from sympy.core.basic import Basic
from sympy.vector.scalar import BaseScalar
from sympy import eye, trigsimp, ImmutableMatrix as Matrix, Symbol
from sympy.core.compatibility import string_types, range
from sympy.core.cache import cacheit
from sympy.vector.orienters import (Orienter, AxisOrienter, BodyOrienter,
SpaceOrienter, QuaternionOrienter)
import sympy.vector
[docs]class CoordSysCartesian(Basic):
"""
Represents a coordinate system in 3-D space.
"""
def __new__(cls, name, location=None, rotation_matrix=None,
parent=None, vector_names=None, variable_names=None):
"""
The orientation/location parameters are necessary if this system
is being defined at a certain orientation or location wrt another.
Parameters
==========
name : str
The name of the new CoordSysCartesian instance.
location : Vector
The position vector of the new system's origin wrt the parent
instance.
rotation_matrix : SymPy ImmutableMatrix
The rotation matrix of the new coordinate system with respect
to the parent. In other words, the output of
new_system.rotation_matrix(parent).
parent : CoordSysCartesian
The coordinate system wrt which the orientation/location
(or both) is being defined.
vector_names, variable_names : iterable(optional)
Iterables of 3 strings each, with custom names for base
vectors and base scalars of the new system respectively.
Used for simple str printing.
"""
name = str(name)
Vector = sympy.vector.Vector
BaseVector = sympy.vector.BaseVector
Point = sympy.vector.Point
if not isinstance(name, string_types):
raise TypeError("name should be a string")
# If orientation information has been provided, store
# the rotation matrix accordingly
if rotation_matrix is None:
parent_orient = Matrix(eye(3))
else:
if not isinstance(rotation_matrix, Matrix):
raise TypeError("rotation_matrix should be an Immutable" +
"Matrix instance")
parent_orient = rotation_matrix
# If location information is not given, adjust the default
# location as Vector.zero
if parent is not None:
if not isinstance(parent, CoordSysCartesian):
raise TypeError("parent should be a " +
"CoordSysCartesian/None")
if location is None:
location = Vector.zero
else:
if not isinstance(location, Vector):
raise TypeError("location should be a Vector")
# Check that location does not contain base
# scalars
for x in location.free_symbols:
if isinstance(x, BaseScalar):
raise ValueError("location should not contain" +
" BaseScalars")
origin = parent.origin.locate_new(name + '.origin',
location)
else:
location = Vector.zero
origin = Point(name + '.origin')
