859 lines
27 KiB
Python
859 lines
27 KiB
Python
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from sympy.core.backend import (S, sympify, expand, sqrt, Add, zeros, acos,
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ImmutableMatrix as Matrix, _simplify_matrix)
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from sympy.simplify.trigsimp import trigsimp
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from sympy.printing.defaults import Printable
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from sympy.utilities.misc import filldedent
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from sympy.core.evalf import EvalfMixin
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from mpmath.libmp.libmpf import prec_to_dps
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__all__ = ['Vector']
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class Vector(Printable, EvalfMixin):
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"""The class used to define vectors.
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It along with ReferenceFrame are the building blocks of describing a
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classical mechanics system in PyDy and sympy.physics.vector.
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Attributes
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==========
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simp : Boolean
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Let certain methods use trigsimp on their outputs
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"""
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simp = False
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is_number = False
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def __init__(self, inlist):
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"""This is the constructor for the Vector class. You should not be
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calling this, it should only be used by other functions. You should be
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treating Vectors like you would with if you were doing the math by
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hand, and getting the first 3 from the standard basis vectors from a
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ReferenceFrame.
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The only exception is to create a zero vector:
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zv = Vector(0)
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"""
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self.args = []
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if inlist == 0:
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inlist = []
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if isinstance(inlist, dict):
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d = inlist
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else:
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d = {}
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for inp in inlist:
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if inp[1] in d:
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d[inp[1]] += inp[0]
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else:
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d[inp[1]] = inp[0]
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for k, v in d.items():
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if v != Matrix([0, 0, 0]):
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self.args.append((v, k))
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@property
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def func(self):
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"""Returns the class Vector. """
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return Vector
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def __hash__(self):
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return hash(tuple(self.args))
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def __add__(self, other):
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"""The add operator for Vector. """
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if other == 0:
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return self
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other = _check_vector(other)
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return Vector(self.args + other.args)
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def __and__(self, other):
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"""Dot product of two vectors.
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Returns a scalar, the dot product of the two Vectors
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Parameters
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==========
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other : Vector
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The Vector which we are dotting with
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Examples
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========
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>>> from sympy.physics.vector import ReferenceFrame, dot
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>>> from sympy import symbols
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>>> q1 = symbols('q1')
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>>> N = ReferenceFrame('N')
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>>> dot(N.x, N.x)
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1
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>>> dot(N.x, N.y)
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0
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>>> A = N.orientnew('A', 'Axis', [q1, N.x])
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>>> dot(N.y, A.y)
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cos(q1)
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"""
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from sympy.physics.vector.dyadic import Dyadic
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if isinstance(other, Dyadic):
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return NotImplemented
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other = _check_vector(other)
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out = S.Zero
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for i, v1 in enumerate(self.args):
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for j, v2 in enumerate(other.args):
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out += ((v2[0].T)
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* (v2[1].dcm(v1[1]))
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* (v1[0]))[0]
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if Vector.simp:
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return trigsimp(out, recursive=True)
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else:
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return out
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def __truediv__(self, other):
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"""This uses mul and inputs self and 1 divided by other. """
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return self.__mul__(S.One / other)
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def __eq__(self, other):
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"""Tests for equality.
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It is very import to note that this is only as good as the SymPy
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equality test; False does not always mean they are not equivalent
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Vectors.
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If other is 0, and self is empty, returns True.
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If other is 0 and self is not empty, returns False.
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If none of the above, only accepts other as a Vector.
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"""
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if other == 0:
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other = Vector(0)
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try:
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other = _check_vector(other)
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except TypeError:
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return False
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if (self.args == []) and (other.args == []):
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return True
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elif (self.args == []) or (other.args == []):
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return False
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frame = self.args[0][1]
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for v in frame:
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if expand((self - other) & v) != 0:
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return False
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return True
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def __mul__(self, other):
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"""Multiplies the Vector by a sympifyable expression.
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Parameters
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==========
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other : Sympifyable
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The scalar to multiply this Vector with
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Examples
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========
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>>> from sympy.physics.vector import ReferenceFrame
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>>> from sympy import Symbol
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>>> N = ReferenceFrame('N')
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>>> b = Symbol('b')
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>>> V = 10 * b * N.x
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>>> print(V)
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10*b*N.x
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"""
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newlist = list(self.args)
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other = sympify(other)
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for i, v in enumerate(newlist):
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newlist[i] = (other * newlist[i][0], newlist[i][1])
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return Vector(newlist)
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def __neg__(self):
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return self * -1
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def __or__(self, other):
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"""Outer product between two Vectors.
