185 lines
7.9 KiB
Python
185 lines
7.9 KiB
Python
|
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))
|