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| from sympy.vector.vector import Vector | |
| from sympy.vector.coordsysrect import CoordSys3D | |
| from sympy.vector.functions import express, matrix_to_vector, orthogonalize | |
| from sympy.core.numbers import Rational | |
| from sympy.core.singleton import S | |
| from sympy.core.symbol import symbols | |
| from sympy.functions.elementary.miscellaneous import sqrt | |
| from sympy.functions.elementary.trigonometric import (cos, sin) | |
| from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix | |
| from sympy.testing.pytest import raises | |
| N = CoordSys3D('N') | |
| q1, q2, q3, q4, q5 = symbols('q1 q2 q3 q4 q5') | |
| A = N.orient_new_axis('A', q1, N.k) # type: ignore | |
| B = A.orient_new_axis('B', q2, A.i) | |
| C = B.orient_new_axis('C', q3, B.j) | |
| def test_express(): | |
| assert express(Vector.zero, N) == Vector.zero | |
| assert express(S.Zero, N) is S.Zero | |
| assert express(A.i, C) == cos(q3)*C.i + sin(q3)*C.k | |
| assert express(A.j, C) == sin(q2)*sin(q3)*C.i + cos(q2)*C.j - \ | |
| sin(q2)*cos(q3)*C.k | |
| assert express(A.k, C) == -sin(q3)*cos(q2)*C.i + sin(q2)*C.j + \ | |
| cos(q2)*cos(q3)*C.k | |
| assert express(A.i, N) == cos(q1)*N.i + sin(q1)*N.j | |
| assert express(A.j, N) == -sin(q1)*N.i + cos(q1)*N.j | |
| assert express(A.k, N) == N.k | |
| assert express(A.i, A) == A.i | |
| assert express(A.j, A) == A.j | |
| assert express(A.k, A) == A.k | |
| assert express(A.i, B) == B.i | |
| assert express(A.j, B) == cos(q2)*B.j - sin(q2)*B.k | |
| assert express(A.k, B) == sin(q2)*B.j + cos(q2)*B.k | |
| assert express(A.i, C) == cos(q3)*C.i + sin(q3)*C.k | |
| assert express(A.j, C) == sin(q2)*sin(q3)*C.i + cos(q2)*C.j - \ | |
| sin(q2)*cos(q3)*C.k | |
| assert express(A.k, C) == -sin(q3)*cos(q2)*C.i + sin(q2)*C.j + \ | |
| cos(q2)*cos(q3)*C.k | |
| # Check to make sure UnitVectors get converted properly | |
| assert express(N.i, N) == N.i | |
| assert express(N.j, N) == N.j | |
| assert express(N.k, N) == N.k | |
| assert express(N.i, A) == (cos(q1)*A.i - sin(q1)*A.j) | |
| assert express(N.j, A) == (sin(q1)*A.i + cos(q1)*A.j) | |
| assert express(N.k, A) == A.k | |
| assert express(N.i, B) == (cos(q1)*B.i - sin(q1)*cos(q2)*B.j + | |
| sin(q1)*sin(q2)*B.k) | |
| assert express(N.j, B) == (sin(q1)*B.i + cos(q1)*cos(q2)*B.j - | |
| sin(q2)*cos(q1)*B.k) | |
| assert express(N.k, B) == (sin(q2)*B.j + cos(q2)*B.k) | |
| assert express(N.i, C) == ( | |
| (cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*C.i - | |
| sin(q1)*cos(q2)*C.j + | |
| (sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*C.k) | |
| assert express(N.j, C) == ( | |
| (sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.i + | |
| cos(q1)*cos(q2)*C.j + | |