# All systems that are defined as 'roots' are unequal, unless
# they have the same name.
# Systems defined at same orientation/position wrt the same
# 'parent' are equal, irrespective of the name.
# This is true even if the same orientation is provided via
# different methods like Axis/Body/Space/Quaternion.
# However, coincident systems may be seen as unequal if
# positioned/oriented wrt different parents, even though
# they may actually be 'coincident' wrt the root system.
if parent is not None:
obj = super(CoordSysCartesian, cls).__new__(
cls, Symbol(name), location, parent_orient, parent)
else:
obj = super(CoordSysCartesian, cls).__new__(
cls, Symbol(name), location, parent_orient)
obj._name = name
# Initialize the base vectors
if vector_names is None:
vector_names = (name + '.i', name + '.j', name + '.k')
latex_vects = [(r'\mathbf{\hat{i}_{%s}}' % name),
(r'\mathbf{\hat{j}_{%s}}' % name),
(r'\mathbf{\hat{k}_{%s}}' % name)]
pretty_vects = (name + '_i', name + '_j', name + '_k')
else:
_check_strings('vector_names', vector_names)
vector_names = list(vector_names)
latex_vects = [(r'\mathbf{\hat{%s}_{%s}}' % (x, name)) for
x in vector_names]
pretty_vects = [(name + '_' + x) for x in vector_names]
obj._i = BaseVector(vector_names[0], 0, obj,
pretty_vects[0], latex_vects[0])
obj._j = BaseVector(vector_names[1], 1, obj,
pretty_vects[1], latex_vects[1])
obj._k = BaseVector(vector_names[2], 2, obj,
pretty_vects[2], latex_vects[2])
# Initialize the base scalars
if variable_names is None:
variable_names = (name + '.x', name + '.y', name + '.z')
latex_scalars = [(r"\mathbf{{x}_{%s}}" % name),
(r"\mathbf{{y}_{%s}}" % name),
(r"\mathbf{{z}_{%s}}" % name)]
pretty_scalars = (name + '_x', name + '_y', name + '_z')
else:
_check_strings('variable_names', vector_names)
variable_names = list(variable_names)
latex_scalars = [(r"\mathbf{{%s}_{%s}}" % (x, name)) for
x in variable_names]
pretty_scalars = [(name + '_' + x) for x in variable_names]
obj._x = BaseScalar(variable_names[0], 0, obj,
pretty_scalars[0], latex_scalars[0])
obj._y = BaseScalar(variable_names[1], 1, obj,
pretty_scalars[1], latex_scalars[1])
obj._z = BaseScalar(variable_names[2], 2, obj,
pretty_scalars[2], latex_scalars[2])
# Assign a Del operator instance
from sympy.vector.deloperator import Del
obj._delop = Del(obj)
# Assign params
obj._parent = parent
if obj._parent is not None:
obj._root = obj._parent._root
else:
obj._root = obj
obj._parent_rotation_matrix = parent_orient
obj._origin = origin
# Return the instance
return obj
def __str__(self, printer=None):
return self._name
__repr__ = __str__
_sympystr = __str__
def __iter__(self):
return iter([self.i, self.j, self.k])
@property
def origin(self):
return self._origin
@property
def delop(self):
return self._delop
@property
def i(self):
return self._i
@property
def j(self):
return self._j
@property
def k(self):
return self._k
@property
def x(self):
return self._x
@property
def y(self):
return self._y
@property
def z(self):
return self._z
def base_vectors(self):
return self._i, self._j, self._k
def base_scalars(self):
return self._x, self._y, self._z
@cacheit
[docs] def rotation_matrix(self, other):
"""
Returns the direction cosine matrix(DCM), also known as the
'rotation matrix' of this coordinate system with respect to
another system.
If v_a is a vector defined in system 'A' (in matrix format)
and v_b is the same vector defined in system 'B', then
v_a = A.rotation_matrix(B) * v_b.
A SymPy Matrix is returned.
Parameters
==========
other : CoordSysCartesian
The system which the DCM is generated to.
Examples
========
>>> from sympy.vector import CoordSysCartesian
>>> from sympy import symbols
>>> q1 = symbols('q1')
>>> N = CoordSysCartesian('N')
>>> A = N.orient_new_axis('A', q1, N.i)
>>> N.rotation_matrix(A)
Matrix([
[1, 0, 0],
[0, cos(q1), -sin(q1)],
[0, sin(q1), cos(q1)]])
"""
from sympy.vector.functions import _path
if not isinstance(other, CoordSysCartesian):
raise TypeError(str(other) +
" is not a CoordSysCartesian")
# Handle special cases
if other == self:
return eye(3)
elif other == self._parent:
return self._parent_rotation_matrix
elif other._parent == self:
return other._parent_rotation_matrix.T
# Else, use tree to calculate position
rootindex, path = _path(self, other)
result = eye(3)
i = -1
for i in range(rootindex):
result *= path[i]._parent_rotation_matrix
i += 2
while i < len(path):
result *= path[i]._parent_rotation_matrix.T
i += 1
return result
@cacheit
[docs] def position_wrt(self, other):
"""
Returns the position vector of the origin of this coordinate
system with respect to another Point/CoordSysCartesian.
Parameters
==========
other : Point/CoordSysCartesian
If other is a Point, the position of this system's origin
wrt it is returned. If its an instance of CoordSyRect,
the position wrt its origin is returned.
