Source code for sympy.geometry.point3d

"""Geometrical Points.

Contains
========
Point3D

"""

from __future__ import print_function, division

from sympy.core import S, sympify
from sympy.core.compatibility import iterable
from sympy.core.containers import Tuple
from sympy.simplify import simplify, nsimplify
from sympy.geometry.exceptions import GeometryError
from sympy.geometry.point import Point
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.complexes import im
from .entity import GeometryEntity
from sympy.matrices import Matrix
from sympy.core.numbers import Float
from sympy.core.evaluate import global_evaluate

[docs]class Point3D(GeometryEntity):
"""A point in a 3-dimensional Euclidean space.

Parameters
==========

coords : sequence of 3 coordinate values.

Attributes
==========

x
y
z
length

Raises
======

NotImplementedError
When trying to create a point other than 2 or 3 dimensions.
When intersection is called with object other than a Point.
TypeError
When trying to add or subtract points with different dimensions.

Notes
=====

Currently only 2-dimensional and 3-dimensional points are supported.

Examples
========

>>> from sympy import Point3D
>>> from sympy.abc import x
>>> Point3D(1, 2, 3)
Point3D(1, 2, 3)
>>> Point3D([1, 2, 3])
Point3D(1, 2, 3)
>>> Point3D(0, x, 3)
Point3D(0, x, 3)

Floats are automatically converted to Rational unless the
evaluate flag is False:

>>> Point3D(0.5, 0.25, 2)
Point3D(1/2, 1/4, 2)
>>> Point3D(0.5, 0.25, 3, evaluate=False)
Point3D(0.5, 0.25, 3)

"""
def __new__(cls, *args, **kwargs):
eval = kwargs.get('evaluate', global_evaluate[0])
if isinstance(args[0], (Point, Point3D)):
if not eval:
return args[0]
args = args[0].args
elif isinstance(args[0], Point):
args = args[0].args
else:
if iterable(args[0]):
args = args[0]
if len(args) not in (2, 3):
raise TypeError(
"Enter a 2 or 3 dimensional point")
coords = Tuple(*args)
if len(coords) == 2:
coords += (S.Zero,)
if eval:
coords = coords.xreplace(dict(
[(f, simplify(nsimplify(f, rational=True)))
for f in coords.atoms(Float)]))
return GeometryEntity.__new__(cls, *coords)

def __contains__(self, item):
return item == self

@property
[docs]    def x(self):
"""
Returns the X coordinate of the Point.

Examples
========

>>> from sympy import Point3D
>>> p = Point3D(0, 1, 3)
>>> p.x
0
"""
return self.args[0]

@property
[docs]    def y(self):
"""
Returns the Y coordinate of the Point.

Examples
========

>>> from sympy import Point3D
>>> p = Point3D(0, 1, 2)
>>> p.y
1
"""
return self.args[1]

@property
[docs]    def z(self):
"""
Returns the Z coordinate of the Point.

Examples
========

>>> from sympy import Point3D
>>> p = Point3D(0, 1, 1)
>>> p.z
1
"""
return self.args[2]

@property
[docs]    def length(self):
"""
Treating a Point as a Line, this returns 0 for the length of a Point.

Examples
========

>>> from sympy import Point3D
>>> p = Point3D(0, 1, 1)
>>> p.length
0
"""
return S.Zero

[docs]    def direction_ratio(self, point):
"""
Gives the direction ratio between 2 points

Parameters
==========

p : Point3D

Returns
=======

list

Examples
========

>>> from sympy import Point3D
>>> p1 = Point3D(1, 2, 3)
>>> p1.direction_ratio(Point3D(2, 3, 5))
[1, 1, 2]
"""
return [(point.x - self.x),(point.y - self.y),(point.z - self.z)]

[docs]    def direction_cosine(self, point):
"""
Gives the direction cosine between 2 points

Parameters
==========

p : Point3D

Returns
=======

list

Examples
========

>>> from sympy import Point3D
>>> p1 = Point3D(1, 2, 3)
>>> p1.direction_cosine(Point3D(2, 3, 5))
[sqrt(6)/6, sqrt(6)/6, sqrt(6)/3]
"""
a = self.direction_ratio(point)
b = sqrt(sum(i**2 for i in a))
return [(point.x - self.x) / b,(point.y - self.y) / b,
(point.z - self.z) / b]

@staticmethod
[docs]    def are_collinear(*points):
"""Is a sequence of points collinear?

