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"""Geometry objects for use by wrapping pathways."""
from abc import ABC, abstractmethod
from sympy import Integer, acos, pi, sqrt, sympify, tan
from sympy.core.relational import Eq
from sympy.functions.elementary.trigonometric import atan2
from sympy.polys.polytools import cancel
from sympy.physics.vector import Vector, dot
from sympy.simplify.simplify import trigsimp
__all__ = [
'WrappingGeometryBase',
'WrappingCylinder',
'WrappingSphere',
]
class WrappingGeometryBase(ABC):
"""Abstract base class for all geometry classes to inherit from.
Notes
=====
Instances of this class cannot be directly instantiated by users. However,
it can be used to created custom geometry types through subclassing.
"""
@property
@abstractmethod
def point(cls):
"""The point with which the geometry is associated."""
pass
@abstractmethod
def point_on_surface(self, point):
"""Returns ``True`` if a point is on the geometry's surface.
Parameters
==========
point : Point
The point for which it's to be ascertained if it's on the
geometry's surface or not.
"""
pass
@abstractmethod
def geodesic_length(self, point_1, point_2):
"""Returns the shortest distance between two points on a geometry's
surface.
Parameters
==========
point_1 : Point
The point from which the geodesic length should be calculated.
point_2 : Point
The point to which the geodesic length should be calculated.
"""
pass
@abstractmethod
def geodesic_end_vectors(self, point_1, point_2):
"""The vectors parallel to the geodesic at the two end points.
Parameters
==========
point_1 : Point
The point from which the geodesic originates.
point_2 : Point
The point at which the geodesic terminates.
"""
pass
def __repr__(self):
"""Default representation of a geometry object."""
return f'{self.__class__.__name__}()'
class WrappingSphere(WrappingGeometryBase):
"""A solid spherical object.
Explanation
===========
A wrapping geometry that allows for circular arcs to be defined between
pairs of points. These paths are always geodetic (the shortest possible).
Examples
========
To create a ``WrappingSphere`` instance, a ``Symbol`` denoting its radius
and ``Point`` at which its center will be located are needed:
>>> from sympy import symbols
>>> from sympy.physics.mechanics import Point, WrappingSphere
>>> r = symbols('r')
>>> pO = Point('pO')
A sphere with radius ``r`` centered on ``pO`` can be instantiated with:
>>> WrappingSphere(r, pO)
WrappingSphere(radius=r, point=pO)
Parameters
==========
radius : Symbol
Radius of the sphere. This symbol must represent a value that is
positive and constant, i.e. it cannot be a dynamic symbol, nor can it
be an expression.
point : Point
A point at which the sphere is centered.
See Also
========
WrappingCylinder: Cylindrical geometry where the wrapping direction can be
defined.
"""
def __init__(self, radius, point):
"""Initializer for ``WrappingSphere``.
Parameters
==========
radius : Symbol
The radius of the sphere.
point : Point
A point on which the sphere is centered.
"""
self.radius = radius
self.point = point
@property
def radius(self):
"""Radius of the sphere."""
return self._radius
@radius.setter
def radius(self, radius):
self._radius = radius
@property
def point(self):
"""A point on which the sphere is centered."""
return self._point
@point.setter
def point(self, point):
self._point = point
def point_on_surface(self, point):
"""Returns ``True`` if a point is on the sphere's surface.
Parameters
==========
point : Point
The point for which it's to be ascertained if it's on the sphere's
surface or not. This point's position relative to the sphere's
center must be a simple expression involving the radius of the
sphere, otherwise this check will likely not work.
"""
point_vector = point.pos_from(self.point)
if isinstance(point_vector, Vector):
point_radius_squared = dot(point_vector, point_vector)
else:
point_radius_squared = point_vector**2
return Eq(point_radius_squared, self.radius**2) == True
def geodesic_length(self, point_1, point_2):
r"""Returns the shortest distance between two points on the sphere's
surface.
Explanation
===========
The geodesic length, i.e. the shortest arc along the surface of a
sphere, connecting two points can be calculated using the formula:
.. math::
l = \arccos\left(\mathbf{v}_1 \cdot \mathbf{v}_2\right)
where $\mathbf{v}_1$ and $\mathbf{v}_2$ are the unit vectors from the
sphere's center to the first and second points on the sphere's surface
respectively. Note that the actual path that the geodesic will take is
undefined when the two points are directly opposite one another.
