Vehicle-cpp/hkpc/software/sympy/vector/tests/test_coordsysrect.py

from sympy.testing.pytest import raises
from sympy.vector.coordsysrect import CoordSys3D
from sympy.vector.scalar import BaseScalar
from sympy.core.function import expand
from sympy.core.numbers import pi
from sympy.core.symbol import symbols
from sympy.functions.elementary.hyperbolic import (cosh, sinh)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (acos, atan2, cos, sin)
from sympy.matrices.dense import zeros
from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix
from sympy.simplify.simplify import simplify
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_coordsys3d_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 CoordSys3D.
    """
    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_rotation_trans_equations():
    a = CoordSys3D('a')
    from sympy.core.symbol 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)