from sympy.core.function import diff
from sympy.core.function import expand
from sympy.core.numbers import (E, I, Rational, pi)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import (Abs, conjugate, im, re, sign)
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (acos, asin, cos, sin, atan2, atan)
from sympy.integrals.integrals import integrate
from sympy.matrices.dense import Matrix
from sympy.simplify import simplify
from sympy.simplify.trigsimp import trigsimp
from sympy.algebras.quaternion import Quaternion
from sympy.testing.pytest import raises
from itertools import permutations, product
w, x, y, z = symbols('w:z')
phi = symbols('phi')
def test_quaternion_construction():
q = Quaternion(w, x, y, z)
assert q + q == Quaternion(2*w, 2*x, 2*y, 2*z)
q2 = Quaternion.from_axis_angle((sqrt(3)/3, sqrt(3)/3, sqrt(3)/3),
pi*Rational(2, 3))
assert q2 == Quaternion(S.Half, S.Half,
S.Half, S.Half)
M = Matrix([[cos(phi), -sin(phi), 0], [sin(phi), cos(phi), 0], [0, 0, 1]])
q3 = trigsimp(Quaternion.from_rotation_matrix(M))
assert q3 == Quaternion(
sqrt(2)*sqrt(cos(phi) + 1)/2, 0, 0, sqrt(2 - 2*cos(phi))*sign(sin(phi))/2)
nc = Symbol('nc', commutative=False)
raises(ValueError, lambda: Quaternion(w, x, nc, z))
def test_quaternion_construction_norm():
q1 = Quaternion(*symbols('a:d'))
q2 = Quaternion(w, x, y, z)
assert expand((q1*q2).norm()**2 - (q1.norm()**2 * q2.norm()**2)) == 0
q3 = Quaternion(w, x, y, z, norm=1)
assert (q1 * q3).norm() == q1.norm()
def test_to_and_from_Matrix():
q = Quaternion(w, x, y, z)
q_full = Quaternion.from_Matrix(q.to_Matrix())
q_vect = Quaternion.from_Matrix(q.to_Matrix(True))
assert (q - q_full).is_zero_quaternion()
assert (q.vector_part() - q_vect).is_zero_quaternion()
def test_product_matrices():
q1 = Quaternion(w, x, y, z)
q2 = Quaternion(*(symbols("a:d")))
assert (q1 * q2).to_Matrix() == q1.product_matrix_left * q2.to_Matrix()
assert (q1 * q2).to_Matrix() == q2.product_matrix_right * q1.to_Matrix()
R1 = (q1.product_matrix_left * q1.product_matrix_right.T)[1:, 1:]
R2 = simplify(q1.to_rotation_matrix()*q1.norm()**2)
assert R1 == R2
def test_quaternion_axis_angle():
test_data = [ # axis, angle, expected_quaternion
((1, 0, 0), 0, (1, 0, 0, 0)),
((1, 0, 0), pi/2, (sqrt(2)/2, sqrt(2)/2, 0, 0)),
((0, 1, 0), pi/2, (sqrt(2)/2, 0, sqrt(2)/2, 0)),
((0, 0, 1), pi/2, (sqrt(2)/2, 0, 0, sqrt(2)/2)),
((1, 0, 0), pi, (0, 1, 0, 0)),
((0, 1, 0), pi, (0, 0, 1, 0)),
((0, 0, 1), pi, (0, 0, 0, 1)),
((1, 1, 1), pi, (0, 1/sqrt(3),1/sqrt(3),1/sqrt(3))),
((sqrt(3)/3, sqrt(3)/3, sqrt(3)/3), pi*2/3, (S.Half, S.Half, S.Half, S.Half))
]
for axis, angle, expected in test_data:
