def express(self, frame1, frame2=None):
"""Expresses this Dyadic in alternate frame(s)
The first frame is the list side expression, the second frame is the
right side; if Dyadic is in form A.x|B.y, you can express it in two
different frames. If no second frame is given, the Dyadic is
expressed in only one frame.
Calls the global express function
Parameters
==========
frame1 : ReferenceFrame
The frame to express the left side of the Dyadic in
frame2 : ReferenceFrame
If provided, the frame to express the right side of the Dyadic in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> d.express(B, N)
cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x)
"""
from sympy.physics.vector.functions import express
return express(self, frame1, frame2)
def express(self, frame1, frame2=None):
"""Expresses this Dyadic in alternate frame(s)
The first frame is the list side expression, the second frame is the
right side; if Dyadic is in form A.x|B.y, you can express it in two
different frames. If no second frame is given, the Dyadic is
expressed in only one frame.
Calls the global express function
Parameters
==========
frame1 : ReferenceFrame
The frame to express the left side of the Dyadic in
frame2 : ReferenceFrame
If provided, the frame to express the right side of the Dyadic in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> d.express(B, N)
cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x)
"""
from sympy.physics.vector.functions import express
return express(self, frame1, frame2)
def test_express():
assert express(Vector(0), N) == Vector(0)
assert express(S.Zero, N) is S.Zero
assert express(A.x, C) == cos(q3) * C.x + sin(q3) * C.z
assert express(A.y, C) == sin(q2)*sin(q3)*C.x + cos(q2)*C.y - \
sin(q2)*cos(q3)*C.z
assert express(A.z, C) == -sin(q3)*cos(q2)*C.x + sin(q2)*C.y + \
cos(q2)*cos(q3)*C.z
assert express(A.x, N) == cos(q1) * N.x + sin(q1) * N.y
assert express(A.y, N) == -sin(q1) * N.x + cos(q1) * N.y
assert express(A.z, N) == N.z
assert express(A.x, A) == A.x
assert express(A.y, A) == A.y
assert express(A.z, A) == A.z
assert express(A.x, B) == B.x
assert express(A.y, B) == cos(q2) * B.y - sin(q2) * B.z
assert express(A.z, B) == sin(q2) * B.y + cos(q2) * B.z
assert express(A.x, C) == cos(q3) * C.x + sin(q3) * C.z
assert express(A.y, C) == sin(q2)*sin(q3)*C.x + cos(q2)*C.y - \
sin(q2)*cos(q3)*C.z
assert express(A.z, C) == -sin(q3)*cos(q2)*C.x + sin(q2)*C.y + \
cos(q2)*cos(q3)*C.z
# Check to make sure UnitVectors get converted properly
assert express(N.x, N) == N.x
assert express(N.y, N) == N.y
assert express(N.z, N) == N.z
assert express(N.x, A) == (cos(q1) * A.x - sin(q1) * A.y)
assert express(N.y, A) == (sin(q1) * A.x + cos(q1) * A.y)
assert express(N.z, A) == A.z
assert express(N.x, B) == (cos(q1) * B.x - sin(q1) * cos(q2) * B.y +
sin(q1) * sin(q2) * B.z)
assert express(N.y, B) == (sin(q1) * B.x + cos(q1) * cos(q2) * B.y -
sin(q2) * cos(q1) * B.z)
assert express(N.z, B) == (sin(q2) * B.y + cos(q2) * B.z)
assert express(
N.x, C) == ((cos(q1) * cos(q3) - sin(q1) * sin(q2) * sin(q3)) * C.x -
sin(q1) * cos(q2) * C.y +