Examples
========
A geodesic length can only be calculated between two points on the
sphere's surface. Firstly, a ``WrappingSphere`` instance must be
created along with two points that will lie on its surface:
>>> from sympy import symbols
>>> from sympy.physics.mechanics import (Point, ReferenceFrame,
... WrappingSphere)
>>> N = ReferenceFrame('N')
>>> r = symbols('r')
>>> pO = Point('pO')
>>> pO.set_vel(N, 0)
>>> sphere = WrappingSphere(r, pO)
>>> p1 = Point('p1')
>>> p2 = Point('p2')
Let's assume that ``p1`` lies at a distance of ``r`` in the ``N.x``
direction from ``pO`` and that ``p2`` is located on the sphere's
surface in the ``N.y + N.z`` direction from ``pO``. These positions can
be set with:
>>> p1.set_pos(pO, r*N.x)
>>> p1.pos_from(pO)
r*N.x
>>> p2.set_pos(pO, r*(N.y + N.z).normalize())
>>> p2.pos_from(pO)
sqrt(2)*r/2*N.y + sqrt(2)*r/2*N.z
The geodesic length, which is in this case is a quarter of the sphere's
circumference, can be calculated using the ``geodesic_length`` method:
>>> sphere.geodesic_length(p1, p2)
pi*r/2
If the ``geodesic_length`` method is passed an argument, the ``Point``
that doesn't lie on the sphere's surface then a ``ValueError`` is
raised because it's not possible to calculate a value in this case.
Parameters
==========
point_1 : Point
Point from which the geodesic length should be calculated.
point_2 : Point
Point to which the geodesic length should be calculated.
"""
for point in (point_1, point_2):
if not self.point_on_surface(point):
msg = (
f'Geodesic length cannot be calculated as point {point} '
f'with radius {point.pos_from(self.point).magnitude()} '
f'from the sphere\'s center {self.point} does not lie on '
f'the surface of {self} with radius {self.radius}.'
)
raise ValueError(msg)
point_1_vector = point_1.pos_from(self.point).normalize()
point_2_vector = point_2.pos_from(self.point).normalize()
central_angle = acos(point_2_vector.dot(point_1_vector))
geodesic_length = self.radius*central_angle
return geodesic_length
def geodesic_end_vectors(self, point_1, point_2):
"""The vectors parallel to the geodesic at the two end points.
Parameters
==========
point_1 : Point
The point from which the geodesic originates.
point_2 : Point
The point at which the geodesic terminates.
"""
pA, pB = point_1, point_2
pO = self.point
pA_vec = pA.pos_from(pO)
pB_vec = pB.pos_from(pO)
if pA_vec.cross(pB_vec) == 0:
msg = (
f'Can\'t compute geodesic end vectors for the pair of points '
f'{pA} and {pB} on a sphere {self} as they are diametrically '
f'opposed, thus the geodesic is not defined.'
)
raise ValueError(msg)
return (
pA_vec.cross(pB.pos_from(pA)).cross(pA_vec).normalize(),
pB_vec.cross(pA.pos_from(pB)).cross(pB_vec).normalize(),
)
def __repr__(self):
"""Representation of a ``WrappingSphere``."""
return (
f'{self.__class__.__name__}(radius={self.radius}, '
f'point={self.point})'
)
class WrappingCylinder(WrappingGeometryBase):
"""A solid (infinite) cylindrical object.
Explanation
===========
A wrapping geometry that allows for circular arcs to be defined between
pairs of points. These paths are always geodetic (the shortest possible) in
the sense that they will be a straight line on the unwrapped cylinder's
surface. However, it is also possible for a direction to be specified, i.e.
paths can be influenced such that they either wrap along the shortest side
or the longest side of the cylinder. To define these directions, rotations
are in the positive direction following the right-hand rule.
Examples
========
To create a ``WrappingCylinder`` instance, a ``Symbol`` denoting its
radius, a ``Vector`` defining its axis, and a ``Point`` through which its
axis passes are needed:
>>> from sympy import symbols
>>> from sympy.physics.mechanics import (Point, ReferenceFrame,
... WrappingCylinder)
>>> N = ReferenceFrame('N')
>>> r = symbols('r')
>>> pO = Point('pO')
>>> ax = N.x
A cylinder with radius ``r``, and axis parallel to ``N.x`` passing through
``pO`` can be instantiated with:
>>> WrappingCylinder(r, pO, ax)
WrappingCylinder(radius=r, point=pO, axis=N.x)
Parameters
==========
radius : Symbol
The radius of the cylinder.
point : Point
A point through which the cylinder's axis passes.
axis : Vector
The axis along which the cylinder is aligned.