assert Quaternion.from_axis_angle(axis, angle) == Quaternion(*expected)
def test_quaternion_axis_angle_simplification():
result = Quaternion.from_axis_angle((1, 2, 3), asin(4))
assert result.a == cos(asin(4)/2)
assert result.b == sqrt(14)*sin(asin(4)/2)/14
assert result.c == sqrt(14)*sin(asin(4)/2)/7
assert result.d == 3*sqrt(14)*sin(asin(4)/2)/14
def test_quaternion_complex_real_addition():
a = symbols("a", complex=True)
b = symbols("b", real=True)
# This symbol is not complex:
c = symbols("c", commutative=False)
q = Quaternion(w, x, y, z)
assert a + q == Quaternion(w + re(a), x + im(a), y, z)
assert 1 + q == Quaternion(1 + w, x, y, z)
assert I + q == Quaternion(w, 1 + x, y, z)
assert b + q == Quaternion(w + b, x, y, z)
raises(ValueError, lambda: c + q)
raises(ValueError, lambda: q * c)
raises(ValueError, lambda: c * q)
assert -q == Quaternion(-w, -x, -y, -z)
q1 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False)
q2 = Quaternion(1, 4, 7, 8)
assert q1 + (2 + 3*I) == Quaternion(5 + 7*I, 2 + 5*I, 0, 7 + 8*I)
assert q2 + (2 + 3*I) == Quaternion(3, 7, 7, 8)
assert q1 * (2 + 3*I) == \
Quaternion((2 + 3*I)*(3 + 4*I), (2 + 3*I)*(2 + 5*I), 0, (2 + 3*I)*(7 + 8*I))
assert q2 * (2 + 3*I) == Quaternion(-10, 11, 38, -5)
q1 = Quaternion(1, 2, 3, 4)
q0 = Quaternion(0, 0, 0, 0)
assert q1 + q0 == q1
assert q1 - q0 == q1
assert q1 - q1 == q0
def test_quaternion_evalf():
assert (Quaternion(sqrt(2), 0, 0, sqrt(3)).evalf() ==
Quaternion(sqrt(2).evalf(), 0, 0, sqrt(3).evalf()))
assert (Quaternion(1/sqrt(2), 0, 0, 1/sqrt(2)).evalf() ==
Quaternion((1/sqrt(2)).evalf(), 0, 0, (1/sqrt(2)).evalf()))
def test_quaternion_functions():
q = Quaternion(w, x, y, z)
q1 = Quaternion(1, 2, 3, 4)
q0 = Quaternion(0, 0, 0, 0)
assert conjugate(q) == Quaternion(w, -x, -y, -z)
assert q.norm() == sqrt(w**2 + x**2 + y**2 + z**2)
assert q.normalize() == Quaternion(w, x, y, z) / sqrt(w**2 + x**2 + y**2 + z**2)
assert q.inverse() == Quaternion(w, -x, -y, -z) / (w**2 + x**2 + y**2 + z**2)
assert q.inverse() == q.pow(-1)
raises(ValueError, lambda: q0.inverse())
assert q.pow(2) == Quaternion(w**2 - x**2 - y**2 - z**2, 2*w*x, 2*w*y, 2*w*z)
assert q**(2) == Quaternion(w**2 - x**2 - y**2 - z**2, 2*w*x, 2*w*y, 2*w*z)
assert q1.pow(-2) == Quaternion(
Rational(-7, 225), Rational(-1, 225), Rational(-1, 150), Rational(-2, 225))
assert q1**(-2) == Quaternion(
Rational(-7, 225), Rational(-1, 225), Rational(-1, 150), Rational(-2, 225))
assert q1.pow(-0.5) == NotImplemented
raises(TypeError, lambda: q1**(-0.5))
assert q1.exp() == \
Quaternion(E * cos(sqrt(29)),
2 * sqrt(29) * E * sin(sqrt(29)) / 29,
