from sympy.testing.pytest import raises, warns_deprecated_sympy 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, y, z), (x, x, 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 warns_deprecated_sympy(): 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)