| (sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.k) | |
| assert express(N.k, C) == (-sin(q3)*cos(q2)*C.i + sin(q2)*C.j + | |
| cos(q2)*cos(q3)*C.k) | |
| assert express(A.i, N) == (cos(q1)*N.i + sin(q1)*N.j) | |
| assert express(A.j, N) == (-sin(q1)*N.i + cos(q1)*N.j) | |
| assert express(A.k, N) == N.k | |
| assert express(A.i, A) == A.i | |
| assert express(A.j, A) == A.j | |
| assert express(A.k, A) == A.k | |
| assert express(A.i, B) == B.i | |
| assert express(A.j, B) == (cos(q2)*B.j - sin(q2)*B.k) | |
| assert express(A.k, B) == (sin(q2)*B.j + cos(q2)*B.k) | |
| assert express(A.i, C) == (cos(q3)*C.i + sin(q3)*C.k) | |
| assert express(A.j, C) == (sin(q2)*sin(q3)*C.i + cos(q2)*C.j - | |
| sin(q2)*cos(q3)*C.k) | |
| assert express(A.k, C) == (-sin(q3)*cos(q2)*C.i + sin(q2)*C.j + | |
| cos(q2)*cos(q3)*C.k) | |
| assert express(B.i, N) == (cos(q1)*N.i + sin(q1)*N.j) | |
| assert express(B.j, N) == (-sin(q1)*cos(q2)*N.i + | |
| cos(q1)*cos(q2)*N.j + sin(q2)*N.k) | |
| assert express(B.k, N) == (sin(q1)*sin(q2)*N.i - | |
| sin(q2)*cos(q1)*N.j + cos(q2)*N.k) | |
| assert express(B.i, A) == A.i | |
| assert express(B.j, A) == (cos(q2)*A.j + sin(q2)*A.k) | |
| assert express(B.k, A) == (-sin(q2)*A.j + cos(q2)*A.k) | |
| assert express(B.i, B) == B.i | |
| assert express(B.j, B) == B.j | |
| assert express(B.k, B) == B.k | |
| assert express(B.i, C) == (cos(q3)*C.i + sin(q3)*C.k) | |
| assert express(B.j, C) == C.j | |
| assert express(B.k, C) == (-sin(q3)*C.i + cos(q3)*C.k) | |
| assert express(C.i, N) == ( | |
| (cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*N.i + | |
| (sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*N.j - | |
| sin(q3)*cos(q2)*N.k) | |
| assert express(C.j, N) == ( | |
| -sin(q1)*cos(q2)*N.i + cos(q1)*cos(q2)*N.j + sin(q2)*N.k) | |
| assert express(C.k, N) == ( | |
| (sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*N.i + | |
| (sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*N.j + | |
| cos(q2)*cos(q3)*N.k) | |
| assert express(C.i, A) == (cos(q3)*A.i + sin(q2)*sin(q3)*A.j - | |
| sin(q3)*cos(q2)*A.k) | |
| assert express(C.j, A) == (cos(q2)*A.j + sin(q2)*A.k) | |
| assert express(C.k, A) == (sin(q3)*A.i - sin(q2)*cos(q3)*A.j + | |
| cos(q2)*cos(q3)*A.k) | |
| assert express(C.i, B) == (cos(q3)*B.i - sin(q3)*B.k) | |
| assert express(C.j, B) == B.j | |
| assert express(C.k, B) == (sin(q3)*B.i + cos(q3)*B.k) | |
| assert express(C.i, C) == C.i | |
| assert express(C.j, C) == C.j | |
| assert express(C.k, C) == C.k == (C.k) | |
| # Check to make sure Vectors get converted back to UnitVectors | |
| assert N.i == express((cos(q1)*A.i - sin(q1)*A.j), N).simplify() | |
| assert N.j == express((sin(q1)*A.i + cos(q1)*A.j), N).simplify() | |
| assert N.i == express((cos(q1)*B.i - sin(q1)*cos(q2)*B.j + | |
