"""Geometrical Points.
Contains
========
Point
Point2D
Point3D
"""
from __future__ import division, print_function
from sympy.core import S, sympify
from sympy.core.compatibility import iterable
from sympy.core.containers import Tuple
from sympy.simplify import nsimplify, simplify
from sympy.geometry.exceptions import GeometryError
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.complexes import im
from sympy.matrices import Matrix
from sympy.core.numbers import Float
from sympy.core.evaluate import global_evaluate
from sympy.core.add import Add
from .entity import GeometryEntity
[docs]class Point(GeometryEntity):
"""A point in a n-dimensional Euclidean space.
Parameters
==========
coords : sequence of n-coordinate values. In the special
case where n=2 or 3, a Point2D or Point3D will be created
as appropriate.
Attributes
==========
length
origin: A `Point` representing the origin of the
appropriately-dimensioned space.
Raises
======
TypeError
When trying to add or subtract points with different dimensions.
When `intersection` is called with object other than a Point.
See Also
========
sympy.geometry.line.Segment : Connects two Points
Examples
========
>>> from sympy.geometry import Point
>>> from sympy.abc import x
>>> Point(1, 2, 3)
Point3D(1, 2, 3)
>>> Point([1, 2])
Point2D(1, 2)
>>> Point(0, x)
Point2D(0, x)
Floats are automatically converted to Rational unless the
evaluate flag is False:
>>> Point(0.5, 0.25)
Point2D(1/2, 1/4)
>>> Point(0.5, 0.25, evaluate=False)
Point2D(0.5, 0.25)
"""
def __new__(cls, *args, **kwargs):
evaluate = kwargs.get('evaluate', global_evaluate[0])
if iterable(args[0]):
if isinstance(args[0], Point) and not evaluate:
return args[0]
args = args[0]
# unpack the arguments into a friendly Tuple
# if we were already a Point, we're doing an excess
# iteration, but we'll worry about efficiency later
coords = Tuple(*args)
if any(a.is_number and im(a) for a in coords):
raise ValueError('Imaginary coordinates not permitted.')
# Turn any Floats into rationals and simplify
# any expressions before we instantiate
if evaluate:
coords = coords.xreplace(dict(
[(f, simplify(nsimplify(f, rational=True)))
for f in coords.atoms(Float)]))
if len(coords) == 2:
return Point2D(coords, **kwargs)
if len(coords) == 3:
return Point3D(coords, **kwargs)
return GeometryEntity.__new__(cls, *coords)
is_Point = True
def __contains__(self, item):
return item == self
def is_concyclic(*args):
# Coincident points are irrelevant and can confuse this algorithm.
# Use only unique points.
args = list(set(args))
if not all(isinstance(p, Point) for p in args):
raise TypeError('Must pass only Point objects')
return args[0].is_concyclic(*args[1:])
[docs] def is_collinear(*args):
"""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
=======
is_collinear : boolean
Notes
=====
Slope is preserved everywhere on a line, so the slope between
any two points on the line should be the same. Take the first
two points, p1 and p2, and create a translated point v1
with p1 as the origin. Now for every other point we create
a translated point, vi with p1 also as the origin. Note that
these translations preserve slope since everything is
consistently translated to a new origin of p1. Since slope
is preserved then we have the following equality:
* v1_slope = vi_slope
* v1.y/v1.x = vi.y/vi.x (due to translation)
* v1.y*vi.x = vi.y*v1.x
* v1.y*vi.x - vi.y*v1.x = 0 (*)
Hence, if we have a vi such that the equality in (*) is False
then the points are not collinear. We do this test for every
point in the list, and if all pass then they are collinear.
See Also
========
sympy.geometry.line.Line
Examples
========
>>> from sympy import Point
>>> from sympy.abc import x
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> p3, p4, p5 = Point(2, 2), Point(x, x), Point(1, 2)
>>> Point.is_collinear(p1, p2, p3, p4)
True
>>> Point.is_collinear(p1, p2, p3, p5)
False
"""
# Coincident points are irrelevant; use only unique points.
args = list(set(args))
if not all(isinstance(p, Point) for p in args):
raise TypeError('Must pass only Point objects')
if len(args) == 0:
return False
if len(args) <= 2:
return True
# translate our points
points = [p - args[0] for p in args[1:]]
for p in points[1:]:
if not Point.is_scalar_multiple(points[0], p):
return False
return True
[docs] def is_scalar_multiple(p1, p2):
"""Returns whether `p1` and `p2` are scalar multiples
of eachother.
