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from sympy import (S, sympify, expand, sqrt, Add, zeros, acos,
ImmutableMatrix as Matrix, simplify)
from sympy.simplify.trigsimp import trigsimp
from sympy.printing.defaults import Printable
from sympy.utilities.misc import filldedent
from sympy.core.evalf import EvalfMixin
from mpmath.libmp.libmpf import prec_to_dps
__all__ = ['Vector']
class Vector(Printable, EvalfMixin):
"""The class used to define vectors.
It along with ReferenceFrame are the building blocks of describing a
classical mechanics system in PyDy and sympy.physics.vector.
Attributes
==========
simp : Boolean
Let certain methods use trigsimp on their outputs
"""
simp = False
is_number = False
def __init__(self, inlist):
"""This is the constructor for the Vector class. You should not be
calling this, it should only be used by other functions. You should be
treating Vectors like you would with if you were doing the math by
hand, and getting the first 3 from the standard basis vectors from a
ReferenceFrame.
The only exception is to create a zero vector:
zv = Vector(0)
"""
self.args = []
if inlist == 0:
inlist = []
if isinstance(inlist, dict):
d = inlist
else:
d = {}
for inp in inlist:
if inp[1] in d:
d[inp[1]] += inp[0]
else:
d[inp[1]] = inp[0]
for k, v in d.items():
if v != Matrix([0, 0, 0]):
self.args.append((v, k))
@property
def func(self):
"""Returns the class Vector. """
return Vector
def __hash__(self):
return hash(tuple(self.args))
def __add__(self, other):
"""The add operator for Vector. """
if other == 0:
return self
other = _check_vector(other)
return Vector(self.args + other.args)
def dot(self, other):
"""Dot product of two vectors.
Returns a scalar, the dot product of the two Vectors
Parameters
==========
other : Vector
The Vector which we are dotting with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dot
>>> from sympy import symbols
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> dot(N.x, N.x)
1
>>> dot(N.x, N.y)
0
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
>>> dot(N.y, A.y)
cos(q1)
"""
from sympy.physics.vector.dyadic import Dyadic, _check_dyadic
if isinstance(other, Dyadic):
other = _check_dyadic(other)
ol = Vector(0)
for v in other.args:
ol += v[0] * v[2] * (v[1].dot(self))
return ol
other = _check_vector(other)
out = S.Zero
for v1 in self.args:
for v2 in other.args:
out += ((v2[0].T) * (v2[1].dcm(v1[1])) * (v1[0]))[0]
if Vector.simp:
return trigsimp(out, recursive=True)
else:
return out
def __truediv__(self, other):
"""This uses mul and inputs self and 1 divided by other. """
return self.__mul__(S.One / other)
def __eq__(self, other):
"""Tests for equality.
It is very import to note that this is only as good as the SymPy
equality test; False does not always mean they are not equivalent
Vectors.
If other is 0, and self is empty, returns True.
If other is 0 and self is not empty, returns False.
If none of the above, only accepts other as a Vector.
"""
if other == 0:
other = Vector(0)
try:
other = _check_vector(other)
except TypeError:
return False
if (self.args == []) and (other.args == []):
return True
elif (self.args == []) or (other.args == []):
return False
frame = self.args[0][1]
for v in frame:
if expand((self - other).dot(v)) != 0:
return False
return True
def __mul__(self, other):
"""Multiplies the Vector by a sympifyable expression.
Parameters
==========
other : Sympifyable
The scalar to multiply this Vector with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy import Symbol
>>> N = ReferenceFrame('N')
>>> b = Symbol('b')
>>> V = 10 * b * N.x
>>> print(V)
10*b*N.x
"""
newlist = list(self.args)
other = sympify(other)
for i, v in enumerate(newlist):
newlist[i] = (other * newlist[i][0], newlist[i][1])
return Vector(newlist)
def __neg__(self):
return self * -1
def outer(self, other):
"""Outer product between two Vectors.
