from sympy.core.backend import (S, sympify, expand, sqrt, Add, zeros,
ImmutableMatrix as Matrix)
from sympy import trigsimp
from sympy.core.compatibility import unicode
from sympy.utilities.misc import filldedent
__all__ = ['Vector']
[docs]class Vector(object):
"""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
def __init__(self, inlist):
"""This is the constructor for the Vector class. You shouldn't 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))
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 __and__(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
if isinstance(other, Dyadic):
return NotImplemented
other = _check_vector(other)
out = S(0)
for i, v1 in enumerate(self.args):
for j, v2 in enumerate(other.args):
out += ((v2[0].T)
* (v2[1].dcm(v1[1]))
* (v1[0]))[0]
if Vector.simp:
return trigsimp(sympify(out), recursive=True)
else:
return sympify(out)
def __div__(self, other):
"""This uses mul and inputs self and 1 divided by other. """
return self.__mul__(sympify(1) / other)
__truediv__ = __div__
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) & 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 = [v for v in self.args]
for i, v in enumerate(newlist):
newlist[i] = (sympify(other) * newlist[i][0], newlist[i][1])
return Vector(newlist)
def __ne__(self, other):
return not self == other
def __neg__(self):
return self * -1
def __or__(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 i, v in enumerate(self.args):
for i2, v2 in enumerate(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=None):
"""Latex Printing method. """
from sympy.physics.vector.printing import VectorLatexPrinter
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 = VectorLatexPrinter().doprint(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=None):
"""Pretty Printing method. """
from sympy.physics.vector.printing import VectorPrettyPrinter
from sympy.printing.pretty.stringpict import prettyForm
e = self
class Fake(object):
def render(self, *args, **kwargs):
ar = e.args # just to shorten things
if len(ar) == 0:
return unicode(0)
settings = printer._settings if printer else {}
vp = printer if printer else VectorPrettyPrinter(settings)
pforms = [] # 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:
pform = vp._print(ar[i][1].pretty_vecs[j])
# if the coef of the basis vector is -1, we skip the 1
elif ar[i][0][j] == -1:
pform = vp._print(ar[i][1].pretty_vecs[j])
pform = prettyForm(*pform.left(" - "))
bin = prettyForm.NEG
pform = prettyForm(binding=bin, *pform)
elif ar[i][0][j] != 0:
# If the basis vector coeff is not 1 or -1,
# we might wrap it in parentheses, for readability.
pform = vp._print(ar[i][0][j])
if isinstance(ar[i][0][j], Add):
tmp = pform.parens()
pform = prettyForm(tmp[0], tmp[1])
pform = prettyForm(*pform.right(" ",
ar[i][1].pretty_vecs[j]))
else:
continue
pforms.append(pform)
pform = prettyForm.__add__(*pforms)
kwargs["wrap_line"] = kwargs.get("wrap_line")
kwargs["num_columns"] = kwargs.get("num_columns")
out_str = pform.render(*args, **kwargs)
mlines = [line.rstrip() for line in out_str.split("\n")]
return "\n".join(mlines)
return Fake()
def __ror__(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 i, v in enumerate(other.args):
for i2, v2 in enumerate(self.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 __rsub__(self, other):
return (-1 * self) + other
def __str__(self, printer=None, order=True):
"""Printing method. """
from sympy.physics.vector.printing import VectorStrPrinter
if not order or len(self.args) == 1:
ar = list(self.args)
elif len(self.args) == 0:
return str(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 = VectorStrPrinter().doprint(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 subraction operator. """
return self.__add__(other * -1)
def __xor__(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.physics.vector import ReferenceFrame, Vector
>>> from sympy import symbols
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> N.x ^ N.y
N.z
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
>>> A.x ^ N.y
N.z
>>> N.y ^ A.x
- sin(q1)*A.y - cos(q1)*A.z
"""
from sympy.physics.vector.dyadic import Dyadic
if isinstance(other, Dyadic):
return NotImplemented
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 won't take in Vector, so need a custom function.
You shouldn't 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 & tempx, self & tempy,
self & tempz], [Vector([ar[i]]) & tempx,
Vector([ar[i]]) & tempy, Vector([ar[i]]) & tempz]])
outlist += _det(tempm).args
return Vector(outlist)
# We don't define _repr_png_ here because it would add a large amount of
# data to any notebook containing SymPy expressions, without adding
# anything useful to the notebook. It can still enabled manually, e.g.,
# for the qtconsole, with init_printing().
def _repr_latex_(self):
"""
IPython/Jupyter LaTeX printing
To change the behavior of this (e.g., pass in some settings to LaTeX),
use init_printing(). init_printing() will also enable LaTeX printing
for built in numeric types like ints and container types that contain
SymPy objects, like lists and dictionaries of expressions.
"""
from sympy.printing.latex import latex
s = latex(self, mode='plain')
return "$\\displaystyle %s$" % s
_repr_latex_orig = _repr_latex_
_sympystr = __str__
_sympyrepr = _sympystr
__repr__ = __str__
__radd__ = __add__
__rand__ = __and__
__rmul__ = __mul__
[docs] 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
[docs] def dot(self, other):
return self & other
dot.__doc__ = __and__.__doc__
[docs] def cross(self, other):
return self ^ other
cross.__doc__ = __xor__.__doc__
[docs] def outer(self, other):
return self | other
outer.__doc__ = __or__.__doc__
[docs] 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 Vector
>>> Vector.simp = True
>>> t = Symbol('t')
>>> q1 = dynamicsymbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.y])
>>> A.x.diff(t, N)
- 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
var = sympify(var)
_check_frame(frame)
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)]).express(component_frame).args
return Vector(inlist)
[docs] 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, Vector, dynamicsymbols
>>> 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)
[docs] 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
>>> from sympy.physics.mechanics.functions import inertia
>>> 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)
[docs] 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)
[docs] 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)
[docs] def simplify(self):
"""Returns a simplified Vector."""
d = {}
for v in self.args:
d[v[1]] = v[0].simplify()
return Vector(d)
[docs] 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)
[docs] def magnitude(self):
"""Returns the magnitude (Euclidean norm) of self."""
return sqrt(self & self)
[docs] def normalize(self):
"""Returns a Vector of magnitude 1, codirectional with self."""
return Vector(self.args + []) / self.magnitude()
[docs] 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)
[docs] def free_symbols(self, reference_frame):
"""
Returns the free symbols in the measure numbers of the vector
expressed in the given reference frame.
Parameter
=========
reference_frame : ReferenceFrame
The frame with respect to which the free symbols of the
given vector is to be determined.
"""
return self.to_matrix(reference_frame).free_symbols
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(VectorTypeError, self).__init__(msg)
def _check_vector(other):
if not isinstance(other, Vector):
raise TypeError('A Vector must be supplied')
return other
