410 lines
16 KiB
Python
410 lines
16 KiB
Python
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from sympy.core.function import diff
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from sympy.core.function import expand
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from sympy.core.numbers import (E, I, Rational, pi)
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from sympy.core.singleton import S
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from sympy.core.symbol import (Symbol, symbols)
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from sympy.functions.elementary.complexes import (Abs, conjugate, im, re, sign)
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from sympy.functions.elementary.exponential import log
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.trigonometric import (acos, asin, cos, sin, atan2, atan)
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from sympy.integrals.integrals import integrate
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from sympy.matrices.dense import Matrix
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from sympy.simplify import simplify
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from sympy.simplify.trigsimp import trigsimp
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from sympy.algebras.quaternion import Quaternion
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from sympy.testing.pytest import raises
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from itertools import permutations, product
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w, x, y, z = symbols('w:z')
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phi = symbols('phi')
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def test_quaternion_construction():
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q = Quaternion(w, x, y, z)
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assert q + q == Quaternion(2*w, 2*x, 2*y, 2*z)
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q2 = Quaternion.from_axis_angle((sqrt(3)/3, sqrt(3)/3, sqrt(3)/3),
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pi*Rational(2, 3))
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assert q2 == Quaternion(S.Half, S.Half,
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S.Half, S.Half)
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M = Matrix([[cos(phi), -sin(phi), 0], [sin(phi), cos(phi), 0], [0, 0, 1]])
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q3 = trigsimp(Quaternion.from_rotation_matrix(M))
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assert q3 == Quaternion(
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sqrt(2)*sqrt(cos(phi) + 1)/2, 0, 0, sqrt(2 - 2*cos(phi))*sign(sin(phi))/2)
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nc = Symbol('nc', commutative=False)
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raises(ValueError, lambda: Quaternion(w, x, nc, z))
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def test_quaternion_construction_norm():
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q1 = Quaternion(*symbols('a:d'))
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q2 = Quaternion(w, x, y, z)
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assert expand((q1*q2).norm()**2 - (q1.norm()**2 * q2.norm()**2)) == 0
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q3 = Quaternion(w, x, y, z, norm=1)
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assert (q1 * q3).norm() == q1.norm()
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def test_to_and_from_Matrix():
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q = Quaternion(w, x, y, z)
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q_full = Quaternion.from_Matrix(q.to_Matrix())
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q_vect = Quaternion.from_Matrix(q.to_Matrix(True))
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assert (q - q_full).is_zero_quaternion()
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assert (q.vector_part() - q_vect).is_zero_quaternion()
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def test_product_matrices():
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q1 = Quaternion(w, x, y, z)
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q2 = Quaternion(*(symbols("a:d")))
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assert (q1 * q2).to_Matrix() == q1.product_matrix_left * q2.to_Matrix()
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assert (q1 * q2).to_Matrix() == q2.product_matrix_right * q1.to_Matrix()
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R1 = (q1.product_matrix_left * q1.product_matrix_right.T)[1:, 1:]
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R2 = simplify(q1.to_rotation_matrix()*q1.norm()**2)
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assert R1 == R2
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def test_quaternion_axis_angle():
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test_data = [ # axis, angle, expected_quaternion
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((1, 0, 0), 0, (1, 0, 0, 0)),
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((1, 0, 0), pi/2, (sqrt(2)/2, sqrt(2)/2, 0, 0)),
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((0, 1, 0), pi/2, (sqrt(2)/2, 0, sqrt(2)/2, 0)),
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((0, 0, 1), pi/2, (sqrt(2)/2, 0, 0, sqrt(2)/2)),
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((1, 0, 0), pi, (0, 1, 0, 0)),
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((0, 1, 0), pi, (0, 0, 1, 0)),
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((0, 0, 1), pi, (0, 0, 0, 1)),
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((1, 1, 1), pi, (0, 1/sqrt(3),1/sqrt(3),1/sqrt(3))),
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((sqrt(3)/3, sqrt(3)/3, sqrt(3)/3), pi*2/3, (S.Half, S.Half, S.Half, S.Half))
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]
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for axis, angle, expected in test_data:
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assert Quaternion.from_axis_angle(axis, angle) == Quaternion(*expected)
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def test_quaternion_axis_angle_simplification():