A rank increasing operation, which returns a Dyadic from two Vectors
Parameters
==========
other : Vector
The Vector to take the outer product with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> outer(N.x, N.x)
(N.x|N.x)
"""
from sympy.physics.vector.dyadic import Dyadic
other = _check_vector(other)
ol = Dyadic(0)
for v in self.args:
for v2 in other.args:
# it looks this way because if we are in the same frame and
# use the enumerate function on the same frame in a nested
# fashion, then bad things happen
ol += Dyadic([(v[0][0] * v2[0][0], v[1].x, v2[1].x)])
ol += Dyadic([(v[0][0] * v2[0][1], v[1].x, v2[1].y)])
ol += Dyadic([(v[0][0] * v2[0][2], v[1].x, v2[1].z)])
ol += Dyadic([(v[0][1] * v2[0][0], v[1].y, v2[1].x)])
ol += Dyadic([(v[0][1] * v2[0][1], v[1].y, v2[1].y)])
ol += Dyadic([(v[0][1] * v2[0][2], v[1].y, v2[1].z)])
ol += Dyadic([(v[0][2] * v2[0][0], v[1].z, v2[1].x)])
ol += Dyadic([(v[0][2] * v2[0][1], v[1].z, v2[1].y)])
ol += Dyadic([(v[0][2] * v2[0][2], v[1].z, v2[1].z)])
return ol
def _latex(self, printer):
"""Latex Printing method. """
ar = self.args # just to shorten things
if len(ar) == 0:
return str(0)
ol = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
for j in 0, 1, 2:
# if the coef of the basis vector is 1, we skip the 1
if ar[i][0][j] == 1:
ol.append(' + ' + ar[i][1].latex_vecs[j])
# if the coef of the basis vector is -1, we skip the 1
elif ar[i][0][j] == -1:
ol.append(' - ' + ar[i][1].latex_vecs[j])
elif ar[i][0][j] != 0:
# If the coefficient of the basis vector is not 1 or -1;
# also, we might wrap it in parentheses, for readability.
arg_str = printer._print(ar[i][0][j])
if isinstance(ar[i][0][j], Add):
arg_str = "(%s)" % arg_str
if arg_str[0] == '-':
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + ar[i][1].latex_vecs[j])
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr
def _pretty(self, printer):
"""Pretty Printing method. """
from sympy.printing.pretty.stringpict import prettyForm
terms = []
def juxtapose(a, b):
pa = printer._print(a)
pb = printer._print(b)
if a.is_Add:
pa = prettyForm(*pa.parens())
return printer._print_seq([pa, pb], delimiter=' ')
for M, N in self.args:
for i in range(3):
if M[i] == 0:
continue
elif M[i] == 1:
terms.append(prettyForm(N.pretty_vecs[i]))
elif M[i] == -1:
terms.append(prettyForm("-1") * prettyForm(N.pretty_vecs[i]))
else:
terms.append(juxtapose(M[i], N.pretty_vecs[i]))
if terms:
pretty_result = prettyForm.__add__(*terms)
else:
pretty_result = prettyForm("0")
return pretty_result
def __rsub__(self, other):
return (-1 * self) + other
def _sympystr(self, printer, order=True):
"""Printing method. """
if not order or len(self.args) == 1:
ar = list(self.args)
elif len(self.args) == 0:
return printer._print(0)
else:
d = {v[1]: v[0] for v in self.args}
keys = sorted(d.keys(), key=lambda x: x.index)
ar = []
for key in keys:
ar.append((d[key], key))
ol = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
for j in 0, 1, 2:
# if the coef of the basis vector is 1, we skip the 1
if ar[i][0][j] == 1:
ol.append(' + ' + ar[i][1].str_vecs[j])
# if the coef of the basis vector is -1, we skip the 1
elif ar[i][0][j] == -1:
ol.append(' - ' + ar[i][1].str_vecs[j])
elif ar[i][0][j] != 0:
# If the coefficient of the basis vector is not 1 or -1;
# also, we might wrap it in parentheses, for readability.
arg_str = printer._print(ar[i][0][j])
if isinstance(ar[i][0][j], Add):
arg_str = "(%s)" % arg_str
if arg_str[0] == '-':
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + '*' + ar[i][1].str_vecs[j])
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr
def __sub__(self, other):
"""The subtraction operator. """
return self.__add__(other * -1)
def cross(self, other):
"""The cross product operator for two Vectors.