Examples
========
>>> from sympy.vector import CoordSysCartesian
>>> N = CoordSysCartesian('N')
>>> N1 = N.locate_new('N1', 10 * N.i)
>>> N.position_wrt(N1)
(-10)*N.i
"""
return self.origin.position_wrt(other)
[docs] def scalar_map(self, other):
"""
Returns a dictionary which expresses the coordinate variables
(base scalars) of this frame in terms of the variables of
otherframe.
Parameters
==========
otherframe : CoordSysCartesian
The other system to map the variables to.
Examples
========
>>> from sympy.vector import CoordSysCartesian
>>> from sympy import Symbol
>>> A = CoordSysCartesian('A')
>>> q = Symbol('q')
>>> B = A.orient_new_axis('B', q, A.k)
>>> A.scalar_map(B)
{A.x: -sin(q)*B.y + cos(q)*B.x, A.y: sin(q)*B.x + cos(q)*B.y, A.z: B.z}
"""
relocated_scalars = []
origin_coords = tuple(self.position_wrt(other).to_matrix(other))
for i, x in enumerate(other.base_scalars()):
relocated_scalars.append(x - origin_coords[i])
vars_matrix = (self.rotation_matrix(other) *
Matrix(relocated_scalars))
mapping = {}
for i, x in enumerate(self.base_scalars()):
mapping[x] = trigsimp(vars_matrix[i])
return mapping
[docs] def locate_new(self, name, position, vector_names=None,
variable_names=None):
"""
Returns a CoordSysCartesian with its origin located at the given
position wrt this coordinate system's origin.
Parameters
==========
name : str
The name of the new CoordSysCartesian instance.
position : Vector
The position vector of the new system's origin wrt this
one.
vector_names, variable_names : iterable(optional)
Iterables of 3 strings each, with custom names for base
vectors and base scalars of the new system respectively.
Used for simple str printing.
Examples
========
>>> from sympy.vector import CoordSysCartesian
>>> A = CoordSysCartesian('A')
>>> B = A.locate_new('B', 10 * A.i)
>>> B.origin.position_wrt(A.origin)
10*A.i
"""
return CoordSysCartesian(name, location=position,
vector_names=vector_names,
variable_names=variable_names,
parent=self)
[docs] def orient_new(self, name, orienters, location=None,
vector_names=None, variable_names=None):
"""
Creates a new CoordSysCartesian oriented in the user-specified way
with respect to this system.
Please refer to the documentation of the orienter classes
for more information about the orientation procedure.
Parameters
==========
name : str
The name of the new CoordSysCartesian instance.
orienters : iterable/Orienter
An Orienter or an iterable of Orienters for orienting the
new coordinate system.
If an Orienter is provided, it is applied to get the new
system.
If an iterable is provided, the orienters will be applied
in the order in which they appear in the iterable.
location : Vector(optional)
The location of the new coordinate system's origin wrt this
system's origin. If not specified, the origins are taken to
be coincident.
vector_names, variable_names : iterable(optional)
Iterables of 3 strings each, with custom names for base
vectors and base scalars of the new system respectively.
Used for simple str printing.
Examples
========
>>> from sympy.vector import CoordSysCartesian
>>> from sympy import symbols
>>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
>>> N = CoordSysCartesian('N')
Using an AxisOrienter
>>> from sympy.vector import AxisOrienter
>>> axis_orienter = AxisOrienter(q1, N.i + 2 * N.j)
>>> A = N.orient_new('A', (axis_orienter, ))
Using a BodyOrienter
>>> from sympy.vector import BodyOrienter
>>> body_orienter = BodyOrienter(q1, q2, q3, '123')
>>> B = N.orient_new('B', (body_orienter, ))
Using a SpaceOrienter
>>> from sympy.vector import SpaceOrienter
>>> space_orienter = SpaceOrienter(q1, q2, q3, '312')
>>> C = N.orient_new('C', (space_orienter, ))
Using a QuaternionOrienter
>>> from sympy.vector import QuaternionOrienter
>>> q_orienter = QuaternionOrienter(q0, q1, q2, q3)
>>> D = N.orient_new('D', (q_orienter, ))
"""
if isinstance(orienters, Orienter):
if isinstance(orienters, AxisOrienter):
final_matrix = orienters.rotation_matrix(self)
else:
final_matrix = orienters.rotation_matrix()