"""
# if the vectors p1 and p2 are linearly dependent, then they must
# be scalar multiples of eachother
m = Matrix([p1.args, p2.args])
# XXX: issue #9480 we need `simplify=True` otherwise the
# rank may be computed incorrectly
return m.rank(simplify=True) < 2
@property
def length(self):
"""
Treating a Point as a Line, this returns 0 for the length of a Point.
Examples
========
>>> from sympy import Point
>>> p = Point(0, 1)
>>> p.length
0
"""
return S.Zero
@property
def origin(self):
"""A point of all zeros of the same ambient dimension
as the current point"""
return Point([0]*len(self))
@property
def is_zero(self):
"""True if every coordinate is zero, otherwise False."""
return all(x == S.Zero for x in self.args)
@property
def ambient_dimension(self):
"""The dimension of the ambient space the point is in.
I.e., if the point is in R^n, the ambient dimension
will be n"""
return len(self)
[docs] def distance(self, p):
"""The Euclidean distance from self to point p.
Parameters
==========
p : Point
Returns
=======
distance : number or symbolic expression.
See Also
========
sympy.geometry.line.Segment.length
Examples
========
>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(4, 5)
>>> p1.distance(p2)
5
>>> from sympy.abc import x, y
>>> p3 = Point(x, y)
>>> p3.distance(Point(0, 0))
sqrt(x**2 + y**2)
"""
return sqrt(sum([(a - b)**2 for a, b in zip(
self.args, p.args if isinstance(p, Point) else p)]))
[docs] def taxicab_distance(self, p):
"""The Taxicab Distance from self to point p.
Returns the sum of the horizontal and vertical distances to point p.
Parameters
==========
p : Point
Returns
=======
taxicab_distance : The sum of the horizontal
and vertical distances to point p.
See Also
========
sympy.geometry.Point.distance
Examples
========
>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(4, 5)
>>> p1.taxicab_distance(p2)
7
"""
p = Point(p)
return sum(abs(a - b) 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
See Also
========
sympy.geometry.line.Segment.midpoint
Examples
========
>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(13, 5)
>>> p1.midpoint(p2)
Point2D(7, 3)
"""
return Point([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 Point, Rational
>>> p1 = Point(Rational(1, 2), Rational(3, 2))
>>> p1
Point2D(1/2, 3/2)
>>> p1.evalf()
Point2D(0.5, 1.5)
"""
coords = [x.evalf(prec, **options) for x in self.args]
return Point(*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 Point
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 0)
>>> p1.intersection(p2)
[]
>>> p1.intersection(p3)
[Point2D(0, 0)]
"""
if isinstance(o, Point):
if len(self) != len(o):
raise ValueError("Points must be of the same dimension to intersect")
if self == o:
return [self]
return []
return o.intersection(self)
[docs] def dot(self, p2):
"""Return dot product of self with another Point."""
p2 = Point(p2)
return Add(*[a*b for a,b in zip(self, p2)])
[docs] def equals(self, other):
"""Returns whether the coordinates of self and other agree."""
# a point is equal to another point if all its components are equal
if not isinstance(other, Point) or len(self.args) != len(other.args):
return False
return all(a.equals(b) for a,b in zip(self.args, other.args))
def __len__(self):
return len(self.args)
def __iter__(self):
return self.args.__iter__()
def __eq__(self, other):
if not isinstance(other, Point) or len(self.args) != len(other.args):
return False
return self.args == other.args
def __hash__(self):
return hash(self.args)
def __getitem__(self, key):
return self.args[key]
def __add__(self, other):
"""Add other to self by incrementing self's coordinates by those of other.