3 * sqrt(29) * E * sin(sqrt(29)) / 29,
4 * sqrt(29) * E * sin(sqrt(29)) / 29)
assert q1._ln() == \
Quaternion(log(sqrt(30)),
2 * sqrt(29) * acos(sqrt(30)/30) / 29,
3 * sqrt(29) * acos(sqrt(30)/30) / 29,
4 * sqrt(29) * acos(sqrt(30)/30) / 29)
assert q1.pow_cos_sin(2) == \
Quaternion(30 * cos(2 * acos(sqrt(30)/30)),
60 * sqrt(29) * sin(2 * acos(sqrt(30)/30)) / 29,
90 * sqrt(29) * sin(2 * acos(sqrt(30)/30)) / 29,
120 * sqrt(29) * sin(2 * acos(sqrt(30)/30)) / 29)
assert diff(Quaternion(x, x, x, x), x) == Quaternion(1, 1, 1, 1)
assert integrate(Quaternion(x, x, x, x), x) == \
Quaternion(x**2 / 2, x**2 / 2, x**2 / 2, x**2 / 2)
assert Quaternion.rotate_point((1, 1, 1), q1) == (S.One / 5, 1, S(7) / 5)
n = Symbol('n')
raises(TypeError, lambda: q1**n)
n = Symbol('n', integer=True)
raises(TypeError, lambda: q1**n)
assert Quaternion(22, 23, 55, 8).scalar_part() == 22
assert Quaternion(w, x, y, z).scalar_part() == w
assert Quaternion(22, 23, 55, 8).vector_part() == Quaternion(0, 23, 55, 8)
assert Quaternion(w, x, y, z).vector_part() == Quaternion(0, x, y, z)
assert q1.axis() == Quaternion(0, 2*sqrt(29)/29, 3*sqrt(29)/29, 4*sqrt(29)/29)
assert q1.axis().pow(2) == Quaternion(-1, 0, 0, 0)
assert q0.axis().scalar_part() == 0
assert (q.axis() == Quaternion(0,
x/sqrt(x**2 + y**2 + z**2),
y/sqrt(x**2 + y**2 + z**2),
z/sqrt(x**2 + y**2 + z**2)))
assert q0.is_pure() is True
assert q1.is_pure() is False
assert Quaternion(0, 0, 0, 3).is_pure() is True
assert Quaternion(0, 2, 10, 3).is_pure() is True
assert Quaternion(w, 2, 10, 3).is_pure() is None
assert q1.angle() == atan(sqrt(29))
assert q.angle() == atan2(sqrt(x**2 + y**2 + z**2), w)
assert Quaternion.arc_coplanar(q1, Quaternion(2, 4, 6, 8)) is True
assert Quaternion.arc_coplanar(q1, Quaternion(1, -2, -3, -4)) is True
assert Quaternion.arc_coplanar(q1, Quaternion(1, 8, 12, 16)) is True
assert Quaternion.arc_coplanar(q1, Quaternion(1, 2, 3, 4)) is True
assert Quaternion.arc_coplanar(q1, Quaternion(w, 4, 6, 8)) is True
assert Quaternion.arc_coplanar(q1, Quaternion(2, 7, 4, 1)) is False
assert Quaternion.arc_coplanar(q1, Quaternion(w, x, y, z)) is None
raises(ValueError, lambda: Quaternion.arc_coplanar(q1, q0))
assert Quaternion.vector_coplanar(
Quaternion(0, 8, 12, 16),
Quaternion(0, 4, 6, 8),
Quaternion(0, 2, 3, 4)) is True
assert Quaternion.vector_coplanar(
Quaternion(0, 0, 0, 0), Quaternion(0, 4, 6, 8), Quaternion(0, 2, 3, 4)) is True
assert Quaternion.vector_coplanar(
Quaternion(0, 8, 2, 6), Quaternion(0, 1, 6, 6), Quaternion(0, 0, 3, 4)) is False
assert Quaternion.vector_coplanar(
Quaternion(0, 1, 3, 4),
Quaternion(0, 4, w, 6),