# TODO: trigsimp is needed here so that the matrix becomes
# canonical (scalar_map also calls trigsimp; without this, you can
# end up with the same CoordinateSystem that compares differently
# due to a differently formatted matrix). However, this is
# probably not so good for performance.
final_matrix = trigsimp(final_matrix)
else:
final_matrix = Matrix(eye(3))
for orienter in orienters:
if isinstance(orienter, AxisOrienter):
final_matrix *= orienter.rotation_matrix(self)
else:
final_matrix *= orienter.rotation_matrix()
return CoordSysCartesian(name, rotation_matrix=final_matrix,
vector_names=vector_names,
variable_names=variable_names,
location=location,
parent=self)
[docs] def orient_new_axis(self, name, angle, axis, location=None,
vector_names=None, variable_names=None):
"""
Axis rotation is a rotation about an arbitrary axis by
some angle. The angle is supplied as a SymPy expr scalar, and
the axis is supplied as a Vector.
Parameters
==========
name : string
The name of the new coordinate system
angle : Expr
The angle by which the new system is to be rotated
axis : Vector
The axis around which the rotation has to be performed
location : Vector(optional)
The location of the new coordinate system's origin wrt this
system's origin. If not specified, the origins are taken to
be coincident.
vector_names, variable_names : iterable(optional)
Iterables of 3 strings each, with custom names for base
vectors and base scalars of the new system respectively.
Used for simple str printing.
Examples
========
>>> from sympy.vector import CoordSysCartesian
>>> from sympy import symbols
>>> q1 = symbols('q1')
>>> N = CoordSysCartesian('N')
>>> B = N.orient_new_axis('B', q1, N.i + 2 * N.j)
"""
orienter = AxisOrienter(angle, axis)
return self.orient_new(name, orienter,
location=location,
vector_names=vector_names,
variable_names=variable_names)
[docs] def orient_new_body(self, name, angle1, angle2, angle3,
rotation_order, location=None,
vector_names=None, variable_names=None):
"""
Body orientation takes this coordinate system through three
successive simple rotations.
Body fixed rotations include both Euler Angles and
Tait-Bryan Angles, see http://en.wikipedia.org/wiki/Euler_angles.
Parameters
==========
name : string
The name of the new coordinate system
angle1, angle2, angle3 : Expr
Three successive angles to rotate the coordinate system by
rotation_order : string
String defining the order of axes for rotation
location : Vector(optional)
The location of the new coordinate system's origin wrt this
system's origin. If not specified, the origins are taken to
be coincident.
vector_names, variable_names : iterable(optional)
Iterables of 3 strings each, with custom names for base
vectors and base scalars of the new system respectively.
Used for simple str printing.
Examples
========
>>> from sympy.vector import CoordSysCartesian
>>> from sympy import symbols
>>> q1, q2, q3 = symbols('q1 q2 q3')
>>> N = CoordSysCartesian('N')
A 'Body' fixed rotation is described by three angles and
three body-fixed rotation axes. To orient a coordinate system D
with respect to N, each sequential rotation is always about
the orthogonal unit vectors fixed to D. For example, a '123'
rotation will specify rotations about N.i, then D.j, then
D.k. (Initially, D.i is same as N.i)
Therefore,
>>> D = N.orient_new_body('D', q1, q2, q3, '123')
is same as
>>> D = N.orient_new_axis('D', q1, N.i)
>>> D = D.orient_new_axis('D', q2, D.j)
>>> D = D.orient_new_axis('D', q3, D.k)
Acceptable rotation orders are of length 3, expressed in XYZ or
123, and cannot have a rotation about about an axis twice in a row.