Returns a Vector, expressed in the same ReferenceFrames as self.
Parameters
==========
other : Vector
The Vector which we are crossing with
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame, cross
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> cross(N.x, N.y)
N.z
>>> A = ReferenceFrame('A')
>>> A.orient_axis(N, q1, N.x)
>>> cross(A.x, N.y)
N.z
>>> cross(N.y, A.x)
- sin(q1)*A.y - cos(q1)*A.z
"""
from sympy.physics.vector.dyadic import Dyadic, _check_dyadic
if isinstance(other, Dyadic):
other = _check_dyadic(other)
ol = Dyadic(0)
for i, v in enumerate(other.args):
ol += v[0] * ((self.cross(v[1])).outer(v[2]))
return ol
other = _check_vector(other)
if other.args == []:
return Vector(0)
def _det(mat):
"""This is needed as a little method for to find the determinant
of a list in python; needs to work for a 3x3 list.
SymPy's Matrix will not take in Vector, so need a custom function.
You should not be calling this.
"""
return (mat[0][0] * (mat[1][1] * mat[2][2] - mat[1][2] * mat[2][1])
+ mat[0][1] * (mat[1][2] * mat[2][0] - mat[1][0] *
mat[2][2]) + mat[0][2] * (mat[1][0] * mat[2][1] -
mat[1][1] * mat[2][0]))
outlist = []
ar = other.args # For brevity
for i, v in enumerate(ar):
tempx = v[1].x
tempy = v[1].y
tempz = v[1].z
tempm = ([[tempx, tempy, tempz],
[self.dot(tempx), self.dot(tempy), self.dot(tempz)],
[Vector([ar[i]]).dot(tempx), Vector([ar[i]]).dot(tempy),
Vector([ar[i]]).dot(tempz)]])
outlist += _det(tempm).args
return Vector(outlist)
__radd__ = __add__
__rmul__ = __mul__
def separate(self):
"""
The constituents of this vector in different reference frames,
as per its definition.
Returns a dict mapping each ReferenceFrame to the corresponding
constituent Vector.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> R1 = ReferenceFrame('R1')
>>> R2 = ReferenceFrame('R2')
>>> v = R1.x + R2.x
>>> v.separate() == {R1: R1.x, R2: R2.x}
True
"""
components = {}
for x in self.args:
components[x[1]] = Vector([x])
return components
def __and__(self, other):
return self.dot(other)
__and__.__doc__ = dot.__doc__
__rand__ = __and__
def __xor__(self, other):
return self.cross(other)
__xor__.__doc__ = cross.__doc__
def __or__(self, other):
return self.outer(other)
__or__.__doc__ = outer.__doc__
def diff(self, var, frame, var_in_dcm=True):
"""Returns the partial derivative of the vector with respect to a
variable in the provided reference frame.
Parameters
==========
var : Symbol
What the partial derivative is taken with respect to.
frame : ReferenceFrame
The reference frame that the partial derivative is taken in.
var_in_dcm : boolean
If true, the differentiation algorithm assumes that the variable
may be present in any of the direction cosine matrices that relate
the frame to the frames of any component of the vector. But if it
is known that the variable is not present in the direction cosine
matrices, false can be set to skip full reexpression in the desired
frame.