Test whether or not a set of points are collinear. Returns True if
the set of points are collinear, or False otherwise.

Parameters
==========

points : sequence of Point

Returns
=======

are_collinear : boolean

========

sympy.geometry.line3d.Line3D

Examples
========

>>> from sympy import Point3D, Matrix
>>> from sympy.abc import x
>>> p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1)
>>> p3, p4, p5 = Point3D(2, 2, 2), Point3D(x, x, x), Point3D(1, 2, 6)
>>> Point3D.are_collinear(p1, p2, p3, p4)
True
>>> Point3D.are_collinear(p1, p2, p3, p5)
False
"""
# Coincident points are irrelevant and can confuse this algorithm.
# Use only unique points.
points = list(set(points))
if not all(isinstance(p, Point3D) for p in points):
raise TypeError('Must pass only 3D Point objects')

if len(points) < 2:
return False
if len(points) == 2:
return True  # two points always form a line
if len(points) == 3:
a = (points[0].direction_cosine(points[1]))
b = (points[0].direction_cosine(points[2]))
a = [abs(i) for i in a]
b = [abs(i) for i in b]
if a == b:
return True
else:
return False
# XXX Cross product is used now,
# If the concept needs to extend to more than three
# dimensions then another method would have to be used
for i in range(len(points) - 2):
pv1 = [j - k for j, k in zip(points[i].args,   \
points[i + 1].args)]
pv2 = [j - k for j, k in zip(points[i + 1].args,
points[i + 2].args)]
rank = Matrix([pv1, pv2]).rank()
if (rank != 1):
return False
return True

@staticmethod
[docs]    def are_coplanar(*points):
"""

This function tests whether passed points are coplanar or not.
It uses the fact that the triple scalar product of three vectors
vanishes if the vectors are coplanar. Which means that the volume
of the solid described by them will have to be zero for coplanarity.

Parameters
==========

A set of points 3D points

Returns
=======

boolean

Examples
========

>>> from sympy import Point3D
>>> p1 = Point3D(1, 2, 2)
>>> p2 = Point3D(2, 7, 2)
>>> p3 = Point3D(0, 0, 2)
>>> p4 = Point3D(1, 1, 2)
>>> Point3D.are_coplanar(p1, p2, p3, p4)
True
>>> p5 = Point3D(0, 1, 3)
>>> Point3D.are_coplanar(p1, p2, p3, p5)
False

"""
from sympy.geometry.plane import Plane
points = list(set(points))
if len(points) < 3:
raise ValueError('At least 3 points are needed to define a plane.')
a, b = points[:2]
for i, c in enumerate(points[2:]):
try:
p = Plane(a, b, c)
for j in (0, 1, i):
points.pop(j)
return all(p.is_coplanar(i) for i in points)
except NotImplementedError:  # XXX should be ValueError
pass
raise ValueError('At least 3 non-collinear points needed to define plane.')

[docs]    def distance(self, p):
"""The Euclidean distance from self to point p.

Parameters
==========

p : Point

Returns
=======

distance : number or symbolic expression.

========

sympy.geometry.line.Segment.length

Examples
========

>>> from sympy import Point3D
>>> p1, p2 = Point3D(1, 1, 1), Point3D(4, 5, 0)
>>> p1.distance(p2)
sqrt(26)

>>> from sympy.abc import x, y, z
>>> p3 = Point3D(x, y, z)
>>> p3.distance(Point3D(0, 0, 0))
sqrt(x**2 + y**2 + z**2)

"""
p = Point3D(p)
return sqrt(sum([(a - b)**2 for a, b in zip(self.args, p.args)]))

[docs]    def midpoint(self, p):
"""The midpoint between self and point p.

Parameters
==========

p : Point

Returns
=======

midpoint : Point

========

sympy.geometry.line.Segment.midpoint

Examples
========

>>> from sympy import Point3D
>>> p1, p2 = Point3D(1, 1, 1), Point3D(13, 5, 1)
>>> p1.midpoint(p2)
Point3D(7, 3, 1)

"""
p = Point3D(p)
return Point3D([simplify((a + b)*S.Half) for a, b in
zip(self.args, p.args)])

[docs]    def evalf(self, prec=None, **options):
"""Evaluate the coordinates of the point.

This method will, where possible, create and return a new Point
where the coordinates are evaluated as floating point numbers to
the precision indicated (default=15).

