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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 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() | |