(sin(q3) * cos(q1) + sin(q1) * sin(q2) * cos(q3)) * C.z)
assert express(
N.y, C) == ((sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1)) * C.x +
cos(q1) * cos(q2) * C.y +
(sin(q1) * sin(q3) - sin(q2) * cos(q1) * cos(q3)) * C.z)
assert express(N.z, C) == (-sin(q3) * cos(q2) * C.x + sin(q2) * C.y +
cos(q2) * cos(q3) * C.z)
assert express(A.x, N) == (cos(q1) * N.x + sin(q1) * N.y)
assert express(A.y, N) == (-sin(q1) * N.x + cos(q1) * N.y)
assert express(A.z, N) == N.z
assert express(A.x, A) == A.x
assert express(A.y, A) == A.y
assert express(A.z, A) == A.z
assert express(A.x, B) == B.x
assert express(A.y, B) == (cos(q2) * B.y - sin(q2) * B.z)
assert express(A.z, B) == (sin(q2) * B.y + cos(q2) * B.z)
assert express(A.x, C) == (cos(q3) * C.x + sin(q3) * C.z)
assert express(A.y, C) == (sin(q2) * sin(q3) * C.x + cos(q2) * C.y -
sin(q2) * cos(q3) * C.z)
assert express(A.z, C) == (-sin(q3) * cos(q2) * C.x + sin(q2) * C.y +
cos(q2) * cos(q3) * C.z)
assert express(B.x, N) == (cos(q1) * N.x + sin(q1) * N.y)
assert express(B.y, N) == (-sin(q1) * cos(q2) * N.x +
cos(q1) * cos(q2) * N.y + sin(q2) * N.z)
assert express(B.z, N) == (sin(q1) * sin(q2) * N.x -
sin(q2) * cos(q1) * N.y + cos(q2) * N.z)
assert express(B.x, A) == A.x
assert express(B.y, A) == (cos(q2) * A.y + sin(q2) * A.z)
assert express(B.z, A) == (-sin(q2) * A.y + cos(q2) * A.z)
assert express(B.x, B) == B.x
assert express(B.y, B) == B.y
assert express(B.z, B) == B.z
assert express(B.x, C) == (cos(q3) * C.x + sin(q3) * C.z)
assert express(B.y, C) == C.y
assert express(B.z, C) == (-sin(q3) * C.x + cos(q3) * C.z)
assert express(
C.x, N) == ((cos(q1) * cos(q3) - sin(q1) * sin(q2) * sin(q3)) * N.x +
(sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1)) * N.y -
sin(q3) * cos(q2) * N.z)
assert express(C.y, N) == (-sin(q1) * cos(q2) * N.x +
cos(q1) * cos(q2) * N.y + sin(q2) * N.z)
assert express(
C.z, N) == ((sin(q3) * cos(q1) + sin(q1) * sin(q2) * cos(q3)) * N.x +
(sin(q1) * sin(q3) - sin(q2) * cos(q1) * cos(q3)) * N.y +
cos(q2) * cos(q3) * N.z)
assert express(C.x, A) == (cos(q3) * A.x + sin(q2) * sin(q3) * A.y -
sin(q3) * cos(q2) * A.z)
assert express(C.y, A) == (cos(q2) * A.y + sin(q2) * A.z)
assert express(C.z, A) == (sin(q3) * A.x - sin(q2) * cos(q3) * A.y +
cos(q2) * cos(q3) * A.z)
assert express(C.x, B) == (cos(q3) * B.x - sin(q3) * B.z)
assert express(C.y, B) == B.y
assert express(C.z, B) == (sin(q3) * B.x + cos(q3) * B.z)
assert express(C.x, C) == C.x
assert express(C.y, C) == C.y
assert express(C.z, C) == C.z == (C.z)
# Check to make sure Vectors get converted back to UnitVectors
assert N.x == express((cos(q1) * A.x - sin(q1) * A.y), N)
assert N.y == express((sin(q1) * A.x + cos(q1) * A.y), N)
assert N.x == express(
(cos(q1) * B.x - sin(q1) * cos(q2) * B.y + sin(q1) * sin(q2) * B.z), N)
assert N.y == express(
(sin(q1) * B.x + cos(q1) * cos(q2) * B.y - sin(q2) * cos(q1) * B.z), N)
assert N.z == express((sin(q2) * B.y + cos(q2) * B.z), N)
"""
These don't really test our code, they instead test the auto simplification
(or lack thereof) of SymPy.
