='#5555FF'>></font>
	<b>{</b>
	<font color='#009900'>/*!
	REQUIREMENTS ON T
	T should be some object that provides an interface that is
	compatible with double, float, int, long and the like.

	REQUIREMENTS ON NR
	NR == 3 \|\| NR == 2

	INITIAL VALUE
	x() == 0
	y() == 0
	z() == 0

	WHAT THIS OBJECT REPRESENTS
	This object represents a three dimensional vector. If NR == 2 then
	this object is limited to representing points on the XY plane where
	Z is set to 0.

	Also note that this object performs the appropriate integer and
	floating point conversions and promotions when vectors of mixed
	type are used together. For example:
	vector<int,3> vi;
	vector<double,2> vd;
	vd + vi == a vector<double,3> object type since that is what
	is needed to contain the result of vi+vd without
	any loss of information.
	!*/</font>

	<font color='#0000FF'>public</font>:

	<font color='#0000FF'>typedef</font> T type;

	<b><a name='vector'></a>vector</b> <font face='Lucida Console'>(</font>
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- #*this has been properly initialized
	!*/</font>

	<b><a name='vector'></a>vector</b> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> T _x,
	<font color='#0000FF'>const</font> T _y,
	<font color='#0000FF'>const</font> T _z
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	requires
	- NR == 3
	ensures
	- #x() == _x
	- #y() == _y
	- #z() == _z
	!*/</font>

	<b><a name='vector'></a>vector</b> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> T _x,
	<font color='#0000FF'>const</font> T _y
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	requires
	- NR == 2
	ensures
	- #x() == _x
	- #y() == _y
	- #z() == 0
	!*/</font>

	<font color='#0000FF'>template</font> <font color='#5555FF'><</font><font color='#0000FF'>typename</font> U, <font color='#0000FF'><u>long</u></font> NRv<font color='#5555FF'>></font>
	<b><a name='vector'></a>vector</b> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> vector<font color='#5555FF'><</font>U,NRv<font color='#5555FF'>></font><font color='#5555FF'>&</font> v
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- Initializes *this with the contents of v and does any rounding if necessary and also
	takes care of converting between 2 and 3 dimensional vectors.
	- if (U is a real valued type like float or double and T is an integral type like long) then
	- if (NR == 3) then
	- #x() == floor(v.x() + 0.5)
	- #y() == floor(v.y() + 0.5)
	- #z() == floor(v.z() + 0.5)
	- else // NR == 2
	- #x() == floor(v.x() + 0.5)
	- #y() == floor(v.y() + 0.5)
	- #z() == 0
	- else
	- if (NR == 3) then
	- #x() == v.x()
	- #y() == v.y()
	- #z() == v.z()
	- else // NR == 2
	- #x() == v.x()
	- #y() == v.y()
	- #z() == 0
	!*/</font>

	<font color='#0000FF'>template</font> <font color='#5555FF'><</font><font color='#0000FF'>typename</font> EXP<font color='#5555FF'>></font>
	<b><a name='vector'></a>vector</b> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> matrix_exp<font color='#5555FF'><</font>EXP<font color='#5555FF'>></font><font color='#5555FF'>&</font> m
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	requires
	- m.size() == NR
	- m.nr() == 1 \|\| m.nc() == 1 (i.e. m must be a row or column matrix)
	ensures
	- Initializes *this with the contents of m and does any rounding if necessary and also
	takes care of converting between 2 and 3 dimensional vectors.
	- if (m contains real valued values like float or double and T is an integral type like long) then
	- #x() == floor(m(0) + 0.5)
	- #y() == floor(m(1) + 0.5)
	- if (NR == 3) then
	- #z() == floor(m(2) + 0.5)
	- else
	- #z() == 0
	- else
	- #x() == m(0)
	- #y() == m(1)
	- if (NR == 3) then
	- #z() == m(2)
	- else
	- #z() == 0
	!*/</font>

	~<b><a name='vector'></a>vector</b> <font face='Lucida Console'>(</font>
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- all resources associated with *this have been released
	!*/</font>


	<font color='#0000FF'><u>double</u></font> <b><a name='length'></a>length</b><font face='Lucida Console'>(</font>
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- returns the length of the vector
	!*/</font>

	<font color='#0000FF'><u>double</u></font> <b><a name='length_squared'></a>length_squared</b><font face='Lucida Console'>(</font>
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- returns length()*length()
	!*/</font>