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result = Quaternion.from_axis_angle((1, 2, 3), asin(4))
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assert result.a == cos(asin(4)/2)
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assert result.b == sqrt(14)*sin(asin(4)/2)/14
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assert result.c == sqrt(14)*sin(asin(4)/2)/7
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assert result.d == 3*sqrt(14)*sin(asin(4)/2)/14
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def test_quaternion_complex_real_addition():
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a = symbols("a", complex=True)
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b = symbols("b", real=True)
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# This symbol is not complex:
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c = symbols("c", commutative=False)
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q = Quaternion(w, x, y, z)
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assert a + q == Quaternion(w + re(a), x + im(a), y, z)
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assert 1 + q == Quaternion(1 + w, x, y, z)
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assert I + q == Quaternion(w, 1 + x, y, z)
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assert b + q == Quaternion(w + b, x, y, z)
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raises(ValueError, lambda: c + q)
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raises(ValueError, lambda: q * c)
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raises(ValueError, lambda: c * q)
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assert -q == Quaternion(-w, -x, -y, -z)
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q1 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False)
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q2 = Quaternion(1, 4, 7, 8)
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assert q1 + (2 + 3*I) == Quaternion(5 + 7*I, 2 + 5*I, 0, 7 + 8*I)
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assert q2 + (2 + 3*I) == Quaternion(3, 7, 7, 8)
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assert q1 * (2 + 3*I) == \
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Quaternion((2 + 3*I)*(3 + 4*I), (2 + 3*I)*(2 + 5*I), 0, (2 + 3*I)*(7 + 8*I))
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assert q2 * (2 + 3*I) == Quaternion(-10, 11, 38, -5)
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q1 = Quaternion(1, 2, 3, 4)
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q0 = Quaternion(0, 0, 0, 0)
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assert q1 + q0 == q1
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assert q1 - q0 == q1
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assert q1 - q1 == q0
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def test_quaternion_evalf():
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assert (Quaternion(sqrt(2), 0, 0, sqrt(3)).evalf() ==
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Quaternion(sqrt(2).evalf(), 0, 0, sqrt(3).evalf()))
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assert (Quaternion(1/sqrt(2), 0, 0, 1/sqrt(2)).evalf() ==
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Quaternion((1/sqrt(2)).evalf(), 0, 0, (1/sqrt(2)).evalf()))
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def test_quaternion_functions():
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q = Quaternion(w, x, y, z)
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q1 = Quaternion(1, 2, 3, 4)
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q0 = Quaternion(0, 0, 0, 0)
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assert conjugate(q) == Quaternion(w, -x, -y, -z)
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assert q.norm() == sqrt(w**2 + x**2 + y**2 + z**2)
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assert q.normalize() == Quaternion(w, x, y, z) / sqrt(w**2 + x**2 + y**2 + z**2)
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assert q.inverse() == Quaternion(w, -x, -y, -z) / (w**2 + x**2 + y**2 + z**2)
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assert q.inverse() == q.pow(-1)
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raises(ValueError, lambda: q0.inverse())
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assert q.pow(2) == Quaternion(w**2 - x**2 - y**2 - z**2, 2*w*x, 2*w*y, 2*w*z)
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assert q**(2) == Quaternion(w**2 - x**2 - y**2 - z**2, 2*w*x, 2*w*y, 2*w*z)
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assert q1.pow(-2) == Quaternion(
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Rational(-7, 225), Rational(-1, 225), Rational(-1, 150), Rational(-2, 225))
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assert q1**(-2) == Quaternion(
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Rational(-7, 225), Rational(-1, 225), Rational(-1, 150), Rational(-2, 225))
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assert q1.pow(-0.5) == NotImplemented
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raises(TypeError, lambda: q1**(-0.5))
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assert q1.exp() == \
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Quaternion(E * cos(sqrt(29)),
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2 * sqrt(29) * E * sin(sqrt(29)) / 29,
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3 * sqrt(29) * E * sin(sqrt(29)) / 29,
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4 * sqrt(29) * E * sin(sqrt(29)) / 29)
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assert q1._ln() == \
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Quaternion(log(sqrt(30)),
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2 * sqrt(29) * acos(sqrt(30)/30) / 29,
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3 * sqrt(29) * acos(sqrt(30)/30) / 29,
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4 * sqrt(29) * acos(sqrt(30)/30) / 29)
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assert q1.pow_cos_sin(2) == \
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Quaternion(30 * cos(2 * acos(sqrt(30)/30)),