>>> B = N.orient_new_body('B', q1, q2, q3, '123')
>>> B = N.orient_new_body('B', q1, q2, 0, 'ZXZ')
>>> B = N.orient_new_body('B', 0, 0, 0, 'XYX')
"""
orienter = BodyOrienter(angle1, angle2, angle3, rotation_order)
return self.orient_new(name, orienter,
location=location,
vector_names=vector_names,
variable_names=variable_names)
[docs] def orient_new_space(self, name, angle1, angle2, angle3,
rotation_order, location=None,
vector_names=None, variable_names=None):
"""
Space rotation is similar to Body rotation, but the rotations
are applied in the opposite order.
Parameters
==========
name : string
The name of the new coordinate system
angle1, angle2, angle3 : Expr
Three successive angles to rotate the coordinate system by
rotation_order : string
String defining the order of axes for rotation
location : Vector(optional)
The location of the new coordinate system's origin wrt this
system's origin. If not specified, the origins are taken to
be coincident.
vector_names, variable_names : iterable(optional)
Iterables of 3 strings each, with custom names for base
vectors and base scalars of the new system respectively.
Used for simple str printing.
See Also
========
CoordSysCartesian.orient_new_body : method to orient via Euler
angles
Examples
========
>>> from sympy.vector import CoordSysCartesian
>>> from sympy import symbols
>>> q1, q2, q3 = symbols('q1 q2 q3')
>>> N = CoordSysCartesian('N')
To orient a coordinate system D with respect to N, each
sequential rotation is always about N's orthogonal unit vectors.
For example, a '123' rotation will specify rotations about
N.i, then N.j, then N.k.
Therefore,
>>> D = N.orient_new_space('D', q1, q2, q3, '312')
is same as
>>> B = N.orient_new_axis('B', q1, N.i)
>>> C = B.orient_new_axis('C', q2, N.j)
>>> D = C.orient_new_axis('D', q3, N.k)
"""
orienter = SpaceOrienter(angle1, angle2, angle3, rotation_order)
return self.orient_new(name, orienter,
location=location,
vector_names=vector_names,
variable_names=variable_names)
[docs] def orient_new_quaternion(self, name, q0, q1, q2, q3, location=None,
vector_names=None, variable_names=None):
"""
Quaternion orientation orients the new CoordSysCartesian with
Quaternions, defined as a finite rotation about lambda, a unit
vector, by some amount theta.
This orientation is described by four parameters:
q0 = cos(theta/2)
q1 = lambda_x sin(theta/2)
q2 = lambda_y sin(theta/2)
q3 = lambda_z sin(theta/2)
Quaternion does not take in a rotation order.
Parameters
==========
name : string
The name of the new coordinate system
q0, q1, q2, q3 : Expr
The quaternions to rotate the coordinate system by
location : Vector(optional)
The location of the new coordinate system's origin wrt this
system's origin. If not specified, the origins are taken to
be coincident.
vector_names, variable_names : iterable(optional)
Iterables of 3 strings each, with custom names for base
vectors and base scalars of the new system respectively.
Used for simple str printing.
Examples
========
>>> from sympy.vector import CoordSysCartesian
>>> from sympy import symbols
>>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
>>> N = CoordSysCartesian('N')
>>> B = N.orient_new_quaternion('B', q0, q1, q2, q3)
"""
orienter = QuaternionOrienter(q0, q1, q2, q3)
return self.orient_new(name, orienter,
location=location,
vector_names=vector_names,
variable_names=variable_names)
[docs] def __init__(self, name, location=None, rotation_matrix=None,
parent=None, vector_names=None, variable_names=None,
latex_vects=None, pretty_vects=None, latex_scalars=None,
pretty_scalars=None):
# Dummy initializer for setting docstring
pass
__init__.__doc__ = __new__.__doc__
def _check_strings(arg_name, arg):
errorstr = arg_name + " must be an iterable of 3 string-types"
if len(arg) != 3:
raise ValueError(errorstr)
try:
for s in arg:
if not isinstance(s, string_types):
raise TypeError(errorstr)
except:
raise TypeError(errorstr)