See Also
========
WrappingSphere: Spherical geometry where the wrapping direction is always
geodetic.
"""
def __init__(self, radius, point, axis):
"""Initializer for ``WrappingCylinder``.
Parameters
==========
radius : Symbol
The radius of the cylinder. This symbol must represent a value that
is positive and constant, i.e. it cannot be a dynamic symbol.
point : Point
A point through which the cylinder's axis passes.
axis : Vector
The axis along which the cylinder is aligned.
"""
self.radius = radius
self.point = point
self.axis = axis
@property
def radius(self):
"""Radius of the cylinder."""
return self._radius
@radius.setter
def radius(self, radius):
self._radius = radius
@property
def point(self):
"""A point through which the cylinder's axis passes."""
return self._point
@point.setter
def point(self, point):
self._point = point
@property
def axis(self):
"""Axis along which the cylinder is aligned."""
return self._axis
@axis.setter
def axis(self, axis):
self._axis = axis.normalize()
def point_on_surface(self, point):
"""Returns ``True`` if a point is on the cylinder's surface.
Parameters
==========
point : Point
The point for which it's to be ascertained if it's on the
cylinder's surface or not. This point's position relative to the
cylinder's axis must be a simple expression involving the radius of
the sphere, otherwise this check will likely not work.
"""
relative_position = point.pos_from(self.point)
parallel = relative_position.dot(self.axis) * self.axis
point_vector = relative_position - parallel
if isinstance(point_vector, Vector):
point_radius_squared = dot(point_vector, point_vector)
else:
point_radius_squared = point_vector**2
return Eq(trigsimp(point_radius_squared), self.radius**2) == True
def geodesic_length(self, point_1, point_2):
"""The shortest distance between two points on a geometry's surface.
Explanation
===========
The geodesic length, i.e. the shortest arc along the surface of a
cylinder, connecting two points. It can be calculated using Pythagoras'
theorem. The first short side is the distance between the two points on
the cylinder's surface parallel to the cylinder's axis. The second
short side is the arc of a circle between the two points of the
cylinder's surface perpendicular to the cylinder's axis. The resulting
hypotenuse is the geodesic length.
Examples
========
A geodesic length can only be calculated between two points on the
cylinder's surface. Firstly, a ``WrappingCylinder`` instance must be
created along with two points that will lie on its surface:
>>> from sympy import symbols, cos, sin
>>> from sympy.physics.mechanics import (Point, ReferenceFrame,
... WrappingCylinder, dynamicsymbols)
>>> N = ReferenceFrame('N')
>>> r = symbols('r')
>>> pO = Point('pO')
>>> pO.set_vel(N, 0)
>>> cylinder = WrappingCylinder(r, pO, N.x)
>>> p1 = Point('p1')
>>> p2 = Point('p2')
Let's assume that ``p1`` is located at ``N.x + r*N.y`` relative to
``pO`` and that ``p2`` is located at ``r*(cos(q)*N.y + sin(q)*N.z)``
relative to ``pO``, where ``q(t)`` is a generalized coordinate
specifying the angle rotated around the ``N.x`` axis according to the
right-hand rule where ``N.y`` is zero. These positions can be set with:
>>> q = dynamicsymbols('q')
>>> p1.set_pos(pO, N.x + r*N.y)
>>> p1.pos_from(pO)
N.x + r*N.y
>>> p2.set_pos(pO, r*(cos(q)*N.y + sin(q)*N.z).normalize())
>>> p2.pos_from(pO).simplify()
r*cos(q(t))*N.y + r*sin(q(t))*N.z
The geodesic length, which is in this case a is the hypotenuse of a
right triangle where the other two side lengths are ``1`` (parallel to
the cylinder's axis) and ``r*q(t)`` (parallel to the cylinder's cross
section), can be calculated using the ``geodesic_length`` method:
>>> cylinder.geodesic_length(p1, p2).simplify()
sqrt(r**2*q(t)**2 + 1)
If the ``geodesic_length`` method is passed an argument ``Point`` that
doesn't lie on the sphere's surface then a ``ValueError`` is raised
because it's not possible to calculate a value in this case.
Parameters
==========
point_1 : Point
Point from which the geodesic length should be calculated.
point_2 : Point
Point to which the geodesic length should be calculated.
"""
for point in (point_1, point_2):
if not self.point_on_surface(point):
msg = (
f'Geodesic length cannot be calculated as point {point} '
f'with radius {point.pos_from(self.point).magnitude()} '
f'from the cylinder\'s center {self.point} does not lie on '
f'the surface of {self} with radius {self.radius} and axis '
f'{self.axis}.'