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A rank increasing operation, which returns a Dyadic from two Vectors
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Parameters
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==========
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other : Vector
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The Vector to take the outer product with
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Examples
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========
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>>> from sympy.physics.vector import ReferenceFrame, outer
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>>> N = ReferenceFrame('N')
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>>> outer(N.x, N.x)
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(N.x|N.x)
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"""
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from sympy.physics.vector.dyadic import Dyadic
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other = _check_vector(other)
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ol = Dyadic(0)
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for i, v in enumerate(self.args):
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for i2, v2 in enumerate(other.args):
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# it looks this way because if we are in the same frame and
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# use the enumerate function on the same frame in a nested
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# fashion, then bad things happen
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ol += Dyadic([(v[0][0] * v2[0][0], v[1].x, v2[1].x)])
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ol += Dyadic([(v[0][0] * v2[0][1], v[1].x, v2[1].y)])
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ol += Dyadic([(v[0][0] * v2[0][2], v[1].x, v2[1].z)])
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ol += Dyadic([(v[0][1] * v2[0][0], v[1].y, v2[1].x)])
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ol += Dyadic([(v[0][1] * v2[0][1], v[1].y, v2[1].y)])
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ol += Dyadic([(v[0][1] * v2[0][2], v[1].y, v2[1].z)])
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ol += Dyadic([(v[0][2] * v2[0][0], v[1].z, v2[1].x)])
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ol += Dyadic([(v[0][2] * v2[0][1], v[1].z, v2[1].y)])
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ol += Dyadic([(v[0][2] * v2[0][2], v[1].z, v2[1].z)])
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return ol
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def _latex(self, printer):
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"""Latex Printing method. """
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ar = self.args # just to shorten things
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if len(ar) == 0:
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return str(0)
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ol = [] # output list, to be concatenated to a string
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for i, v in enumerate(ar):
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for j in 0, 1, 2:
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# if the coef of the basis vector is 1, we skip the 1
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if ar[i][0][j] == 1:
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ol.append(' + ' + ar[i][1].latex_vecs[j])
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# if the coef of the basis vector is -1, we skip the 1
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elif ar[i][0][j] == -1:
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ol.append(' - ' + ar[i][1].latex_vecs[j])
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elif ar[i][0][j] != 0:
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# If the coefficient of the basis vector is not 1 or -1;
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# also, we might wrap it in parentheses, for readability.
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arg_str = printer._print(ar[i][0][j])
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if isinstance(ar[i][0][j], Add):
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arg_str = "(%s)" % arg_str
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if arg_str[0] == '-':
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arg_str = arg_str[1:]
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str_start = ' - '
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else:
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str_start = ' + '
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ol.append(str_start + arg_str + ar[i][1].latex_vecs[j])
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outstr = ''.join(ol)
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if outstr.startswith(' + '):
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outstr = outstr[3:]
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elif outstr.startswith(' '):
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outstr = outstr[1:]
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return outstr
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def _pretty(self, printer):
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"""Pretty Printing method. """
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from sympy.printing.pretty.stringpict import prettyForm
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e = self
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class Fake:
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def render(self, *args, **kwargs):
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ar = e.args # just to shorten things
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if len(ar) == 0:
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return str(0)
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pforms = [] # output list, to be concatenated to a string
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for i, v in enumerate(ar):
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for j in 0, 1, 2:
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# if the coef of the basis vector is 1, we skip the 1
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if ar[i][0][j] == 1:
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pform = printer._print(ar[i][1].pretty_vecs[j])
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# if the coef of the basis vector is -1, we skip the 1
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elif ar[i][0][j] == -1:
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pform = printer._print(ar[i][1].pretty_vecs[j])
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pform = prettyForm(*pform.left(" - "))
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bin = prettyForm.NEG
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pform = prettyForm(binding=bin, *pform)
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elif ar[i][0][j] != 0:
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# If the basis vector coeff is not 1 or -1,
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# we might wrap it in parentheses, for readability.