| sin(q1)*sin(q2)*B.k), N).simplify() | |
| assert N.j == express((sin(q1)*B.i + cos(q1)*cos(q2)*B.j - | |
| sin(q2)*cos(q1)*B.k), N).simplify() | |
| assert N.k == express((sin(q2)*B.j + cos(q2)*B.k), N).simplify() | |
| assert A.i == express((cos(q1)*N.i + sin(q1)*N.j), A).simplify() | |
| assert A.j == express((-sin(q1)*N.i + cos(q1)*N.j), A).simplify() | |
| assert A.j == express((cos(q2)*B.j - sin(q2)*B.k), A).simplify() | |
| assert A.k == express((sin(q2)*B.j + cos(q2)*B.k), A).simplify() | |
| assert A.i == express((cos(q3)*C.i + sin(q3)*C.k), A).simplify() | |
| assert A.j == express((sin(q2)*sin(q3)*C.i + cos(q2)*C.j - | |
| sin(q2)*cos(q3)*C.k), A).simplify() | |
| assert A.k == express((-sin(q3)*cos(q2)*C.i + sin(q2)*C.j + | |
| cos(q2)*cos(q3)*C.k), A).simplify() | |
| assert B.i == express((cos(q1)*N.i + sin(q1)*N.j), B).simplify() | |
| assert B.j == express((-sin(q1)*cos(q2)*N.i + | |
| cos(q1)*cos(q2)*N.j + sin(q2)*N.k), B).simplify() | |
| assert B.k == express((sin(q1)*sin(q2)*N.i - | |
| sin(q2)*cos(q1)*N.j + cos(q2)*N.k), B).simplify() | |
| assert B.j == express((cos(q2)*A.j + sin(q2)*A.k), B).simplify() | |
| assert B.k == express((-sin(q2)*A.j + cos(q2)*A.k), B).simplify() | |
| assert B.i == express((cos(q3)*C.i + sin(q3)*C.k), B).simplify() | |
| assert B.k == express((-sin(q3)*C.i + cos(q3)*C.k), B).simplify() | |
| assert C.i == express((cos(q3)*A.i + sin(q2)*sin(q3)*A.j - | |
| sin(q3)*cos(q2)*A.k), C).simplify() | |
| assert C.j == express((cos(q2)*A.j + sin(q2)*A.k), C).simplify() | |
| assert C.k == express((sin(q3)*A.i - sin(q2)*cos(q3)*A.j + | |
| cos(q2)*cos(q3)*A.k), C).simplify() | |
| assert C.i == express((cos(q3)*B.i - sin(q3)*B.k), C).simplify() | |
| assert C.k == express((sin(q3)*B.i + cos(q3)*B.k), C).simplify() | |
| def test_matrix_to_vector(): | |
| m = Matrix([[1], [2], [3]]) | |
| assert matrix_to_vector(m, C) == C.i + 2*C.j + 3*C.k | |
| m = Matrix([[0], [0], [0]]) | |
| assert matrix_to_vector(m, N) == matrix_to_vector(m, C) == \ | |
| Vector.zero | |
| m = Matrix([[q1], [q2], [q3]]) | |
| assert matrix_to_vector(m, N) == q1*N.i + q2*N.j + q3*N.k | |
| def test_orthogonalize(): | |
| C = CoordSys3D('C') | |
| a, b = symbols('a b', integer=True) | |
| i, j, k = C.base_vectors() | |
| v1 = i + 2*j | |
| v2 = 2*i + 3*j | |
| v3 = 3*i + 5*j | |
| v4 = 3*i + j | |
| v5 = 2*i + 2*j | |
| v6 = a*i + b*j | |
| v7 = 4*a*i + 4*b*j | |
| assert orthogonalize(v1, v2) == [C.i + 2*C.j, C.i*Rational(2, 5) + -C.j/5] | |
| # from wikipedia | |
| assert orthogonalize(v4, v5, orthonormal=True) == \ | |
| [(3*sqrt(10))*C.i/10 + (sqrt(10))*C.j/10, (-sqrt(10))*C.i/10 + (3*sqrt(10))*C.j/10] | |
| raises(ValueError, lambda: orthogonalize(v1, v2, v3)) | |
| raises(ValueError, lambda: orthogonalize(v6, v7)) | |