)
raise ValueError(msg)
relative_position = point_2.pos_from(point_1)
parallel_length = relative_position.dot(self.axis)
point_1_relative_position = point_1.pos_from(self.point)
point_1_perpendicular_vector = (
point_1_relative_position
- point_1_relative_position.dot(self.axis)*self.axis
).normalize()
point_2_relative_position = point_2.pos_from(self.point)
point_2_perpendicular_vector = (
point_2_relative_position
- point_2_relative_position.dot(self.axis)*self.axis
).normalize()
central_angle = _directional_atan(
cancel(point_1_perpendicular_vector
.cross(point_2_perpendicular_vector)
.dot(self.axis)),
cancel(point_1_perpendicular_vector.dot(point_2_perpendicular_vector)),
)
planar_arc_length = self.radius*central_angle
geodesic_length = sqrt(parallel_length**2 + planar_arc_length**2)
return geodesic_length
def geodesic_end_vectors(self, point_1, point_2):
"""The vectors parallel to the geodesic at the two end points.
Parameters
==========
point_1 : Point
The point from which the geodesic originates.
point_2 : Point
The point at which the geodesic terminates.
"""
point_1_from_origin_point = point_1.pos_from(self.point)
point_2_from_origin_point = point_2.pos_from(self.point)
if point_1_from_origin_point == point_2_from_origin_point:
msg = (
f'Cannot compute geodesic end vectors for coincident points '
f'{point_1} and {point_2} as no geodesic exists.'
)
raise ValueError(msg)
point_1_parallel = point_1_from_origin_point.dot(self.axis) * self.axis
point_2_parallel = point_2_from_origin_point.dot(self.axis) * self.axis
point_1_normal = (point_1_from_origin_point - point_1_parallel)
point_2_normal = (point_2_from_origin_point - point_2_parallel)
if point_1_normal == point_2_normal:
point_1_perpendicular = Vector(0)
point_2_perpendicular = Vector(0)
else:
point_1_perpendicular = self.axis.cross(point_1_normal).normalize()
point_2_perpendicular = -self.axis.cross(point_2_normal).normalize()
geodesic_length = self.geodesic_length(point_1, point_2)
relative_position = point_2.pos_from(point_1)
parallel_length = relative_position.dot(self.axis)
planar_arc_length = sqrt(geodesic_length**2 - parallel_length**2)
point_1_vector = (
planar_arc_length * point_1_perpendicular
+ parallel_length * self.axis
).normalize()
point_2_vector = (
planar_arc_length * point_2_perpendicular
- parallel_length * self.axis
).normalize()
return (point_1_vector, point_2_vector)
def __repr__(self):
"""Representation of a ``WrappingCylinder``."""
return (
f'{self.__class__.__name__}(radius={self.radius}, '
f'point={self.point}, axis={self.axis})'
)
def _directional_atan(numerator, denominator):
"""Compute atan in a directional sense as required for geodesics.
Explanation
===========
To be able to control the direction of the geodesic length along the
surface of a cylinder a dedicated arctangent function is needed that
properly handles the directionality of different case. This function
ensures that the central angle is always positive but shifting the case
where ``atan2`` would return a negative angle to be centered around
``2*pi``.
Notes
=====
This function only handles very specific cases, i.e. the ones that are
expected to be encountered when calculating symbolic geodesics on uniformly
curved surfaces. As such, ``NotImplemented`` errors can be raised in many
cases. This function is named with a leader underscore to indicate that it
only aims to provide very specific functionality within the private scope
of this module.
"""
if numerator.is_number and denominator.is_number:
angle = atan2(numerator, denominator)
if angle < 0:
angle += 2 * pi
elif numerator.is_number:
msg = (
f'Cannot compute a directional atan when the numerator {numerator} '
f'is numeric and the denominator {denominator} is symbolic.'
)
raise NotImplementedError(msg)
elif denominator.is_number:
msg = (
f'Cannot compute a directional atan when the numerator {numerator} '
f'is symbolic and the denominator {denominator} is numeric.'
)
raise NotImplementedError(msg)
else:
ratio = sympify(trigsimp(numerator / denominator))
if isinstance(ratio, tan):
angle = ratio.args[0]
elif (
ratio.is_Mul
and ratio.args[0] == Integer(-1)
and isinstance(ratio.args[1], tan)
):
angle = 2 * pi - ratio.args[1].args[0]
else:
msg = f'Cannot compute a directional atan for the value {ratio}.'
raise NotImplementedError(msg)
return angle