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pform = printer._print(ar[i][0][j])
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if isinstance(ar[i][0][j], Add):
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tmp = pform.parens()
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pform = prettyForm(tmp[0], tmp[1])
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pform = prettyForm(*pform.right(
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" ", ar[i][1].pretty_vecs[j]))
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else:
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continue
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pforms.append(pform)
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pform = prettyForm.__add__(*pforms)
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kwargs["wrap_line"] = kwargs.get("wrap_line")
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kwargs["num_columns"] = kwargs.get("num_columns")
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out_str = pform.render(*args, **kwargs)
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mlines = [line.rstrip() for line in out_str.split("\n")]
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return "\n".join(mlines)
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return Fake()
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def __ror__(self, other):
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"""Outer product between two Vectors.
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A rank increasing operation, which returns a Dyadic from two Vectors
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Parameters
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==========
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other : Vector
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The Vector to take the outer product with
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Examples
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========
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>>> from sympy.physics.vector import ReferenceFrame, outer
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>>> N = ReferenceFrame('N')
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>>> outer(N.x, N.x)
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(N.x|N.x)
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"""
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from sympy.physics.vector.dyadic import Dyadic
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other = _check_vector(other)
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ol = Dyadic(0)
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for i, v in enumerate(other.args):
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for i2, v2 in enumerate(self.args):
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# it looks this way because if we are in the same frame and
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# use the enumerate function on the same frame in a nested
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# fashion, then bad things happen
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ol += Dyadic([(v[0][0] * v2[0][0], v[1].x, v2[1].x)])
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ol += Dyadic([(v[0][0] * v2[0][1], v[1].x, v2[1].y)])
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ol += Dyadic([(v[0][0] * v2[0][2], v[1].x, v2[1].z)])
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ol += Dyadic([(v[0][1] * v2[0][0], v[1].y, v2[1].x)])
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ol += Dyadic([(v[0][1] * v2[0][1], v[1].y, v2[1].y)])
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ol += Dyadic([(v[0][1] * v2[0][2], v[1].y, v2[1].z)])
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ol += Dyadic([(v[0][2] * v2[0][0], v[1].z, v2[1].x)])
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ol += Dyadic([(v[0][2] * v2[0][1], v[1].z, v2[1].y)])
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ol += Dyadic([(v[0][2] * v2[0][2], v[1].z, v2[1].z)])
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return ol
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def __rsub__(self, other):
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return (-1 * self) + other
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def _sympystr(self, printer, order=True):
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"""Printing method. """
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if not order or len(self.args) == 1:
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ar = list(self.args)
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elif len(self.args) == 0:
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return printer._print(0)
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else:
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d = {v[1]: v[0] for v in self.args}
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keys = sorted(d.keys(), key=lambda x: x.index)
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ar = []
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for key in keys:
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ar.append((d[key], key))
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ol = [] # output list, to be concatenated to a string
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for i, v in enumerate(ar):
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for j in 0, 1, 2:
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# if the coef of the basis vector is 1, we skip the 1
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if ar[i][0][j] == 1:
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ol.append(' + ' + ar[i][1].str_vecs[j])
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# if the coef of the basis vector is -1, we skip the 1
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elif ar[i][0][j] == -1:
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ol.append(' - ' + ar[i][1].str_vecs[j])
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elif ar[i][0][j] != 0:
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# 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 __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 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
|
||
|
|
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 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 & tempx, self & tempy, self & tempz],
|
||
|
|
[Vector([ar[i]]) & tempx, Vector([ar[i]]) & tempy,
|
||
|
|
Vector([ar[i]]) & tempz]])
|
||
|
|
outlist += _det(tempm).args
|
||
|
|
return Vector(outlist)
|
||
|
|
|
||
|
|
__radd__ = __add__
|
||
|
|
__rand__ = __and__
|
||
|
|
__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 dot(self, other):
|
||
|
|
return self & other
|
||
|
|
dot.__doc__ = __and__.__doc__
|
||
|
|
|
||
|
|
def cross(self, other):
|
||
|
|
return self ^ other
|
||
|
|
cross.__doc__ = __xor__.__doc__
|
||
|
|
|
||
|
|
def outer(self, other):
|
||
|
|
return self | other
|
||
|
|
outer.__doc__ = __or__.__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 Vector
|
||
|
|
>>> from sympy.physics.vector import init_vprinting
|
||
|
|
>>> init_vprinting(pretty_print=False)
|
||
|
|
>>> 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)
|
||
|
|
- sin(q1)*q1'*N.x - cos(q1)*q1'*N.z
|
||
|
|
>>> A.x.diff(t, N).express(A)
|
||
|
|
- 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_matrix(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 & 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
|