assert N.x == express((
(cos(q1)*cos(q3)-sin(q1)*sin(q2)*sin(q3))*C.x -
sin(q1)*cos(q2)*C.y +
(sin(q3)*cos(q1)+sin(q1)*sin(q2)*cos(q3))*C.z), N)
assert N.y == express((
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.x +
cos(q1)*cos(q2)*C.y +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.z), N)
assert N.z == express((-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z), N)
"""
assert A.x == express((cos(q1) * N.x + sin(q1) * N.y), A)
assert A.y == express((-sin(q1) * N.x + cos(q1) * N.y), A)
assert A.y == express((cos(q2) * B.y - sin(q2) * B.z), A)
assert A.z == express((sin(q2) * B.y + cos(q2) * B.z), A)
assert A.x == express((cos(q3) * C.x + sin(q3) * C.z), A)
# Tripsimp messes up here too.
#print express((sin(q2)*sin(q3)*C.x + cos(q2)*C.y -
# sin(q2)*cos(q3)*C.z), A)
assert A.y == express(
(sin(q2) * sin(q3) * C.x + cos(q2) * C.y - sin(q2) * cos(q3) * C.z), A)
assert A.z == express(
(-sin(q3) * cos(q2) * C.x + sin(q2) * C.y + cos(q2) * cos(q3) * C.z),
A)
assert B.x == express((cos(q1) * N.x + sin(q1) * N.y), B)
assert B.y == express(
(-sin(q1) * cos(q2) * N.x + cos(q1) * cos(q2) * N.y + sin(q2) * N.z),
B)
assert B.z == express(
(sin(q1) * sin(q2) * N.x - sin(q2) * cos(q1) * N.y + cos(q2) * N.z), B)
assert B.y == express((cos(q2) * A.y + sin(q2) * A.z), B)
assert B.z == express((-sin(q2) * A.y + cos(q2) * A.z), B)
assert B.x == express((cos(q3) * C.x + sin(q3) * C.z), B)
assert B.z == express((-sin(q3) * C.x + cos(q3) * C.z), B)
"""
assert C.x == express((
(cos(q1)*cos(q3)-sin(q1)*sin(q2)*sin(q3))*N.x +
(sin(q1)*cos(q3)+sin(q2)*sin(q3)*cos(q1))*N.y -
sin(q3)*cos(q2)*N.z), C)
assert C.y == express((
-sin(q1)*cos(q2)*N.x + cos(q1)*cos(q2)*N.y + sin(q2)*N.z), C)
assert C.z == express((
(sin(q3)*cos(q1)+sin(q1)*sin(q2)*cos(q3))*N.x +
(sin(q1)*sin(q3)-sin(q2)*cos(q1)*cos(q3))*N.y +
cos(q2)*cos(q3)*N.z), C)
"""
assert C.x == express(
(cos(q3) * A.x + sin(q2) * sin(q3) * A.y - sin(q3) * cos(q2) * A.z), C)
assert C.y == express((cos(q2) * A.y + sin(q2) * A.z), C)
assert C.z == express(
(sin(q3) * A.x - sin(q2) * cos(q3) * A.y + cos(q2) * cos(q3) * A.z), C)
assert C.x == express((cos(q3) * B.x - sin(q3) * B.z), C)
assert C.z == express((sin(q3) * B.x + cos(q3) * B.z), C)
def test_express():
assert express(Vector(0), N) == Vector(0)
assert express(S(0), N) == S(0)
assert express(A.x, C) == cos(q3)*C.x + sin(q3)*C.z
assert express(A.y, C) == sin(q2)*sin(q3)*C.x + cos(q2)*C.y - \
sin(q2)*cos(q3)*C.z
assert express(A.z, C) == -sin(q3)*cos(q2)*C.x + sin(q2)*C.y + \
cos(q2)*cos(q3)*C.z
assert express(A.x, N) == cos(q1)*N.x + sin(q1)*N.y
assert express(A.y, N) == -sin(q1)*N.x + cos(q1)*N.y
assert express(A.z, N) == N.z
assert express(A.x, A) == A.x
assert express(A.y, A) == A.y
assert express(A.z, A) == A.z
assert express(A.x, B) == B.x
assert express(A.y, B) == cos(q2)*B.y - sin(q2)*B.z
assert express(A.z, B) == sin(q2)*B.y + cos(q2)*B.z
assert express(A.x, C) == cos(q3)*C.x + sin(q3)*C.z
assert express(A.y, C) == sin(q2)*sin(q3)*C.x + cos(q2)*C.y - \
sin(q2)*cos(q3)*C.z
assert express(A.z, C) == -sin(q3)*cos(q2)*C.x + sin(q2)*C.y + \
cos(q2)*cos(q3)*C.z
# Check to make sure UnitVectors get converted properly
assert express(N.x, N) == N.x
assert express(N.y, N) == N.y
assert express(N.z, N) == N.z
assert express(N.x, A) == (cos(q1)*A.x - sin(q1)*A.y)