	T<font color='#5555FF'>&</font> <b><a name='x'></a>x</b> <font face='Lucida Console'>(</font>
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- returns a reference to the x component of the vector
	!*/</font>

	T<font color='#5555FF'>&</font> <b><a name='y'></a>y</b> <font face='Lucida Console'>(</font>
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- returns a reference to the y component of the vector
	!*/</font>

	T<font color='#5555FF'>&</font> <b><a name='z'></a>z</b> <font face='Lucida Console'>(</font>
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	requires
	- NR == 3 (this function actually doesn't exist when NR != 3)
	ensures
	- returns a reference to the z component of the vector
	!*/</font>

	<font color='#0000FF'>const</font> T<font color='#5555FF'>&</font> <b><a name='x'></a>x</b> <font face='Lucida Console'>(</font>
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- returns a const reference to the x component of the vector
	!*/</font>

	<font color='#0000FF'>const</font> T<font color='#5555FF'>&</font> <b><a name='y'></a>y</b> <font face='Lucida Console'>(</font>
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- returns a const reference to the y component of the vector
	!*/</font>

	<font color='#0000FF'>const</font> T<font color='#5555FF'>&</font> <b><a name='z'></a>z</b> <font face='Lucida Console'>(</font>
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- if (NR == 3) then
	- returns a const reference to the z component of the vector
	- else
	- return 0
	(there isn't really a z in this case so we just return 0)
	!*/</font>

	T <b><a name='dot'></a>dot</b> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> vector<font color='#5555FF'>&</font> rhs
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- returns the result of the dot product between *this and rhs
	!*/</font>

	vector<font color='#5555FF'><</font>T,<font color='#979000'>3</font><font color='#5555FF'>></font> <b><a name='cross'></a>cross</b> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> vector<font color='#5555FF'>&</font> rhs
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- returns the result of the cross product between *this and rhs
	!*/</font>

	vector<font color='#5555FF'><</font><font color='#0000FF'><u>double</u></font>,NR<font color='#5555FF'>></font> <b><a name='normalize'></a>normalize</b> <font face='Lucida Console'>(</font>
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- returns a vector with length() == 1 and in the same direction as *this
	!*/</font>

	vector <b><a name='operator'></a>operator</b><font color='#5555FF'>+</font> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> vector<font color='#5555FF'>&</font> rhs
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- returns the result of adding *this to rhs
	!*/</font>

	vector <b><a name='operator'></a>operator</b><font color='#5555FF'>-</font> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> vector<font color='#5555FF'>&</font> rhs
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- returns the result of subtracting rhs from *this
	!*/</font>

	vector <b><a name='operator'></a>operator</b><font color='#5555FF'>-</font> <font face='Lucida Console'>(</font>
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- returns -1(this)
	!*/</font>

	vector <b><a name='operator'></a>operator</b><font color='#5555FF'>/</font> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> T rhs
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- returns the result of dividing *this by rhs
	!*/</font>

	vector<font color='#5555FF'>&</font> <b><a name='operator'></a>operator</b><font color='#5555FF'>=</font> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> vector<font color='#5555FF'>&</font> rhs
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- #x() == rhs.x()
	- #y() == rhs.y()
	- #z() == rhs.z()
	- returns #*this
	!*/</font>

	vector<font color='#5555FF'>&</font> <b><a name='operator'></a>operator</b> <font color='#5555FF'>+</font><font color='#5555FF'>=</font> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> vector<font color='#5555FF'>&</font> rhs
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- #this == this + rhs
	- returns #*this
	!*/</font>

	vector<font color='#5555FF'>&</font> <b><a name='operator'></a>operator</b> <font color='#5555FF'>-</font><font color='#5555FF'>=</font> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> vector<font color='#5555FF'>&</font> rhs
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- #this == this - rhs
	- returns #*this
	!*/</font>

	vector<font color='#5555FF'>&</font> <b><a name='operator'></a>operator</b> <font color='#5555FF'>*</font><font color='#5555FF'>=</font> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> T rhs
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- #this == this * rhs
	- returns #*this
	!*/</font>

	vector<font color='#5555FF'>&</font> <b><a name='operator'></a>operator</b> <font color='#5555FF'>/</font><font color='#5555FF'>=</font> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> T rhs
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- #this == this / rhs
	- returns #*this
	!*/</font>

