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227 lines
9.2 KiB
227 lines
9.2 KiB
from sympy.core.function import diff
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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)
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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.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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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(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_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() == Quaternion(sqrt(2).evalf(), 0, 0, sqrt(3).evalf())
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assert Quaternion(1/sqrt(2), 0, 0, 1/sqrt(2)).evalf() == 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(Rational(-7, 225), Rational(-1, 225), Rational(-1, 150), Rational(-2, 225))
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assert q1**(-2) == Quaternion(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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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() == 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)) == Matrix([[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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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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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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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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[0, cos(phi), -sin(phi)],
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[0, sin(phi), cos(phi)]]))
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def test_quaternion_multiplication():
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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)
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q3 = Quaternion(1, 1, 1, y)
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assert Quaternion._generic_mul(4, 1) == 4
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assert Quaternion._generic_mul(4, q1) == Quaternion(12 + 16*I, 8 + 20*I, 0, 28 + 32*I)
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assert q2.mul(2) == Quaternion(2, 4, 6, 10)
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assert q2.mul(q3) == Quaternion(-5*y - 4, 3*y - 2, 9 - 2*y, y + 4)
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assert q2.mul(q3) == q2*q3
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z = symbols('z', complex=True)
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z_quat = Quaternion(re(z), im(z), 0, 0)
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q = Quaternion(*symbols('q:4', real=True))
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assert z * q == z_quat * q
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assert q * z == q * z_quat
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def test_issue_16318():
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#for rtruediv
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q0 = Quaternion(0, 0, 0, 0)
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raises(ValueError, lambda: 1/q0)
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#for rotate_point
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q = Quaternion(1, 2, 3, 4)
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(axis, angle) = q.to_axis_angle()
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assert Quaternion.rotate_point((1, 1, 1), (axis, angle)) == (S.One / 5, 1, S(7) / 5)
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#test for to_axis_angle
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q = Quaternion(-1, 1, 1, 1)
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axis = (-sqrt(3)/3, -sqrt(3)/3, -sqrt(3)/3)
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angle = 2*pi/3
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assert (axis, angle) == q.to_axis_angle()
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