assert express(N.y, A) == (sin(q1)*A.x + cos(q1)*A.y)
assert express(N.z, A) == A.z
assert express(N.x, B) == (cos(q1)*B.x - sin(q1)*cos(q2)*B.y +
sin(q1)*sin(q2)*B.z)
assert express(N.y, B) == (sin(q1)*B.x + cos(q1)*cos(q2)*B.y -
sin(q2)*cos(q1)*B.z)
assert express(N.z, B) == (sin(q2)*B.y + cos(q2)*B.z)
assert express(N.x, C) == (
(cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*C.x -
sin(q1)*cos(q2)*C.y +
(sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*C.z)
assert express(N.y, C) == (
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.x +
cos(q1)*cos(q2)*C.y +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.z)
assert express(N.z, C) == (-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z)
assert express(A.x, N) == (cos(q1)*N.x + sin(q1)*N.y)
assert express(A.y, N) == (-sin(q1)*N.x + cos(q1)*N.y)
assert express(A.z, N) == N.z
assert express(A.x, A) == A.x
assert express(A.y, A) == A.y
assert express(A.z, A) == A.z
assert express(A.x, B) == B.x
assert express(A.y, B) == (cos(q2)*B.y - sin(q2)*B.z)
assert express(A.z, B) == (sin(q2)*B.y + cos(q2)*B.z)
assert express(A.x, C) == (cos(q3)*C.x + sin(q3)*C.z)
assert express(A.y, C) == (sin(q2)*sin(q3)*C.x + cos(q2)*C.y -
sin(q2)*cos(q3)*C.z)
assert express(A.z, C) == (-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z)
assert express(B.x, N) == (cos(q1)*N.x + sin(q1)*N.y)
assert express(B.y, N) == (-sin(q1)*cos(q2)*N.x +
cos(q1)*cos(q2)*N.y + sin(q2)*N.z)
assert express(B.z, N) == (sin(q1)*sin(q2)*N.x -
sin(q2)*cos(q1)*N.y + cos(q2)*N.z)
assert express(B.x, A) == A.x
assert express(B.y, A) == (cos(q2)*A.y + sin(q2)*A.z)
assert express(B.z, A) == (-sin(q2)*A.y + cos(q2)*A.z)
assert express(B.x, B) == B.x
assert express(B.y, B) == B.y
assert express(B.z, B) == B.z
assert express(B.x, C) == (cos(q3)*C.x + sin(q3)*C.z)
assert express(B.y, C) == C.y
assert express(B.z, C) == (-sin(q3)*C.x + cos(q3)*C.z)
assert express(C.x, N) == (
(cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*N.x +
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*N.y -
sin(q3)*cos(q2)*N.z)
assert express(C.y, N) == (
-sin(q1)*cos(q2)*N.x + cos(q1)*cos(q2)*N.y + sin(q2)*N.z)
assert express(C.z, N) == (
(sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*N.x +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*N.y +
cos(q2)*cos(q3)*N.z)
assert express(C.x, A) == (cos(q3)*A.x + sin(q2)*sin(q3)*A.y -
sin(q3)*cos(q2)*A.z)
assert express(C.y, A) == (cos(q2)*A.y + sin(q2)*A.z)
assert express(C.z, A) == (sin(q3)*A.x - sin(q2)*cos(q3)*A.y +
cos(q2)*cos(q3)*A.z)
assert express(C.x, B) == (cos(q3)*B.x - sin(q3)*B.z)
assert express(C.y, B) == B.y
assert express(C.z, B) == (sin(q3)*B.x + cos(q3)*B.z)
assert express(C.x, C) == C.x
assert express(C.y, C) == C.y
assert express(C.z, C) == C.z == (C.z)
# Check to make sure Vectors get converted back to UnitVectors
assert N.x == express((cos(q1)*A.x - sin(q1)*A.y), N)
assert N.y == express((sin(q1)*A.x + cos(q1)*A.y), N)
assert N.x == express((cos(q1)*B.x - sin(q1)*cos(q2)*B.y +
sin(q1)*sin(q2)*B.z), N)
assert N.y == express((sin(q1)*B.x + cos(q1)*cos(q2)*B.y -
sin(q2)*cos(q1)*B.z), N)
assert N.z == express((sin(q2)*B.y + cos(q2)*B.z), N)
"""
These don't really test our code, they instead test the auto simplification
(or lack thereof) of SymPy.
