Spaces:
Sleeping
Sleeping
File size: 6,264 Bytes
6a86ad5 |
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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 |
"""Predefined R^n manifolds together with common coord. systems.
Coordinate systems are predefined as well as the transformation laws between
them.
Coordinate functions can be accessed as attributes of the manifold (eg `R2.x`),
as attributes of the coordinate systems (eg `R2_r.x` and `R2_p.theta`), or by
using the usual `coord_sys.coord_function(index, name)` interface.
"""
from typing import Any
import warnings
from sympy.core.symbol import (Dummy, symbols)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (acos, atan2, cos, sin)
from .diffgeom import Manifold, Patch, CoordSystem
__all__ = [
'R2', 'R2_origin', 'relations_2d', 'R2_r', 'R2_p',
'R3', 'R3_origin', 'relations_3d', 'R3_r', 'R3_c', 'R3_s'
]
###############################################################################
# R2
###############################################################################
R2: Any = Manifold('R^2', 2)
R2_origin: Any = Patch('origin', R2)
x, y = symbols('x y', real=True)
r, theta = symbols('rho theta', nonnegative=True)
relations_2d = {
('rectangular', 'polar'): [(x, y), (sqrt(x**2 + y**2), atan2(y, x))],
('polar', 'rectangular'): [(r, theta), (r*cos(theta), r*sin(theta))],
}
R2_r: Any = CoordSystem('rectangular', R2_origin, (x, y), relations_2d)
R2_p: Any = CoordSystem('polar', R2_origin, (r, theta), relations_2d)
# support deprecated feature
with warnings.catch_warnings():
warnings.simplefilter("ignore")
x, y, r, theta = symbols('x y r theta', cls=Dummy)
R2_r.connect_to(R2_p, [x, y],
[sqrt(x**2 + y**2), atan2(y, x)],
inverse=False, fill_in_gaps=False)
R2_p.connect_to(R2_r, [r, theta],
[r*cos(theta), r*sin(theta)],
inverse=False, fill_in_gaps=False)
# Defining the basis coordinate functions and adding shortcuts for them to the
# manifold and the patch.
R2.x, R2.y = R2_origin.x, R2_origin.y = R2_r.x, R2_r.y = R2_r.coord_functions()
R2.r, R2.theta = R2_origin.r, R2_origin.theta = R2_p.r, R2_p.theta = R2_p.coord_functions()
# Defining the basis vector fields and adding shortcuts for them to the
# manifold and the patch.
R2.e_x, R2.e_y = R2_origin.e_x, R2_origin.e_y = R2_r.e_x, R2_r.e_y = R2_r.base_vectors()
R2.e_r, R2.e_theta = R2_origin.e_r, R2_origin.e_theta = R2_p.e_r, R2_p.e_theta = R2_p.base_vectors()
# Defining the basis oneform fields and adding shortcuts for them to the
# manifold and the patch.
R2.dx, R2.dy = R2_origin.dx, R2_origin.dy = R2_r.dx, R2_r.dy = R2_r.base_oneforms()
R2.dr, R2.dtheta = R2_origin.dr, R2_origin.dtheta = R2_p.dr, R2_p.dtheta = R2_p.base_oneforms()
###############################################################################
# R3
###############################################################################
R3: Any = Manifold('R^3', 3)
R3_origin: Any = Patch('origin', R3)
x, y, z = symbols('x y z', real=True)
rho, psi, r, theta, phi = symbols('rho psi r theta phi', nonnegative=True)
relations_3d = {
('rectangular', 'cylindrical'): [(x, y, z),
(sqrt(x**2 + y**2), atan2(y, x), z)],
('cylindrical', 'rectangular'): [(rho, psi, z),
(rho*cos(psi), rho*sin(psi), z)],
('rectangular', 'spherical'): [(x, y, z),
(sqrt(x**2 + y**2 + z**2),
acos(z/sqrt(x**2 + y**2 + z**2)),
atan2(y, x))],
('spherical', 'rectangular'): [(r, theta, phi),
(r*sin(theta)*cos(phi),
r*sin(theta)*sin(phi),
r*cos(theta))],
('cylindrical', 'spherical'): [(rho, psi, z),
(sqrt(rho**2 + z**2),
acos(z/sqrt(rho**2 + z**2)),
psi)],
('spherical', 'cylindrical'): [(r, theta, phi),
(r*sin(theta), phi, r*cos(theta))],
}
R3_r: Any = CoordSystem('rectangular', R3_origin, (x, y, z), relations_3d)
R3_c: Any = CoordSystem('cylindrical', R3_origin, (rho, psi, z), relations_3d)
R3_s: Any = CoordSystem('spherical', R3_origin, (r, theta, phi), relations_3d)
# support deprecated feature
with warnings.catch_warnings():
warnings.simplefilter("ignore")
x, y, z, rho, psi, r, theta, phi = symbols('x y z rho psi r theta phi', cls=Dummy)
R3_r.connect_to(R3_c, [x, y, z],
[sqrt(x**2 + y**2), atan2(y, x), z],
inverse=False, fill_in_gaps=False)
R3_c.connect_to(R3_r, [rho, psi, z],
[rho*cos(psi), rho*sin(psi), z],
inverse=False, fill_in_gaps=False)
## rectangular <-> spherical
R3_r.connect_to(R3_s, [x, y, z],
[sqrt(x**2 + y**2 + z**2), acos(z/
sqrt(x**2 + y**2 + z**2)), atan2(y, x)],
inverse=False, fill_in_gaps=False)
R3_s.connect_to(R3_r, [r, theta, phi],
[r*sin(theta)*cos(phi), r*sin(
theta)*sin(phi), r*cos(theta)],
inverse=False, fill_in_gaps=False)
## cylindrical <-> spherical
R3_c.connect_to(R3_s, [rho, psi, z],
[sqrt(rho**2 + z**2), acos(z/sqrt(rho**2 + z**2)), psi],
inverse=False, fill_in_gaps=False)
R3_s.connect_to(R3_c, [r, theta, phi],
[r*sin(theta), phi, r*cos(theta)],
inverse=False, fill_in_gaps=False)
# Defining the basis coordinate functions.
R3_r.x, R3_r.y, R3_r.z = R3_r.coord_functions()
R3_c.rho, R3_c.psi, R3_c.z = R3_c.coord_functions()
R3_s.r, R3_s.theta, R3_s.phi = R3_s.coord_functions()
# Defining the basis vector fields.
R3_r.e_x, R3_r.e_y, R3_r.e_z = R3_r.base_vectors()
R3_c.e_rho, R3_c.e_psi, R3_c.e_z = R3_c.base_vectors()
R3_s.e_r, R3_s.e_theta, R3_s.e_phi = R3_s.base_vectors()
# Defining the basis oneform fields.
R3_r.dx, R3_r.dy, R3_r.dz = R3_r.base_oneforms()
R3_c.drho, R3_c.dpsi, R3_c.dz = R3_c.base_oneforms()
R3_s.dr, R3_s.dtheta, R3_s.dphi = R3_s.base_oneforms()
|