Returns
=======

point : Point

Examples
========

>>> from sympy import Point3D, Rational
>>> p1 = Point3D(Rational(1, 2), Rational(3, 2), Rational(5, 2))
>>> p1
Point3D(1/2, 3/2, 5/2)
>>> p1.evalf()
Point3D(0.5, 1.5, 2.5)

"""
coords = [x.evalf(prec, **options) for x in self.args]
return Point3D(*coords, evaluate=False)

n = evalf

[docs]    def intersection(self, o):
"""The intersection between this point and another point.

Parameters
==========

other : Point

Returns
=======

intersection : list of Points

Notes
=====

The return value will either be an empty list if there is no
intersection, otherwise it will contain this point.

Examples
========

>>> from sympy import Point3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 0, 0)
>>> p1.intersection(p2)
[]
>>> p1.intersection(p3)
[Point3D(0, 0, 0)]

"""
if isinstance(o, Point3D):
if self == o:
return [self]
return []

return o.intersection(self)

[docs]    def scale(self, x=1, y=1, z=1, pt=None):
"""Scale the coordinates of the Point by multiplying by
x and y after subtracting pt -- default is (0, 0) --
and then adding pt back again (i.e. pt is the point of
reference for the scaling).

========

translate

Examples
========

>>> from sympy import Point3D
>>> t = Point3D(1, 1, 1)
>>> t.scale(2)
Point3D(2, 1, 1)
>>> t.scale(2, 2)
Point3D(2, 2, 1)

"""
if pt:
pt = Point3D(pt)
return
self.translate(*(-pt).args).scale(x, y, z).translate(*pt.args)
return Point3D(self.x*x, self.y*y, self.z*z)

[docs]    def translate(self, x=0, y=0, z=0):
"""Shift the Point by adding x and y to the coordinates of the Point.

========

rotate, scale

Examples
========

>>> from sympy import Point3D
>>> t = Point3D(0, 1, 1)
>>> t.translate(2)
Point3D(2, 1, 1)
>>> t.translate(2, 2)
Point3D(2, 3, 1)
>>> t + Point3D(2, 2, 2)
Point3D(2, 3, 3)

"""
return Point3D(self.x + x, self.y + y, self.z + z)

[docs]    def transform(self, matrix):
"""Return the point after applying the transformation described
by the 3x3 Matrix, matrix.

========
geometry.entity.rotate
geometry.entity.scale
geometry.entity.translate
"""
x, y, z = self.args
return Point3D(*(Matrix(1, 3, [x, y, z])*matrix).tolist()[0][:2])

[docs]    def dot(self, p2):
"""Return dot product of self with another Point."""
p2 = Point3D(p2)
x1, y1, z1 = self.args
x2, y2, z2 = p2.args
return x1*x2 + y1*y2 + z1*z2

def equals(self, other):
if not isinstance(other, Point3D):
return False
return all(a.equals(b) for a, b in zip(self.args, other.args))

"""Add other to self by incrementing self's coordinates by those of
other.

========

sympy.geometry.entity.translate

"""

if isinstance(other, Point3D):
if len(other.args) == len(self.args):
return Point3D(*[simplify(a + b) for a, b in
zip(self.args, other.args)])
else:
raise TypeError(
"Points must have the same number of dimensions")
else:
raise ValueError('Cannot add non-Point, %s, to a Point' % other)

def __sub__(self, other):
"""Subtract two points, or subtract a factor from this point's
coordinates."""
if isinstance(other, Point3D):
if len(other.args) == len(self.args):
return Point3D(*[simplify(a - b) for a, b in
zip(self.args, other.args)])
else:
raise TypeError(
"Points must have the same number of dimensions")
else:
raise ValueError('Cannot subtract non-Point, %s, to a Point'
% other)

def __mul__(self, factor):
"""Multiply point's coordinates by a factor."""
factor = sympify(factor)
return Point3D([x*factor for x in self.args])

def __div__(self, divisor):
"""Divide point's coordinates by a factor."""
divisor = sympify(divisor)
return Point3D([x/divisor for x in self.args])

__truediv__ = __div__

def __neg__(self):
"""Negate the point."""
return Point3D([-x for x in self.args])

def __abs__(self):
"""Returns the distance between this point and the origin."""
origin = Point3D([0]*len(self.args))
return Point3D.distance(origin, self)