from sympy.utilities.exceptions import SymPyDeprecationWarning from sympy.utilities.pytest import raises import warnings from sympy.vector.coordsysrect import CoordSys3D, CoordSysCartesian from sympy.vector.scalar import BaseScalar from sympy import sin, sinh, cos, cosh, sqrt, pi, ImmutableMatrix as Matrix, \ symbols, simplify, zeros, expand, acos, atan2 from sympy.vector.functions import express from sympy.vector.point import Point from sympy.vector.vector import Vector from sympy.vector.orienters import (AxisOrienter, BodyOrienter, SpaceOrienter, QuaternionOrienter) x, y, z = symbols('x y z') a, b, c, q = symbols('a b c q') q1, q2, q3, q4 = symbols('q1 q2 q3 q4') def test_func_args(): A = CoordSys3D('A') assert A.x.func(*A.x.args) == A.x expr = 3*A.x + 4*A.y assert expr.func(*expr.args) == expr assert A.i.func(*A.i.args) == A.i v = A.x*A.i + A.y*A.j + A.z*A.k assert v.func(*v.args) == v assert A.origin.func(*A.origin.args) == A.origin def test_coordsyscartesian_equivalence(): A = CoordSys3D('A') A1 = CoordSys3D('A') assert A1 == A B = CoordSys3D('B') assert A != B def test_orienters(): A = CoordSys3D('A') axis_orienter = AxisOrienter(a, A.k) body_orienter = BodyOrienter(a, b, c, '123') space_orienter = SpaceOrienter(a, b, c, '123') q_orienter = QuaternionOrienter(q1, q2, q3, q4) assert axis_orienter.rotation_matrix(A) == Matrix([ [ cos(a), sin(a), 0], [-sin(a), cos(a), 0], [ 0, 0, 1]]) assert body_orienter.rotation_matrix() == Matrix([ [ cos(b)*cos(c), sin(a)*sin(b)*cos(c) + sin(c)*cos(a), sin(a)*sin(c) - sin(b)*cos(a)*cos(c)], [-sin(c)*cos(b), -sin(a)*sin(b)*sin(c) + cos(a)*cos(c), sin(a)*cos(c) + sin(b)*sin(c)*cos(a)], [ sin(b), -sin(a)*cos(b), cos(a)*cos(b)]]) assert space_orienter.rotation_matrix() == Matrix([ [cos(b)*cos(c), sin(c)*cos(b), -sin(b)], [sin(a)*sin(b)*cos(c) - sin(c)*cos(a), sin(a)*sin(b)*sin(c) + cos(a)*cos(c), sin(a)*cos(b)], [sin(a)*sin(c) + sin(b)*cos(a)*cos(c), -sin(a)*cos(c) + sin(b)*sin(c)*cos(a), cos(a)*cos(b)]]) assert q_orienter.rotation_matrix() == Matrix([ [q1**2 + q2**2 - q3**2 - q4**2, 2*q1*q4 + 2*q2*q3, -2*q1*q3 + 2*q2*q4], [-2*q1*q4 + 2*q2*q3, q1**2 - q2**2 + q3**2 - q4**2, 2*q1*q2 + 2*q3*q4], [2*q1*q3 + 2*q2*q4, -2*q1*q2 + 2*q3*q4, q1**2 - q2**2 - q3**2 + q4**2]]) def test_coordinate_vars(): """ Tests the coordinate variables functionality with respect to reorientation of coordinate systems. """ A = CoordSys3D('A') # Note that the name given on the lhs is different from A.x._name assert BaseScalar(0, A, 'A_x', r'\mathbf{{x}_{A}}') == A.x assert BaseScalar(1, A, 'A_y', r'\mathbf{{y}_{A}}') == A.y assert BaseScalar(2, A, 'A_z', r'\mathbf{{z}_{A}}') == A.z assert BaseScalar(0, A, 'A_x', r'\mathbf{{x}_{A}}').__hash__() == A.x.