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.vector import dynamicsymbols, ReferenceFrame
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> t = Symbol('t')
>>> q1 = dynamicsymbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.y])
>>> A.x.diff(t, N)
- sin(q1)*q1'*N.x - cos(q1)*q1'*N.z
>>> A.x.diff(t, N).express(A).simplify()
- q1'*A.z
>>> B = ReferenceFrame('B')
>>> u1, u2 = dynamicsymbols('u1, u2')
>>> v = u1 * A.x + u2 * B.y
>>> v.diff(u2, N, var_in_dcm=False)
B.y
"""
from sympy.physics.vector.frame import _check_frame
_check_frame(frame)
var = sympify(var)
inlist = []
for vector_component in self.args:
measure_number = vector_component[0]
component_frame = vector_component[1]
if component_frame == frame:
inlist += [(measure_number.diff(var), frame)]
else:
# If the direction cosine matrix relating the component frame
# with the derivative frame does not contain the variable.
if not var_in_dcm or (frame.dcm(component_frame).diff(var) ==
zeros(3, 3)):
inlist += [(measure_number.diff(var), component_frame)]
else: # else express in the frame
reexp_vec_comp = Vector([vector_component]).express(frame)
deriv = reexp_vec_comp.args[0][0].diff(var)
inlist += Vector([(deriv, frame)]).args
return Vector(inlist)
def express(self, otherframe, variables=False):
"""
Returns a Vector equivalent to this one, expressed in otherframe.
Uses the global express method.
Parameters
==========
otherframe : ReferenceFrame
The frame for this Vector to be described in
variables : boolean
If True, the coordinate symbols(if present) in this Vector
are re-expressed in terms otherframe
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> q1 = dynamicsymbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.y])
>>> A.x.express(N)
cos(q1)*N.x - sin(q1)*N.z
"""
from sympy.physics.vector import express
return express(self, otherframe, variables=variables)
def to_matrix(self, reference_frame):
"""Returns the matrix form of the vector with respect to the given
frame.
Parameters
----------
reference_frame : ReferenceFrame
The reference frame that the rows of the matrix correspond to.
Returns
-------
matrix : ImmutableMatrix, shape(3,1)
The matrix that gives the 1D vector.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> a, b, c = symbols('a, b, c')
>>> N = ReferenceFrame('N')
>>> vector = a * N.x + b * N.y + c * N.z
>>> vector.to_matrix(N)
Matrix([
[a],
[b],
[c]])
>>> beta = symbols('beta')
>>> A = N.orientnew('A', 'Axis', (beta, N.x))
>>> vector.to_matrix(A)
Matrix([
[ a],
[ b*cos(beta) + c*sin(beta)],
[-b*sin(beta) + c*cos(beta)]])
"""
return Matrix([self.dot(unit_vec) for unit_vec in
reference_frame]).reshape(3, 1)
def doit(self, **hints):
"""Calls .doit() on each term in the Vector"""
d = {}
for v in self.args:
d[v[1]] = v[0].applyfunc(lambda x: x.doit(**hints))
return Vector(d)
def dt(self, otherframe):
"""
Returns a Vector which is the time derivative of
the self Vector, taken in frame otherframe.
Calls the global time_derivative method
Parameters
==========
otherframe : ReferenceFrame
The frame to calculate the time derivative in
"""
from sympy.physics.vector import time_derivative
return time_derivative(self, otherframe)
def simplify(self):
"""Returns a simplified Vector."""
d = {}
for v in self.args:
d[v[1]] = simplify(v[0])
return Vector(d)
def subs(self, *args, **kwargs):
"""Substitution on the Vector.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy import Symbol
>>> N = ReferenceFrame('N')
>>> s = Symbol('s')
>>> a = N.x * s
>>> a.subs({s: 2})
2*N.x
"""
d = {}
for v in self.args:
d[v[1]] = v[0].subs(*args, **kwargs)
return Vector(d)
def magnitude(self):
"""Returns the magnitude (Euclidean norm) of self.
Warnings
========
Python ignores the leading negative sign so that might
give wrong results.
``-A.x.magnitude()`` would be treated as ``-(A.x.magnitude())``,
instead of ``(-A.x).magnitude()``.
