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| from sympy.testing.pytest import slow | |
| 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 | |
| import math | |
| 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_issue_25254(): | |
| # calculating the inverse cached the norm which caused problems | |
| # when multiplying | |
| p = Quaternion(1, 0, 0, 0) | |
| q = Quaternion.from_axis_angle((1, 1, 1), 3 * math.pi/4) | |
| qi = q.inverse() # this operation cached the norm | |
| test = q * p * qi | |
| assert ((test - p).norm() < 1E-10) | |
| 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_subs(): | |
| q = Quaternion.from_axis_angle((0, 0, 1), phi) | |
| assert q.subs(phi, 0) == Quaternion(1, 0, 0, 0) | |
| 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.log() == \ | |
| 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() == 2*atan(sqrt(29)) | |
| assert q.angle() == 2*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) | |