__hash__() assert isinstance(A.x, BaseScalar) and \ isinstance(A.y, BaseScalar) and \ isinstance(A.z, BaseScalar) assert A.x*A.y == A.y*A.x assert A.scalar_map(A) == {A.x: A.x, A.y: A.y, A.z: A.z} assert A.x.system == A assert A.x.diff(A.x) == 1 B = A.orient_new_axis('B', q, A.k) assert B.scalar_map(A) == {B.z: A.z, B.y: -A.x*sin(q) + A.y*cos(q), B.x: A.x*cos(q) + A.y*sin(q)} assert A.scalar_map(B) == {A.x: B.x*cos(q) - B.y*sin(q), A.y: B.x*sin(q) + B.y*cos(q), A.z: B.z} assert express(B.x, A, variables=True) == A.x*cos(q) + A.y*sin(q) assert express(B.y, A, variables=True) == -A.x*sin(q) + A.y*cos(q) assert express(B.z, A, variables=True) == A.z assert expand(express(B.x*B.y*B.z, A, variables=True)) == \ expand(A.z*(-A.x*sin(q) + A.y*cos(q))*(A.x*cos(q) + A.y*sin(q))) assert express(B.x*B.i + B.y*B.j + B.z*B.k, A) == \ (B.x*cos(q) - B.y*sin(q))*A.i + (B.x*sin(q) + \ B.y*cos(q))*A.j + B.z*A.k assert simplify(express(B.x*B.i + B.y*B.j + B.z*B.k, A, \ variables=True)) == \ A.x*A.i + A.y*A.j + A.z*A.k assert express(A.x*A.i + A.y*A.j + A.z*A.k, B) == \ (A.x*cos(q) + A.y*sin(q))*B.i + \ (-A.x*sin(q) + A.y*cos(q))*B.j + A.z*B.k assert simplify(express(A.x*A.i + A.y*A.j + A.z*A.k, B, \ variables=True)) == \ B.x*B.i + B.y*B.j + B.z*B.k N = B.orient_new_axis('N', -q, B.k) assert N.scalar_map(A) == \ {N.x: A.x, N.z: A.z, N.y: A.y} C = A.orient_new_axis('C', q, A.i + A.j + A.k) mapping = A.scalar_map(C) assert mapping[A.x].equals(C.x*(2*cos(q) + 1)/3 + C.y*(-2*sin(q + pi/6) + 1)/3 + C.z*(-2*cos(q + pi/3) + 1)/3) assert mapping[A.y].equals(C.x*(-2*cos(q + pi/3) + 1)/3 + C.y*(2*cos(q) + 1)/3 + C.z*(-2*sin(q + pi/6) + 1)/3) assert mapping[A.z].equals(C.x*(-2*sin(q + pi/6) + 1)/3 + C.y*(-2*cos(q + pi/3) + 1)/3 + C.z*(2*cos(q) + 1)/3) D = A.locate_new('D', a*A.i + b*A.j + c*A.k) assert D.scalar_map(A) == {D.z: A.z - c, D.x: A.x - a, D.y: A.y - b} E = A.orient_new_axis('E', a, A.k, a*A.i + b*A.j + c*A.k) assert A.scalar_map(E) == {A.z: E.z + c, A.x: E.x*cos(a) - E.y*sin(a) + a, A.y: E.x*sin(a) + E.y*cos(a) + b} assert E.scalar_map(A) == {E.x: (A.x - a)*cos(a) + (A.y - b)*sin(a), E.y: (-A.x + a)*sin(a) + (A.y - b)*cos(a), E.z: A.z - c} F = A.locate_new('F', Vector.zero) assert A.scalar_map(F) == {A.z: F.z, A.x: F.x, A.y: F.y} def test_rotation_matrix(): N = CoordSys3D('N') A = N.orient_new_axis('A', q1, N.k) B = A.orient_new_axis('B', q2, A.i) C = B.orient_new_axis('C', q3, B.j) D = N.orient_new_axis('D', q4, N.j) E = N.orient_new_space('E', q1, q2, q3, '123') F = N.orient_new_quaternion('F', q1, q2, q3, q4) G = N.orient_new_body('G', q1, q2, q3, '123') assert N.rotation_matrix(C) == Matrix([ [- sin(q1) * sin(q2) * sin(q3) + cos(q1) * cos(q3), - sin(q1) * cos(q2), sin(q1) * sin(q2) * cos(q3) + sin(q3) * cos(q1)], \ [sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1), \ cos(q1) * cos(q2), sin(q1) * sin(q3) - sin(q2) * cos(q1) * \ cos(q3)], [- sin(q3) * cos(q2), sin(q2), cos(q2) * cos(q3)]]) test_mat = D.rotation_matrix(C) - Matrix( [[cos(q1) * cos(q3) * cos(q4) - sin(q3) * (- sin(q4) * cos(q2) + sin(q1) * sin(q2) * cos(q4)), - sin(q2) * sin(q4) - sin(q1) * cos(q2) * cos(q4), sin(q3) * cos(q1) * cos(q4) + cos(q3) * \ (- sin(q4) * cos(q2) + sin(q1) * sin(q2) * cos(q4))], \ [sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1), cos(q1) * \ cos(q2), sin(q1) * sin(q3) - sin(q2) * cos(q1) * cos(q3)], \ [sin(q4) * cos(q1) * cos(q3) - sin(q3) * (cos(q2) * cos(q4) + \ sin(q1) * sin(q2) * \ sin(q4)), sin(q2) * cos(q4) - sin(q1) * sin(q4) * cos(q2), sin(q3) * \ sin(q4) * cos(q1) + cos(q3) * (cos(q2) * cos(q4) + \ sin(q1) * sin(q2) * sin(q4))]]) assert test_mat.expand() == zeros(3, 3) assert E.rotation_matrix(N) == Matrix( [[cos(q2)*cos(q3), sin(q3)*cos(q2), -sin(q2)], [sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1), \ sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), sin(q1)*cos(q2)], \ [sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3), - \ sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1), cos(q1)*cos(q2)]]) assert F.rotation_matrix(N) == Matrix([[ q1**2 + q2**2 - q3**2 - q4**2, 2*q1*q4 + 2*q2*q3, -2*q1*q3 + 2*q2*q4],[ -2*q1*q4 + 2*q2*q3, q1**2 - q2**2 + q3**2 - q4**2, 2*q1*q2 + 2*q3*q4], [2*q1*q3 + 2*q2*q4, -2*q1*q2 + 2*q3*q4, q1**2 - q2**2 - q3**2 + q4**2]]) assert G.rotation_matrix(N) == Matrix([[ cos(q2)*cos(q3), sin(q1)*sin(q2)*cos(q3) + sin(q3)*cos(q1), sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3)], [ -sin(q3)*cos(q2), -sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)],[ sin(q2), -sin(q1)*cos(q2), cos(q1)*cos(q2)]]) def test_vector_with_orientation(): """ Tests the effects of orientation of coordinate systems on basic vector operations. """ N = CoordSys3D('N') A = N.orient_new_axis('A', q1, N.k) B = A.orient_new_axis('B', q2, A.i) C = B.orient_new_axis('C', q3, B.j) # Test to_matrix v1 = a*N.i + b*N.j + c*N.k assert v1.to_matrix(A) == Matrix([[ a*cos(q1) + b*sin(q1)], [-a*sin(q1) + b*cos(q1)], [ c]]) # Test dot assert N.i.dot(A.i) == cos(q1) assert N.i.dot(A.j) == -sin(q1) assert N.i.dot(A.k) == 0 assert N.j.dot(A.i) == sin(q1) assert N.j.dot(A.j) == cos(q1) assert N.j.dot(A.k) == 0 assert N.k.dot(A.i) == 0 assert N.k.dot(A.j) == 0 assert N.k.dot(A.k) == 1 assert N.i.dot(A.i + A.j) == -sin(q1) + cos(q1) == \ (A.i + A.j).dot(N.i) assert A.i.dot(C.i) == cos(q3) assert A.i.dot(C.j) == 0 assert A.i.dot(C.k) == sin(q3) assert A.j.dot(C.i) == sin(q2)*sin(q3) assert A.j.dot(C.j) == cos(q2) assert A.j.dot(C.k) == -sin(q2)*cos(q3) assert A.k.dot(C.i) == -cos(q2)*sin(q3) assert A.k.dot(C.j) == sin(q2) assert A.k.dot(C.k) == cos(q2)*cos(q3) # Test cross assert N.i.cross(A.i) == sin(q1)*A.k assert