Spaces:
Running
Running
File size: 8,050 Bytes
6a86ad5 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 |
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))
|