"""
return sqrt(self.dot(self))
def normalize(self):
"""Returns a Vector of magnitude 1, codirectional with self."""
return Vector(self.args + []) / self.magnitude()
def applyfunc(self, f):
"""Apply a function to each component of a vector."""
if not callable(f):
raise TypeError("`f` must be callable.")
d = {}
for v in self.args:
d[v[1]] = v[0].applyfunc(f)
return Vector(d)
def angle_between(self, vec):
"""
Returns the smallest angle between Vector 'vec' and self.
Parameter
=========
vec : Vector
The Vector between which angle is needed.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> A = ReferenceFrame("A")
>>> v1 = A.x
>>> v2 = A.y
>>> v1.angle_between(v2)
pi/2
>>> v3 = A.x + A.y + A.z
>>> v1.angle_between(v3)
acos(sqrt(3)/3)
Warnings
========
Python ignores the leading negative sign so that might give wrong
results. ``-A.x.angle_between()`` would be treated as
``-(A.x.angle_between())``, instead of ``(-A.x).angle_between()``.
"""
vec1 = self.normalize()
vec2 = vec.normalize()
angle = acos(vec1.dot(vec2))
return angle
def free_symbols(self, reference_frame):
"""Returns the free symbols in the measure numbers of the vector
expressed in the given reference frame.
Parameters
==========
reference_frame : ReferenceFrame
The frame with respect to which the free symbols of the given
vector is to be determined.
Returns
=======
set of Symbol
set of symbols present in the measure numbers of
``reference_frame``.
"""
return self.to_matrix(reference_frame).free_symbols
def free_dynamicsymbols(self, reference_frame):
"""Returns the free dynamic symbols (functions of time ``t``) in the
measure numbers of the vector expressed in the given reference frame.
Parameters
==========
reference_frame : ReferenceFrame
The frame with respect to which the free dynamic symbols of the
given vector is to be determined.
Returns
=======
set
Set of functions of time ``t``, e.g.
``Function('f')(me.dynamicsymbols._t)``.
"""
# TODO : Circular dependency if imported at top. Should move
# find_dynamicsymbols into physics.vector.functions.
from sympy.physics.mechanics.functions import find_dynamicsymbols
return find_dynamicsymbols(self, reference_frame=reference_frame)
def _eval_evalf(self, prec):
if not self.args:
return self
new_args = []
dps = prec_to_dps(prec)
for mat, frame in self.args:
new_args.append([mat.evalf(n=dps), frame])
return Vector(new_args)
def xreplace(self, rule):
"""Replace occurrences of objects within the measure numbers of the
vector.
Parameters
==========
rule : dict-like
Expresses a replacement rule.
Returns
=======
Vector
Result of the replacement.
Examples
========
>>> from sympy import symbols, pi
>>> from sympy.physics.vector import ReferenceFrame
>>> A = ReferenceFrame('A')
>>> x, y, z = symbols('x y z')
>>> ((1 + x*y) * A.x).xreplace({x: pi})
(pi*y + 1)*A.x
>>> ((1 + x*y) * A.x).xreplace({x: pi, y: 2})
(1 + 2*pi)*A.x
Replacements occur only if an entire node in the expression tree is
matched:
>>> ((x*y + z) * A.x).xreplace({x*y: pi})
(z + pi)*A.x
>>> ((x*y*z) * A.x).xreplace({x*y: pi})
x*y*z*A.x
"""
new_args = []
for mat, frame in self.args:
mat = mat.xreplace(rule)
new_args.append([mat, frame])
return Vector(new_args)
class VectorTypeError(TypeError):
def __init__(self, other, want):
msg = filldedent("Expected an instance of %s, but received object "
"'%s' of %s." % (type(want), other, type(other)))
super().__init__(msg)
def _check_vector(other):
if not isinstance(other, Vector):
raise TypeError('A Vector must be supplied')
return other
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