	<font color='#0000FF'>template</font> <font color='#5555FF'><</font><font color='#0000FF'>typename</font> U, <font color='#0000FF'><u>long</u></font> NR2<font color='#5555FF'>></font>
	<font color='#0000FF'><u>bool</u></font> <b><a name='operator'></a>operator</b><font color='#5555FF'>=</font><font color='#5555FF'>=</font> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> vector<font color='#5555FF'><</font>U,NR2<font color='#5555FF'>></font><font color='#5555FF'>&</font> rhs
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- if (x() == rhs.x() && y() == rhs.y() && z() == rhs.z()) then
	- returns true
	- else
	- returns false
	!*/</font>

	<font color='#0000FF'>template</font> <font color='#5555FF'><</font><font color='#0000FF'>typename</font> U, <font color='#0000FF'><u>long</u></font> NR2<font color='#5555FF'>></font>
	<font color='#0000FF'><u>bool</u></font> <b><a name='operator'></a>operator</b><font color='#5555FF'>!</font><font color='#5555FF'>=</font> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> vector<font color='#5555FF'><</font>U,NR2<font color='#5555FF'>></font><font color='#5555FF'>&</font> rhs
	<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>;
	<font color='#009900'>/*!
	ensures
	- returns !((*this) == rhs)
	!*/</font>

	<font color='#0000FF'><u>void</u></font> <b><a name='swap'></a>swap</b> <font face='Lucida Console'>(</font>
	vector<font color='#5555FF'>&</font> item
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- swaps *this and item
	!*/</font>

	<b>}</b>;

	<font color='#009900'>// ----------------------------------------------------------------------------------------
	</font>
	<font color='#0000FF'>template</font><font color='#5555FF'><</font><font color='#0000FF'>typename</font> T, <font color='#0000FF'>typename</font> U, <font color='#0000FF'><u>long</u></font> NR<font color='#5555FF'>></font>
	vector <b><a name='operator'></a>operator</b><font color='#5555FF'>*</font> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> vector<font color='#5555FF'><</font>T,NR<font color='#5555FF'>></font> <font color='#5555FF'>&</font> lhs,
	<font color='#0000FF'>const</font> U rhs
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- returns the result of multiplying the scalar rhs by lhs
	!*/</font>

	<font color='#0000FF'>template</font><font color='#5555FF'><</font><font color='#0000FF'>typename</font> T, <font color='#0000FF'>typename</font> U, <font color='#0000FF'><u>long</u></font> NR<font color='#5555FF'>></font>
	vector <b><a name='operator'></a>operator</b><font color='#5555FF'>*</font> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> U lhs,
	<font color='#0000FF'>const</font> vector<font color='#5555FF'><</font>T,NR<font color='#5555FF'>></font> <font color='#5555FF'>&</font> rhs
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- returns the result of multiplying the scalar lhs by rhs
	!*/</font>

	<font color='#0000FF'>template</font><font color='#5555FF'><</font><font color='#0000FF'>typename</font> T, <font color='#0000FF'><u>long</u></font> NR<font color='#5555FF'>></font>
	<font color='#0000FF'>inline</font> <font color='#0000FF'><u>void</u></font> <b><a name='swap'></a>swap</b> <font face='Lucida Console'>(</font>
	vector<font color='#5555FF'><</font>T,NR<font color='#5555FF'>></font> <font color='#5555FF'>&</font> a,
	vector<font color='#5555FF'><</font>T,NR<font color='#5555FF'>></font> <font color='#5555FF'>&</font> b
	<font face='Lucida Console'>)</font> <b>{</b> a.<font color='#BB00BB'>swap</font><font face='Lucida Console'>(</font>b<font face='Lucida Console'>)</font>; <b>}</b>
	<font color='#009900'>/*!
	provides a global swap function
	!*/</font>

	<font color='#0000FF'>template</font><font color='#5555FF'><</font><font color='#0000FF'>typename</font> T, <font color='#0000FF'><u>long</u></font> NR<font color='#5555FF'>></font>
	<font color='#0000FF'><u>void</u></font> <b><a name='serialize'></a>serialize</b> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> vector<font color='#5555FF'><</font>T,NR<font color='#5555FF'>></font><font color='#5555FF'>&</font> item,
	std::ostream<font color='#5555FF'>&</font> out
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	provides serialization support
	!*/</font>