assert N.x == express((
(cos(q1)*cos(q3)-sin(q1)*sin(q2)*sin(q3))*C.x -
sin(q1)*cos(q2)*C.y +
(sin(q3)*cos(q1)+sin(q1)*sin(q2)*cos(q3))*C.z), N)
assert N.y == express((
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.x +
cos(q1)*cos(q2)*C.y +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.z), N)
assert N.z == express((-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z), N)
"""
assert A.x == express((cos(q1)*N.x + sin(q1)*N.y), A)
assert A.y == express((-sin(q1)*N.x + cos(q1)*N.y), A)
assert A.y == express((cos(q2)*B.y - sin(q2)*B.z), A)
assert A.z == express((sin(q2)*B.y + cos(q2)*B.z), A)
assert A.x == express((cos(q3)*C.x + sin(q3)*C.z), A)
# Tripsimp messes up here too.
#print express((sin(q2)*sin(q3)*C.x + cos(q2)*C.y -
# sin(q2)*cos(q3)*C.z), A)
assert A.y == express((sin(q2)*sin(q3)*C.x + cos(q2)*C.y -
sin(q2)*cos(q3)*C.z), A)
assert A.z == express((-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z), A)
assert B.x == express((cos(q1)*N.x + sin(q1)*N.y), B)
assert B.y == express((-sin(q1)*cos(q2)*N.x +
cos(q1)*cos(q2)*N.y + sin(q2)*N.z), B)
assert B.z == express((sin(q1)*sin(q2)*N.x -
sin(q2)*cos(q1)*N.y + cos(q2)*N.z), B)
assert B.y == express((cos(q2)*A.y + sin(q2)*A.z), B)
assert B.z == express((-sin(q2)*A.y + cos(q2)*A.z), B)
assert B.x == express((cos(q3)*C.x + sin(q3)*C.z), B)
assert B.z == express((-sin(q3)*C.x + cos(q3)*C.z), B)
"""
assert C.x == express((
(cos(q1)*cos(q3)-sin(q1)*sin(q2)*sin(q3))*N.x +
(sin(q1)*cos(q3)+sin(q2)*sin(q3)*cos(q1))*N.y -
sin(q3)*cos(q2)*N.z), C)
assert C.y == express((
-sin(q1)*cos(q2)*N.x + cos(q1)*cos(q2)*N.y + sin(q2)*N.z), C)
assert C.z == express((
(sin(q3)*cos(q1)+sin(q1)*sin(q2)*cos(q3))*N.x +
(sin(q1)*sin(q3)-sin(q2)*cos(q1)*cos(q3))*N.y +
cos(q2)*cos(q3)*N.z), C)
"""
assert C.x == express((cos(q3)*A.x + sin(q2)*sin(q3)*A.y -
sin(q3)*cos(q2)*A.z), C)
assert C.y == express((cos(q2)*A.y + sin(q2)*A.z), C)
assert C.z == express((sin(q3)*A.x - sin(q2)*cos(q3)*A.y +
cos(q2)*cos(q3)*A.z), C)
assert C.x == express((cos(q3)*B.x - sin(q3)*B.z), C)
assert C.z == express((sin(q3)*B.x + cos(q3)*B.z), C)