N.i.cross(A.j) == cos(q1)*A.k assert N.i.cross(A.k) == -sin(q1)*A.i - cos(q1)*A.j assert N.j.cross(A.i) == -cos(q1)*A.k assert N.j.cross(A.j) == sin(q1)*A.k assert N.j.cross(A.k) == cos(q1)*A.i - sin(q1)*A.j assert N.k.cross(A.i) == A.j assert N.k.cross(A.j) == -A.i assert N.k.cross(A.k) == Vector.zero assert N.i.cross(A.i) == sin(q1)*A.k assert N.i.cross(A.j) == cos(q1)*A.k assert N.i.cross(A.i + A.j) == sin(q1)*A.k + cos(q1)*A.k assert (A.i + A.j).cross(N.i) == (-sin(q1) - cos(q1))*N.k assert A.i.cross(C.i) == sin(q3)*C.j assert A.i.cross(C.j) == -sin(q3)*C.i + cos(q3)*C.k assert A.i.cross(C.k) == -cos(q3)*C.j assert C.i.cross(A.i) == (-sin(q3)*cos(q2))*A.j + \ (-sin(q2)*sin(q3))*A.k assert C.j.cross(A.i) == (sin(q2))*A.j + (-cos(q2))*A.k assert express(C.k.cross(A.i), C).trigsimp() == cos(q3)*C.j def test_orient_new_methods(): N = CoordSys3D('N') orienter1 = AxisOrienter(q4, N.j) orienter2 = SpaceOrienter(q1, q2, q3, '123') orienter3 = QuaternionOrienter(q1, q2, q3, q4) orienter4 = BodyOrienter(q1, q2, q3, '123') D = N.orient_new('D', (orienter1, )) E = N.orient_new('E', (orienter2, )) F = N.orient_new('F', (orienter3, )) G = N.orient_new('G', (orienter4, )) assert D == N.orient_new_axis('D', q4, N.j) assert E == N.orient_new_space('E', q1, q2, q3, '123') assert F == N.orient_new_quaternion('F', q1, q2, q3, q4) assert G == N.orient_new_body('G', q1, q2, q3, '123') def test_locatenew_point(): """ Tests Point class, and locate_new method in CoordSysCartesian. """ A = CoordSys3D('A') assert isinstance(A.origin, Point) v = a*A.i + b*A.j + c*A.k C = A.locate_new('C', v) assert C.origin.position_wrt(A) == \ C.position_wrt(A) == \ C.origin.position_wrt(A.origin) == v assert A.origin.position_wrt(C) == \ A.position_wrt(C) == \ A.origin.position_wrt(C.origin) == -v assert A.origin.express_coordinates(C) == (-a, -b, -c) p = A.origin.locate_new('p', -v) assert p.express_coordinates(A) == (-a, -b, -c) assert p.position_wrt(C.origin) == p.position_wrt(C) == \ -2 * v p1 = p.locate_new('p1', 2*v) assert p1.position_wrt(C.origin) == Vector.zero assert p1.express_coordinates(C) == (0, 0, 0) p2 = p.locate_new('p2', A.i) assert p1.position_wrt(p2) == 2*v - A.i assert p2.express_coordinates(C) == (-2*a + 1, -2*b, -2*c) def test_create_new(): a = CoordSys3D('a') c = a.create_new('c', transformation='spherical') assert c._parent == a assert c.transformation_to_parent() == \ (c.r*sin(c.theta)*cos(c.phi), c.r*sin(c.theta)*sin(c.phi), c.r*cos(c.theta)) assert c.transformation_from_parent() == \ (sqrt(a.x**2 + a.y**2 + a.z**2), acos(a.z/sqrt(a.x**2 + a.y**2 + a.z**2)), atan2(a.y, a.x)) def test_evalf(): A = CoordSys3D('A') v = 3*A.i + 4*A.j + a*A.k assert v.n() == v.evalf() assert v.evalf(subs={a:1}) == v.subs(a, 1).evalf() def test_lame_coefficients(): a = CoordSys3D('a', 'spherical') assert a.lame_coefficients() == (1, a.r, sin(a.theta)*a.r) a = CoordSys3D('a') assert a.lame_coefficients() == (1, 1, 1) a = CoordSys3D('a', 'cartesian') assert a.lame_coefficients() == (1, 1, 1) a = CoordSys3D('a', 'cylindrical') assert a.lame_coefficients() == (1, a.r, 1) def test_transformation_equations(): x, y, z = symbols('x y z') # Str a = CoordSys3D('a', transformation='spherical', variable_names=["r", "theta", "phi"]) r, theta, phi = a.base_scalars() assert r == a.r assert theta == a.theta assert phi == a.phi raises(AttributeError, lambda: a.x) raises(AttributeError, lambda: a.y) raises(AttributeError, lambda: a.z) assert a.transformation_to_parent() == ( r*sin(theta)*cos(phi), r*sin(theta)*sin(phi), r*cos(theta) ) assert a.lame_coefficients() == (1, r, r*sin(theta)) assert a.transformation_from_parent_function()(x, y, z) == ( sqrt(x ** 2 + y ** 2 + z ** 2), acos((z) / sqrt(x**2 + y**2 + z**2)), atan2(y, x) ) a = CoordSys3D('a', transformation='cylindrical', variable_names=["r", "theta", "z"]) r, theta, z = a.base_scalars() assert a.transformation_to_parent() == ( r*cos(theta), r*sin(theta), z ) assert a.lame_coefficients() == (1, a.r, 1) assert a.transformation_from_parent_function()(x, y, z) == (sqrt(x**2 + y**2), atan2(y, x), z) a = CoordSys3D('a', 'cartesian') assert a.transformation_to_parent() == (a.x, a.y, a.z) assert a.lame_coefficients() == (1, 1, 1) assert a.transformation_from_parent_function()(x, y, z) == (x, y, z) # Variables and expressions # Cartesian with equation tuple: x, y, z = symbols('x y z') a = CoordSys3D('a', ((x, y, z), (x, y, z))) a._calculate_inv_trans_equations() assert a.transformation_to_parent() == (a.x1, a.x2, a.x3) assert a.lame_coefficients() == (1, 1, 1) assert a.transformation_from_parent_function()(x, y, z) == (x, y, z) r, theta, z = symbols("r theta z") # Cylindrical with equation tuple: a = CoordSys3D('a', [(r, theta, z), (r*cos(theta), r*sin(theta), z)], variable_names=["r", "theta", "z"]) r, theta, z = a.base_scalars() assert a.transformation_to_parent() == ( r*cos(theta), r*sin(theta), z ) assert a.lame_coefficients() == ( sqrt(sin(theta)**2 + cos(theta)**2), sqrt(r**2*sin(theta)**2 + r**2*cos(theta)**2), 1 ) # ==> this should simplify to (1, r, 1), tests are too slow with `simplify`. # Definitions with `lambda`: # Cartesian with `lambda` a = CoordSys3D('a', lambda x, y, z: (x, y, z)) assert a.transformation_to_parent() == (a.x1, a.x2, a.x3) assert a.lame_coefficients() == (1, 1, 1) a._calculate_inv_trans_equations() assert a.transformation_from_parent_function()(x, y, z) == (x, y, z) # Spherical with `lambda` a = CoordSys3D('a', lambda r, theta, phi: (r*sin(theta)*cos(phi), r*sin(theta)*sin(phi), r*cos(theta)), variable_names=["r", "theta", "phi"]) r, theta, phi = a.base_scalars() assert a.transformation_to_parent() == ( r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta) ) assert a.lame_coefficients() == ( sqrt(sin(phi)**2*sin(theta)**2 + sin(theta)**2*cos(phi)**2 + cos(theta)**2), sqrt(r**2*sin(phi)**2*cos(theta)**2 + r**2*sin(theta)**2 + r**2*cos(phi)**2*cos(theta)**2), sqrt(r**2*sin(phi)**2*sin(theta)**2 + r**2*sin(theta)**2*cos(phi)**2) ) # ==> this should simplify to (1, r, sin(theta)*r), `simplify` is too slow. # Cylindrical with `lambda` a = CoordSys3D('a', lambda r, theta, z: (r*cos(theta), r*sin(theta), z), variable_names=["r", "theta", "z"] ) r, theta, z = a.base_scalars() assert a.transformation_to_parent() == (r*cos(theta), r*sin(theta), z) assert a.lame_coefficients() == ( sqrt(sin(theta)**2 + cos(theta)**2), sqrt(r**2*sin(theta)**2 + r**2*cos(theta)**2), 1 ) # ==> this should simplify to (1, a.x, 1) raises(TypeError, lambda: CoordSys3D('a', transformation={ x: x*sin(y)*cos(z), y:x*sin(y)*sin(z), z: x*cos(y)})) def test_check_orthogonality(): x, y, z = symbols('x y z') u,v = symbols('u, v') a = CoordSys3D('a', transformation=((x, y, z), (x*sin(y)*cos(z), x*sin(y)*sin(z), x*cos(y)))) assert a._check_orthogonality(a._transformation) is True a = CoordSys3D('a', transformation=((x, y, z), (x * cos(y), x * sin(y), z))) assert a._check_orthogonality(a._transformation) is True a = CoordSys3D('a', transformation=((u, v, z), (cosh(u) * cos(v), sinh(u) * sin(v), z))) assert a._check_orthogonality(a._transformation) is True raises(ValueError, lambda: CoordSys3D('a', transformation=((x, x, z), (x, y, z)))) raises(ValueError, lambda: CoordSys3D('a', transformation=( (x, y, z), (x*sin(y/2)*cos(z), x*sin(y)*sin(z), x*cos(y))))) def test_coordsys3d(): with warnings.catch_warnings(): warnings.filterwarnings("ignore", category=SymPyDeprecationWarning) assert CoordSysCartesian("C") == CoordSys3D("C") def test_rotation_trans_equations(): a = CoordSys3D('a') from sympy import symbols q0 = symbols('q0') assert a._rotation_trans_equations(a._parent_rotation_matrix, a.base_scalars()) == (a.x, a.y, a.z) assert a._rotation_trans_equations(a._inverse_rotation_matrix(), a.base_scalars()) == (a.x, a.y, a.z) b = a.orient_new_axis('b', 0, -a.k) assert b._rotation_trans_equations(b._parent_rotation_matrix, b.base_scalars()) == (b.x, b.y, b.z) assert b._rotation_trans_equations(b._inverse_rotation_matrix(), b.base_scalars()) == (b.x, b.y, b.z) c = a.orient_new_axis('c', q0, -a.k) assert c._rotation_trans_equations(c._parent_rotation_matrix, c.base_scalars()) == \ (-sin(q0) * c.y + cos(q0) * c.x, sin(q0) * c.x + cos(q0) * c.y, c.z) assert c._rotation_trans_equations(c._inverse_rotation_matrix(), c.base_scalars()) == \ (sin(q0) * c.y + cos(q0) * c.x, -sin(q0) * c.x + cos(q0) * c.y, c.z)