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60 * sqrt(29) * sin(2 * acos(sqrt(30)/30)) / 29,
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90 * sqrt(29) * sin(2 * acos(sqrt(30)/30)) / 29,
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120 * sqrt(29) * sin(2 * acos(sqrt(30)/30)) / 29)
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assert diff(Quaternion(x, x, x, x), x) == Quaternion(1, 1, 1, 1)
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assert integrate(Quaternion(x, x, x, x), x) == \
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Quaternion(x**2 / 2, x**2 / 2, x**2 / 2, x**2 / 2)
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assert Quaternion.rotate_point((1, 1, 1), q1) == (S.One / 5, 1, S(7) / 5)
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n = Symbol('n')
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raises(TypeError, lambda: q1**n)
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n = Symbol('n', integer=True)
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raises(TypeError, lambda: q1**n)
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assert Quaternion(22, 23, 55, 8).scalar_part() == 22
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assert Quaternion(w, x, y, z).scalar_part() == w
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assert Quaternion(22, 23, 55, 8).vector_part() == Quaternion(0, 23, 55, 8)
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assert Quaternion(w, x, y, z).vector_part() == Quaternion(0, x, y, z)
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assert q1.axis() == Quaternion(0, 2*sqrt(29)/29, 3*sqrt(29)/29, 4*sqrt(29)/29)
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assert q1.axis().pow(2) == Quaternion(-1, 0, 0, 0)
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assert q0.axis().scalar_part() == 0
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assert (q.axis() == Quaternion(0,
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x/sqrt(x**2 + y**2 + z**2),
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y/sqrt(x**2 + y**2 + z**2),
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z/sqrt(x**2 + y**2 + z**2)))
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assert q0.is_pure() is True
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assert q1.is_pure() is False
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assert Quaternion(0, 0, 0, 3).is_pure() is True
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assert Quaternion(0, 2, 10, 3).is_pure() is True
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assert Quaternion(w, 2, 10, 3).is_pure() is None
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assert q1.angle() == atan(sqrt(29))
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assert q.angle() == atan2(sqrt(x**2 + y**2 + z**2), w)
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assert Quaternion.arc_coplanar(q1, Quaternion(2, 4, 6, 8)) is True
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assert Quaternion.arc_coplanar(q1, Quaternion(1, -2, -3, -4)) is True
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assert Quaternion.arc_coplanar(q1, Quaternion(1, 8, 12, 16)) is True
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assert Quaternion.arc_coplanar(q1, Quaternion(1, 2, 3, 4)) is True
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assert Quaternion.arc_coplanar(q1, Quaternion(w, 4, 6, 8)) is True
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assert Quaternion.arc_coplanar(q1, Quaternion(2, 7, 4, 1)) is False
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assert Quaternion.arc_coplanar(q1, Quaternion(w, x, y, z)) is None
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raises(ValueError, lambda: Quaternion.arc_coplanar(q1, q0))
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assert Quaternion.vector_coplanar(
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Quaternion(0, 8, 12, 16),
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Quaternion(0, 4, 6, 8),
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Quaternion(0, 2, 3, 4)) is True
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assert Quaternion.vector_coplanar(
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Quaternion(0, 0, 0, 0), Quaternion(0, 4, 6, 8), Quaternion(0, 2, 3, 4)) is True
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assert Quaternion.vector_coplanar(
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Quaternion(0, 8, 2, 6), Quaternion(0, 1, 6, 6), Quaternion(0, 0, 3, 4)) is False
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assert Quaternion.vector_coplanar(
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Quaternion(0, 1, 3, 4),
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Quaternion(0, 4, w, 6),
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Quaternion(0, 6, 8, 1)) is None
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raises(ValueError, lambda:
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Quaternion.vector_coplanar(q0, Quaternion(0, 4, 6, 8), q1))
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assert Quaternion(0, 1, 2, 3).parallel(Quaternion(0, 2, 4, 6)) is True
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assert Quaternion(0, 1, 2, 3).parallel(Quaternion(0, 2, 2, 6)) is False
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assert Quaternion(0, 1, 2, 3).parallel(Quaternion(w, x, y, 6)) is None
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raises(ValueError, lambda: q0.parallel(q1))
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assert Quaternion(0, 1, 2, 3).orthogonal(Quaternion(0, -2, 1, 0)) is True
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assert Quaternion(0, 2, 4, 7).orthogonal(Quaternion(0, 2, 2, 6)) is False
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assert Quaternion(0, 2, 4, 7).orthogonal(Quaternion(w, x, y, 6)) is None
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raises(ValueError, lambda: q0.orthogonal(q1))
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assert q1.index_vector() == Quaternion(
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0, 2*sqrt(870)/29,
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3*sqrt(870)/29,
