144 lines
6.1 KiB
Python
144 lines
6.1 KiB
Python
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"""Predefined R^n manifolds together with common coord. systems.
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Coordinate systems are predefined as well as the transformation laws between
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them.
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Coordinate functions can be accessed as attributes of the manifold (eg `R2.x`),
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as attributes of the coordinate systems (eg `R2_r.x` and `R2_p.theta`), or by
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using the usual `coord_sys.coord_function(index, name)` interface.
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"""
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from typing import Any
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import warnings
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from sympy.core.symbol import (Dummy, symbols)
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.trigonometric import (acos, atan2, cos, sin)
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from .diffgeom import Manifold, Patch, CoordSystem
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__all__ = [
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'R2', 'R2_origin', 'relations_2d', 'R2_r', 'R2_p',
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'R3', 'R3_origin', 'relations_3d', 'R3_r', 'R3_c', 'R3_s'
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]
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###############################################################################
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# R2
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###############################################################################
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R2: Any = Manifold('R^2', 2)
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R2_origin: Any = Patch('origin', R2)
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x, y = symbols('x y', real=True)
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r, theta = symbols('rho theta', nonnegative=True)
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relations_2d = {
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('rectangular', 'polar'): [(x, y), (sqrt(x**2 + y**2), atan2(y, x))],
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('polar', 'rectangular'): [(r, theta), (r*cos(theta), r*sin(theta))],
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}
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R2_r: Any = CoordSystem('rectangular', R2_origin, (x, y), relations_2d)
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R2_p: Any = CoordSystem('polar', R2_origin, (r, theta), relations_2d)
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# support deprecated feature
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with warnings.catch_warnings():
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warnings.simplefilter("ignore")
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x, y, r, theta = symbols('x y r theta', cls=Dummy)
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R2_r.connect_to(R2_p, [x, y],
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[sqrt(x**2 + y**2), atan2(y, x)],
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inverse=False, fill_in_gaps=False)
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R2_p.connect_to(R2_r, [r, theta],
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[r*cos(theta), r*sin(theta)],
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inverse=False, fill_in_gaps=False)
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# Defining the basis coordinate functions and adding shortcuts for them to the
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# manifold and the patch.
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R2.x, R2.y = R2_origin.x, R2_origin.y = R2_r.x, R2_r.y = R2_r.coord_functions()
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R2.r, R2.theta = R2_origin.r, R2_origin.theta = R2_p.r, R2_p.theta = R2_p.coord_functions()
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# Defining the basis vector fields and adding shortcuts for them to the
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# manifold and the patch.
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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()
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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()
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# Defining the basis oneform fields and adding shortcuts for them to the
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# manifold and the patch.
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R2.dx, R2.dy = R2_origin.dx, R2_origin.dy = R2_r.dx, R2_r.dy = R2_r.base_oneforms()
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R2.dr, R2.dtheta = R2_origin.dr, R2_origin.dtheta = R2_p.dr, R2_p.dtheta = R2_p.base_oneforms()
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###############################################################################
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# R3
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###############################################################################
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R3: Any = Manifold('R^3', 3)
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R3_origin: Any = Patch('origin', R3)
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x, y, z = symbols('x y z', real=True)
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rho, psi, r, theta, phi = symbols('rho psi r theta phi', nonnegative=True)
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relations_3d = {
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('rectangular', 'cylindrical'): [(x, y, z),
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(sqrt(x**2 + y**2), atan2(y, x), z)],
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('cylindrical', 'rectangular'): [(rho, psi, z),
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(rho*cos(psi), rho*sin(psi), z)],
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('rectangular', 'spherical'): [(x, y, z),
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(sqrt(x**2 + y**2 + z**2),
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acos(z/sqrt(x**2 + y**2 + z**2)),
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atan2(y, x))],
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('spherical', 'rectangular'): [(r, theta, phi),
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(r*sin(theta)*cos(phi),
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r*sin(theta)*sin(phi),
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r*cos(theta))],
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('cylindrical', 'spherical'): [(rho, psi, z),
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(sqrt(rho**2 + z**2),
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acos(z/sqrt(rho**2 + z**2)),
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psi)],
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('spherical', 'cylindrical'): [(r, theta, phi),
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(r*sin(theta), phi, r*cos(theta))],
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}
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R3_r: Any = CoordSystem('rectangular', R3_origin, (x, y, z), relations_3d)
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R3_c: Any = CoordSystem('cylindrical', R3_origin, (rho, psi, z), relations_3d)
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R3_s: Any = CoordSystem('spherical', R3_origin, (r, theta, phi), relations_3d)
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# support deprecated feature
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with warnings.catch_warnings():
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warnings.simplefilter("ignore")
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x, y, z, rho, psi, r, theta, phi = symbols('x y z rho psi r theta phi', cls=Dummy)
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R3_r.connect_to(R3_c, [x, y, z],
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[sqrt(x**2 + y**2), atan2(y, x), z],
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inverse=False, fill_in_gaps=False)
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R3_c.connect_to(R3_r, [rho, psi, z],
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[rho*cos(psi), rho*sin(psi), z],
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inverse=False, fill_in_gaps=False)
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## rectangular <-> spherical
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R3_r.connect_to(R3_s, [x, y, z],
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[sqrt(x**2 + y**2 + z**2), acos(z/
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sqrt(x**2 + y**2 + z**2)), atan2(y, x)],
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inverse=False, fill_in_gaps=False)
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R3_s.connect_to(R3_r, [r, theta, phi],
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[r*sin(theta)*cos(phi), r*sin(
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theta)*sin(phi), r*cos(theta)],
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inverse=False, fill_in_gaps=False)
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## cylindrical <-> spherical
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R3_c.connect_to(R3_s, [rho, psi, z],
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[sqrt(rho**2 + z**2), acos(z/sqrt(rho**2 + z**2)), psi],
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inverse=False, fill_in_gaps=False)
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R3_s.connect_to(R3_c, [r, theta, phi],
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[r*sin(theta), phi, r*cos(theta)],
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inverse=False, fill_in_gaps=False)
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# Defining the basis coordinate functions.
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R3_r.x, R3_r.y, R3_r.z = R3_r.coord_functions()
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R3_c.rho, R3_c.psi, R3_c.z = R3_c.coord_functions()
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R3_s.r, R3_s.theta, R3_s.phi = R3_s.coord_functions()
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# Defining the basis vector fields.
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R3_r.e_x, R3_r.e_y, R3_r.e_z = R3_r.base_vectors()
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R3_c.e_rho, R3_c.e_psi, R3_c.e_z = R3_c.base_vectors()
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R3_s.e_r, R3_s.e_theta, R3_s.e_phi = R3_s.base_vectors()
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# Defining the basis oneform fields.
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R3_r.dx, R3_r.dy, R3_r.dz = R3_r.base_oneforms()
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R3_c.drho, R3_c.dpsi, R3_c.dz = R3_c.base_oneforms()
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R3_s.dr, R3_s.dtheta, R3_s.dphi = R3_s.base_oneforms()
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