Quaternion(0, 6, 8, 1)) is None
raises(ValueError, lambda:
Quaternion.vector_coplanar(q0, Quaternion(0, 4, 6, 8), q1))
assert Quaternion(0, 1, 2, 3).parallel(Quaternion(0, 2, 4, 6)) is True
assert Quaternion(0, 1, 2, 3).parallel(Quaternion(0, 2, 2, 6)) is False
assert Quaternion(0, 1, 2, 3).parallel(Quaternion(w, x, y, 6)) is None
raises(ValueError, lambda: q0.parallel(q1))
assert Quaternion(0, 1, 2, 3).orthogonal(Quaternion(0, -2, 1, 0)) is True
assert Quaternion(0, 2, 4, 7).orthogonal(Quaternion(0, 2, 2, 6)) is False
assert Quaternion(0, 2, 4, 7).orthogonal(Quaternion(w, x, y, 6)) is None
raises(ValueError, lambda: q0.orthogonal(q1))
assert q1.index_vector() == Quaternion(
0, 2*sqrt(870)/29,
3*sqrt(870)/29,
4*sqrt(870)/29)
assert Quaternion(0, 3, 9, 4).index_vector() == Quaternion(0, 3, 9, 4)
assert Quaternion(4, 3, 9, 4).mensor() == log(sqrt(122))
assert Quaternion(3, 3, 0, 2).mensor() == log(sqrt(22))
assert q0.is_zero_quaternion() is True
assert q1.is_zero_quaternion() is False
assert Quaternion(w, 0, 0, 0).is_zero_quaternion() is None
def test_quaternion_conversions():
q1 = Quaternion(1, 2, 3, 4)
assert q1.to_axis_angle() == ((2 * sqrt(29)/29,
3 * sqrt(29)/29,
4 * sqrt(29)/29),
2 * acos(sqrt(30)/30))
assert (q1.to_rotation_matrix() ==
Matrix([[Rational(-2, 3), Rational(2, 15), Rational(11, 15)],
[Rational(2, 3), Rational(-1, 3), Rational(2, 3)],
[Rational(1, 3), Rational(14, 15), Rational(2, 15)]]))
assert (q1.to_rotation_matrix((1, 1, 1)) ==
Matrix([
[Rational(-2, 3), Rational(2, 15), Rational(11, 15), Rational(4, 5)],
[Rational(2, 3), Rational(-1, 3), Rational(2, 3), S.Zero],
[Rational(1, 3), Rational(14, 15), Rational(2, 15), Rational(-2, 5)],
[S.Zero, S.Zero, S.Zero, S.One]]))
theta = symbols("theta", real=True)
q2 = Quaternion(cos(theta/2), 0, 0, sin(theta/2))
assert trigsimp(q2.to_rotation_matrix()) == Matrix([
[cos(theta), -sin(theta), 0],
[sin(theta), cos(theta), 0],
[0, 0, 1]])
assert q2.to_axis_angle() == ((0, 0, sin(theta/2)/Abs(sin(theta/2))),
2*acos(cos(theta/2)))
assert trigsimp(q2.to_rotation_matrix((1, 1, 1))) == Matrix([
[cos(theta), -sin(theta), 0, sin(theta) - cos(theta) + 1],
[sin(theta), cos(theta), 0, -sin(theta) - cos(theta) + 1],
[0, 0, 1, 0],
[0, 0, 0, 1]])
def test_rotation_matrix_homogeneous():
q = Quaternion(w, x, y, z)
R1 = q.to_rotation_matrix(homogeneous=True) * q.norm()**2
R2 = simplify(q.to_rotation_matrix(homogeneous=False) * q.norm()**2)
assert R1 == R2
def test_quaternion_rotation_iss1593():
"""
There was a sign mistake in the definition,
of the rotation matrix. This tests that particular sign mistake.
See issue 1593 for reference.