	<font color='#0000FF'>template</font><font color='#5555FF'><</font><font color='#0000FF'>typename</font> T, <font color='#0000FF'><u>long</u></font> NR<font color='#5555FF'>></font>
	<font color='#0000FF'><u>void</u></font> <b><a name='deserialize'></a>deserialize</b> <font face='Lucida Console'>(</font>
	vector<font color='#5555FF'><</font>T,NR<font color='#5555FF'>></font><font color='#5555FF'>&</font> item,
	std::istream<font color='#5555FF'>&</font> in
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	provides deserialization support
	!*/</font>

	<font color='#0000FF'>template</font><font color='#5555FF'><</font><font color='#0000FF'>typename</font> T<font color='#5555FF'>></font>
	std::ostream<font color='#5555FF'>&</font> <b><a name='operator'></a>operator</b><font color='#5555FF'><</font><font color='#5555FF'><</font> <font face='Lucida Console'>(</font>
	std::ostream<font color='#5555FF'>&</font> out,
	<font color='#0000FF'>const</font> vector<font color='#5555FF'><</font>T,<font color='#979000'>3</font><font color='#5555FF'>></font><font color='#5555FF'>&</font> item
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- writes item to out in the form "(x, y, z)"
	!*/</font>

	<font color='#0000FF'>template</font><font color='#5555FF'><</font><font color='#0000FF'>typename</font> T<font color='#5555FF'>></font>
	std::istream<font color='#5555FF'>&</font> <b><a name='operator'></a>operator</b><font color='#5555FF'>></font><font color='#5555FF'>></font><font face='Lucida Console'>(</font>
	std::istream<font color='#5555FF'>&</font> in,
	vector<font color='#5555FF'><</font>T,<font color='#979000'>3</font><font color='#5555FF'>></font><font color='#5555FF'>&</font> item
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- reads a vector from the input stream in and stores it in #item.
	The data in the input stream should be of the form (x, y, z)
	!*/</font>

	<font color='#0000FF'>template</font><font color='#5555FF'><</font><font color='#0000FF'>typename</font> T<font color='#5555FF'>></font>
	std::ostream<font color='#5555FF'>&</font> <b><a name='operator'></a>operator</b><font color='#5555FF'><</font><font color='#5555FF'><</font> <font face='Lucida Console'>(</font>
	std::ostream<font color='#5555FF'>&</font> out,
	<font color='#0000FF'>const</font> vector<font color='#5555FF'><</font>T,<font color='#979000'>2</font><font color='#5555FF'>></font><font color='#5555FF'>&</font> item
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- writes item to out in the form "(x, y)"
	!*/</font>

	<font color='#0000FF'>template</font><font color='#5555FF'><</font><font color='#0000FF'>typename</font> T<font color='#5555FF'>></font>
	std::istream<font color='#5555FF'>&</font> <b><a name='operator'></a>operator</b><font color='#5555FF'>></font><font color='#5555FF'>></font><font face='Lucida Console'>(</font>
	std::istream<font color='#5555FF'>&</font> in,
	vector<font color='#5555FF'><</font>T,<font color='#979000'>2</font><font color='#5555FF'>></font><font color='#5555FF'>&</font> item
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- reads a vector from the input stream in and stores it in #item.
	The data in the input stream should be of the form (x, y)
	!*/</font>

	<font color='#009900'>// ----------------------------------------------------------------------------------------
	</font>
	<a name='point'/><font color='#009900'>/*!A point
	This is just a typedef of the vector object.
	!*/</font>

	<font color='#0000FF'>typedef</font> vector<font color='#5555FF'><</font><font color='#0000FF'><u>long</u></font>,<font color='#979000'>2</font><font color='#5555FF'>></font> point;

	<a name='dpoint'/><font color='#009900'>/*!A dpoint
	This is just a typedef of the vector object.
	!*/</font>

	<font color='#0000FF'>typedef</font> vector<font color='#5555FF'><</font><font color='#0000FF'><u>double</u></font>,<font color='#979000'>2</font><font color='#5555FF'>></font> dpoint;

	<font color='#009900'>// ----------------------------------------------------------------------------------------
	</font>
	<font color='#0000FF'><u>bool</u></font> <b><a name='is_convex_quadrilateral'></a>is_convex_quadrilateral</b> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> std::array<font color='#5555FF'><</font>dpoint,<font color='#979000'>4</font><font color='#5555FF'>></font><font color='#5555FF'>&</font> pts
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	ensures
	- If you walk the points in pts in order pts[0], pts[1], pts[2], pts[3], pts[0]
	does it draw a convex quadrilateral? This routine returns true if yes and
	false if not.
	!*/</font>

