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rn.py
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rn.py
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"""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 __future__ import print_function, division
from .diffgeom import Manifold, Patch, CoordSystem
from sympy import sqrt, atan2, acos, sin, cos, Dummy
###############################################################################
# R2
###############################################################################
R2 = Manifold('R^2', 2)
# Patch and coordinate systems.
R2_origin = Patch('origin', R2)
R2_r = CoordSystem('rectangular', R2_origin, ['x', 'y'])
R2_p = CoordSystem('polar', R2_origin, ['r', 'theta'])
# Connecting the coordinate charts.
x, y, r, theta = [Dummy(s) for s in ['x', 'y', 'r', 'theta']]
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)
del x, y, r, theta
# 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 = Manifold('R^3', 3)
# Patch and coordinate systems.
R3_origin = Patch('origin', R3)
R3_r = CoordSystem('rectangular', R3_origin, ['x', 'y', 'z'])
R3_c = CoordSystem('cylindrical', R3_origin, ['rho', 'psi', 'z'])
R3_s = CoordSystem('spherical', R3_origin, ['r', 'theta', 'phi'])
# Connecting the coordinate charts.
x, y, z, rho, psi, r, theta, phi = [Dummy(s) for s in ['x', 'y', 'z',
'rho', 'psi', 'r', 'theta', 'phi']]
## rectangular <-> cylindrical
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)
del x, y, z, rho, psi, r, theta, phi
# 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()