See wikipedia
https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Quaternion-derived_rotation_matrix
for the correct definition
"""
q = Quaternion(cos(phi/2), sin(phi/2), 0, 0)
assert(trigsimp(q.to_rotation_matrix()) == Matrix([
[1, 0, 0],
[0, cos(phi), -sin(phi)],
[0, sin(phi), cos(phi)]]))
def test_quaternion_multiplication():
q1 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False)
q2 = Quaternion(1, 2, 3, 5)
q3 = Quaternion(1, 1, 1, y)
assert Quaternion._generic_mul(S(4), S.One) == 4
assert (Quaternion._generic_mul(S(4), q1) ==
Quaternion(12 + 16*I, 8 + 20*I, 0, 28 + 32*I))
assert q2.mul(2) == Quaternion(2, 4, 6, 10)
assert q2.mul(q3) == Quaternion(-5*y - 4, 3*y - 2, 9 - 2*y, y + 4)
assert q2.mul(q3) == q2*q3
z = symbols('z', complex=True)
z_quat = Quaternion(re(z), im(z), 0, 0)
q = Quaternion(*symbols('q:4', real=True))
assert z * q == z_quat * q
assert q * z == q * z_quat
def test_issue_16318():
#for rtruediv
q0 = Quaternion(0, 0, 0, 0)
raises(ValueError, lambda: 1/q0)
#for rotate_point
q = Quaternion(1, 2, 3, 4)
(axis, angle) = q.to_axis_angle()
assert Quaternion.rotate_point((1, 1, 1), (axis, angle)) == (S.One / 5, 1, S(7) / 5)
#test for to_axis_angle
q = Quaternion(-1, 1, 1, 1)
axis = (-sqrt(3)/3, -sqrt(3)/3, -sqrt(3)/3)
angle = 2*pi/3
assert (axis, angle) == q.to_axis_angle()
def test_to_euler():
q = Quaternion(w, x, y, z)
q_normalized = q.normalize()
seqs = ['zxy', 'zyx', 'zyz', 'zxz']
seqs += [seq.upper() for seq in seqs]
for seq in seqs:
euler_from_q = q.to_euler(seq)
q_back = simplify(Quaternion.from_euler(euler_from_q, seq))
assert q_back == q_normalized
def test_to_euler_iss24504():
"""
There was a mistake in the degenerate case testing
See issue 24504 for reference.
"""
q = Quaternion.from_euler((phi, 0, 0), 'zyz')
assert trigsimp(q.to_euler('zyz'), inverse=True) == (phi, 0, 0)
def test_to_euler_numerical_singilarities():
def test_one_case(angles, seq):
q = Quaternion.from_euler(angles, seq)
assert q.to_euler(seq) == angles
# symmetric
test_one_case((pi/2, 0, 0), 'zyz')
test_one_case((pi/2, 0, 0), 'ZYZ')
test_one_case((pi/2, pi, 0), 'zyz')
test_one_case((pi/2, pi, 0), 'ZYZ')
# asymmetric
test_one_case((pi/2, pi/2, 0), 'zyx')
test_one_case((pi/2, -pi/2, 0), 'zyx')
test_one_case((pi/2, pi/2, 0), 'ZYX')
test_one_case((pi/2, -pi/2, 0), 'ZYX')
def test_to_euler_options():
def test_one_case(q):
angles1 = Matrix(q.to_euler(seq, True, True))
angles2 = Matrix(q.to_euler(seq, False, False))
angle_errors = simplify(angles1-angles2).evalf()
for angle_error in angle_errors:
# forcing angles to set {-pi, pi}
angle_error = (angle_error + pi) % (2 * pi) - pi
assert angle_error < 10e-7
for xyz in ('xyz', 'XYZ'):
for seq_tuple in permutations(xyz):
for symmetric in (True, False):
if symmetric:
seq = ''.join([seq_tuple[0], seq_tuple[1], seq_tuple[0]])
else:
seq = ''.join(seq_tuple)
for elements in product([-1, 0, 1], repeat=4):
q = Quaternion(*elements)
if not q.is_zero_quaternion():
test_one_case(q)