Source code for sympy.physics.vector.vector

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. """ 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.__eq__(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. if isinstance(ar[i][0][j], Add): pform = vp._print( ar[i][0][j]).parens() else: pform = vp._print( ar[i][0][j]) 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) _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): """Substituion 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)
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