	<font color='#009900'>// ----------------------------------------------------------------------------------------
	</font>
	<font color='#0000FF'>template</font> <font color='#5555FF'><</font>
	<font color='#0000FF'>typename</font> array_of_dpoints
	<font color='#5555FF'>></font>
	<font color='#0000FF'><u>double</u></font> <b><a name='polygon_area'></a>polygon_area</b> <font face='Lucida Console'>(</font>
	<font color='#0000FF'>const</font> array_of_dpoints<font color='#5555FF'>&</font> pts
	<font face='Lucida Console'>)</font>;
	<font color='#009900'>/*!
	requires
	- array_of_dpoints is something with an interface compatible with
	std::vector<dpoint> or std::array<dpoint,N>.
	ensures
	- If you walk the points pts in order to make a closed polygon, what is its
	area? This function returns that area. It uses the shoelace formula to
	compute the result and so works for general non-self-intersecting polygons.
	!*/</font>

	<font color='#009900'>// ----------------------------------------------------------------------------------------
	</font>
	<b>}</b>

	<font color='#0000FF'>namespace</font> std
	<b>{</b>
	<font color='#009900'>/*!
	Define std::less<vector<T,3> > so that you can use vectors in the associative containers.
	!*/</font>
	<font color='#0000FF'>template</font><font color='#5555FF'><</font><font color='#0000FF'>typename</font> T<font color='#5555FF'>></font>
	<font color='#0000FF'>struct</font> <b><a name='less'></a>less</b><font color='#5555FF'><</font>dlib::vector<font color='#5555FF'><</font>T,<font color='#979000'>3</font><font color='#5555FF'>></font> <font color='#5555FF'>></font> : <font color='#0000FF'>public</font> binary_function<font color='#5555FF'><</font>dlib::vector<font color='#5555FF'><</font>T,<font color='#979000'>3</font><font color='#5555FF'>></font> ,dlib::vector<font color='#5555FF'><</font>T,<font color='#979000'>3</font><font color='#5555FF'>></font> ,<font color='#0000FF'><u>bool</u></font><font color='#5555FF'>></font>
	<b>{</b>
	<font color='#0000FF'>inline</font> <font color='#0000FF'><u>bool</u></font> <b><a name='operator'></a>operator</b><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font> <font face='Lucida Console'>(</font><font color='#0000FF'>const</font> dlib::vector<font color='#5555FF'><</font>T,<font color='#979000'>3</font><font color='#5555FF'>></font> <font color='#5555FF'>&</font> a, <font color='#0000FF'>const</font> dlib::vector<font color='#5555FF'><</font>T,<font color='#979000'>3</font><font color='#5555FF'>></font> <font color='#5555FF'>&</font> b<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>
	<b>{</b>
	<font color='#0000FF'>if</font> <font face='Lucida Console'>(</font>a.<font color='#BB00BB'>x</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font> <font color='#5555FF'><</font> b.<font color='#BB00BB'>x</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font><font face='Lucida Console'>)</font> <font color='#0000FF'>return</font> <font color='#979000'>true</font>;
	<font color='#0000FF'>else</font> <font color='#0000FF'>if</font> <font face='Lucida Console'>(</font>a.<font color='#BB00BB'>x</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font> <font color='#5555FF'>></font> b.<font color='#BB00BB'>x</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font><font face='Lucida Console'>)</font> <font color='#0000FF'>return</font> <font color='#979000'>false</font>;
	<font color='#0000FF'>else</font> <font color='#0000FF'>if</font> <font face='Lucida Console'>(</font>a.<font color='#BB00BB'>y</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font> <font color='#5555FF'><</font> b.<font color='#BB00BB'>y</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font><font face='Lucida Console'>)</font> <font color='#0000FF'>return</font> <font color='#979000'>true</font>;
	<font color='#0000FF'>else</font> <font color='#0000FF'>if</font> <font face='Lucida Console'>(</font>a.<font color='#BB00BB'>y</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font> <font color='#5555FF'>></font> b.<font color='#BB00BB'>y</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font><font face='Lucida Console'>)</font> <font color='#0000FF'>return</font> <font color='#979000'>false</font>;
	<font color='#0000FF'>else</font> <font color='#0000FF'>if</font> <font face='Lucida Console'>(</font>a.<font color='#BB00BB'>z</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font> <font color='#5555FF'><</font> b.<font color='#BB00BB'>z</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font><font face='Lucida Console'>)</font> <font color='#0000FF'>return</font> <font color='#979000'>true</font>;
	<font color='#0000FF'>else</font> <font color='#0000FF'>if</font> <font face='Lucida Console'>(</font>a.<font color='#BB00BB'>z</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font> <font color='#5555FF'>></font> b.<font color='#BB00BB'>z</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font><font face='Lucida Console'>)</font> <font color='#0000FF'>return</font> <font color='#979000'>false</font>;
	<font color='#0000FF'>else</font> <font color='#0000FF'>return</font> <font color='#979000'>false</font>;
	<b>}</b>
	<b>}</b>;