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4*sqrt(870)/29)
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assert Quaternion(0, 3, 9, 4).index_vector() == Quaternion(0, 3, 9, 4)
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assert Quaternion(4, 3, 9, 4).mensor() == log(sqrt(122))
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assert Quaternion(3, 3, 0, 2).mensor() == log(sqrt(22))
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assert q0.is_zero_quaternion() is True
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assert q1.is_zero_quaternion() is False
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assert Quaternion(w, 0, 0, 0).is_zero_quaternion() is None
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def test_quaternion_conversions():
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q1 = Quaternion(1, 2, 3, 4)
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assert q1.to_axis_angle() == ((2 * sqrt(29)/29,
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3 * sqrt(29)/29,
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4 * sqrt(29)/29),
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2 * acos(sqrt(30)/30))
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assert (q1.to_rotation_matrix() ==
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Matrix([[Rational(-2, 3), Rational(2, 15), Rational(11, 15)],
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[Rational(2, 3), Rational(-1, 3), Rational(2, 3)],
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[Rational(1, 3), Rational(14, 15), Rational(2, 15)]]))
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assert (q1.to_rotation_matrix((1, 1, 1)) ==
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Matrix([
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[Rational(-2, 3), Rational(2, 15), Rational(11, 15), Rational(4, 5)],
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[Rational(2, 3), Rational(-1, 3), Rational(2, 3), S.Zero],
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[Rational(1, 3), Rational(14, 15), Rational(2, 15), Rational(-2, 5)],
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[S.Zero, S.Zero, S.Zero, S.One]]))
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theta = symbols("theta", real=True)
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q2 = Quaternion(cos(theta/2), 0, 0, sin(theta/2))
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assert trigsimp(q2.to_rotation_matrix()) == Matrix([
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[cos(theta), -sin(theta), 0],
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[sin(theta), cos(theta), 0],
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[0, 0, 1]])
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assert q2.to_axis_angle() == ((0, 0, sin(theta/2)/Abs(sin(theta/2))),
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2*acos(cos(theta/2)))
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assert trigsimp(q2.to_rotation_matrix((1, 1, 1))) == Matrix([
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[cos(theta), -sin(theta), 0, sin(theta) - cos(theta) + 1],
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[sin(theta), cos(theta), 0, -sin(theta) - cos(theta) + 1],
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[0, 0, 1, 0],
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[0, 0, 0, 1]])
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||
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def test_rotation_matrix_homogeneous():
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q = Quaternion(w, x, y, z)
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R1 = q.to_rotation_matrix(homogeneous=True) * q.norm()**2
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R2 = simplify(q.to_rotation_matrix(homogeneous=False) * q.norm()**2)
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assert R1 == R2
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||
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|
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||
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|
def test_quaternion_rotation_iss1593():
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"""
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|
There was a sign mistake in the definition,
|
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|
|
of the rotation matrix. This tests that particular sign mistake.
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||
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|
See issue 1593 for reference.
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|
See wikipedia
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|
https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Quaternion-derived_rotation_matrix
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|
for the correct definition
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||
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|
"""
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q = Quaternion(cos(phi/2), sin(phi/2), 0, 0)
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assert(trigsimp(q.to_rotation_matrix()) == Matrix([
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||
|
|
[1, 0, 0],
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||
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|
[0, cos(phi), -sin(phi)],
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|
[0, sin(phi), cos(phi)]]))
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||
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|
||
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|
def test_quaternion_multiplication():
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|
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q1 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False)
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|
q2 = Quaternion(1, 2, 3, 5)
|
||
|
|
q3 = Quaternion(1, 1, 1, y)
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||
|
|
|
||
|
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assert Quaternion._generic_mul(S(4), S.One) == 4
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||
|
|
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)
|