	<font color='#009900'>/*!
	Define std::less<vector<T,2> > so that you can use vector<T,2>s in the associative containers.
	!*/</font>
	<font color='#0000FF'>template</font><font color='#5555FF'><</font><font color='#0000FF'>typename</font> T<font color='#5555FF'>></font>
	<font color='#0000FF'>struct</font> <b><a name='less'></a>less</b><font color='#5555FF'><</font>dlib::vector<font color='#5555FF'><</font>T,<font color='#979000'>2</font><font color='#5555FF'>></font> <font color='#5555FF'>></font> : <font color='#0000FF'>public</font> binary_function<font color='#5555FF'><</font>dlib::vector<font color='#5555FF'><</font>T,<font color='#979000'>2</font><font color='#5555FF'>></font> ,dlib::vector<font color='#5555FF'><</font>T,<font color='#979000'>2</font><font color='#5555FF'>></font> ,<font color='#0000FF'><u>bool</u></font><font color='#5555FF'>></font>
	<b>{</b>
	<font color='#0000FF'>inline</font> <font color='#0000FF'><u>bool</u></font> <b><a name='operator'></a>operator</b><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font> <font face='Lucida Console'>(</font><font color='#0000FF'>const</font> dlib::vector<font color='#5555FF'><</font>T,<font color='#979000'>2</font><font color='#5555FF'>></font> <font color='#5555FF'>&</font> a, <font color='#0000FF'>const</font> dlib::vector<font color='#5555FF'><</font>T,<font color='#979000'>2</font><font color='#5555FF'>></font> <font color='#5555FF'>&</font> b<font face='Lucida Console'>)</font> <font color='#0000FF'>const</font>
	<b>{</b>
	<font color='#0000FF'>if</font> <font face='Lucida Console'>(</font>a.<font color='#BB00BB'>x</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font> <font color='#5555FF'><</font> b.<font color='#BB00BB'>x</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font><font face='Lucida Console'>)</font> <font color='#0000FF'>return</font> <font color='#979000'>true</font>;
	<font color='#0000FF'>else</font> <font color='#0000FF'>if</font> <font face='Lucida Console'>(</font>a.<font color='#BB00BB'>x</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font> <font color='#5555FF'>></font> b.<font color='#BB00BB'>x</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font><font face='Lucida Console'>)</font> <font color='#0000FF'>return</font> <font color='#979000'>false</font>;
	<font color='#0000FF'>else</font> <font color='#0000FF'>if</font> <font face='Lucida Console'>(</font>a.<font color='#BB00BB'>y</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font> <font color='#5555FF'><</font> b.<font color='#BB00BB'>y</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font><font face='Lucida Console'>)</font> <font color='#0000FF'>return</font> <font color='#979000'>true</font>;
	<font color='#0000FF'>else</font> <font color='#0000FF'>if</font> <font face='Lucida Console'>(</font>a.<font color='#BB00BB'>y</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font> <font color='#5555FF'>></font> b.<font color='#BB00BB'>y</font><font face='Lucida Console'>(</font><font face='Lucida Console'>)</font><font face='Lucida Console'>)</font> <font color='#0000FF'>return</font> <font color='#979000'>false</font>;
	<font color='#0000FF'>else</font> <font color='#0000FF'>return</font> <font color='#979000'>false</font>;
	<b>}</b>
	<b>}</b>;
	<b>}</b>

	<font color='#0000FF'>#endif</font> <font color='#009900'>// DLIB_VECTOR_ABSTRACT_
	</font>

	</pre></body></html>