See Also
========
sympy.geometry.entity.translate
"""
if iterable(other) and len(other) == len(self):
return Point([simplify(a + b) for a, b in zip(self, other)])
else:
raise ValueError(
"Points must have the same number of dimensions")
def __sub__(self, other):
"""Subtract two points, or subtract a factor from this point's
coordinates."""
return self + (-other)
def __mul__(self, factor):
"""Multiply point's coordinates by a factor."""
factor = sympify(factor)
return Point([simplify(x*factor) for x in self.args])
def __div__(self, divisor):
"""Divide point's coordinates by a factor."""
divisor = sympify(divisor)
return Point([simplify(x/divisor) for x in self.args])
__truediv__ = __div__
def __neg__(self):
"""Negate the point."""
return Point([-x for x in self.args])
def __abs__(self):
"""Returns the distance between this point and the origin."""
origin = Point([0]*len(self))
return Point.distance(origin, self)
[docs]class Point2D(Point):
"""A point in a 2-dimensional Euclidean space.
Parameters
==========
coords : sequence of 2 coordinate values.
Attributes
==========
x
y
length
Raises
======
TypeError
When trying to add or subtract points with different dimensions.
When trying to create a point with more than two dimensions.
When `intersection` is called with object other than a Point.
See Also
========
sympy.geometry.line.Segment : Connects two Points
Examples
========
>>> from sympy.geometry import Point2D
>>> from sympy.abc import x
>>> Point2D(1, 2)
Point2D(1, 2)
>>> Point2D([1, 2])
Point2D(1, 2)
>>> Point2D(0, x)
Point2D(0, x)
Floats are automatically converted to Rational unless the
evaluate flag is False:
>>> Point2D(0.5, 0.25)
Point2D(1/2, 1/4)
>>> Point2D(0.5, 0.25, evaluate=False)
Point2D(0.5, 0.25)
"""
def __new__(cls, *args, **kwargs):
eval = kwargs.get('evaluate', global_evaluate[0])
check = True
if isinstance(args[0], Point2D):
if not eval:
return args[0]
args = args[0].args
check = False
else:
if iterable(args[0]):
args = args[0]
if len(args) != 2:
raise ValueError(
"Only two dimensional points currently supported")
coords = Tuple(*args)
if check:
if any(a.is_number and im(a) for a in coords):
raise ValueError('Imaginary args not permitted.')
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
def x(self):
"""
Returns the X coordinate of the Point.
Examples
========
>>> from sympy import Point2D
>>> p = Point2D(0, 1)
>>> p.x
0
"""
return self.args[0]
@property
def y(self):
"""
Returns the Y coordinate of the Point.
Examples
========
>>> from sympy import Point2D
>>> p = Point2D(0, 1)
>>> p.y
1
"""
return self.args[1]
@property
def bounds(self):
"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
rectangle for the geometric figure.
"""
return (self.x, self.y, self.x, self.y)
[docs] def is_concyclic(*points):
"""Is a sequence of points concyclic?
Test whether or not a sequence of points are concyclic (i.e., they lie
on a circle).
Parameters
==========
points : sequence of Points
Returns
=======
is_concyclic : boolean
True if points are concyclic, False otherwise.
See Also
========
sympy.geometry.ellipse.Circle
Notes
=====
No points are not considered to be concyclic. One or two points
are definitely concyclic and three points are conyclic iff they
are not collinear.
For more than three points, create a circle from the first three
points. If the circle cannot be created (i.e., they are collinear)
then all of the points cannot be concyclic. If the circle is created
successfully then simply check the remaining points for containment
in the circle.
Examples
========
>>> from sympy.geometry import Point
>>> p1, p2 = Point(-1, 0), Point(1, 0)
>>> p3, p4 = Point(0, 1), Point(-1, 2)
>>> Point.is_concyclic(p1, p2, p3)
True
>>> Point.is_concyclic(p1, p2, p3, p4)
False
"""
if len(points) == 0:
return False
if len(points) <= 2:
return True
points = [Point(p) for p in points]
if len(points) == 3:
return (not Point.is_collinear(*points))
try:
from .ellipse import Circle
c = Circle(points[0], points[1], points[2])
for point in points[3:]:
if point not in c:
return False
return True
except GeometryError:
# Circle could not be created, because of collinearity of the
# three points passed in, hence they are not concyclic.
return False
[docs] def rotate(self, angle, pt=None):
"""Rotate ``angle`` radians counterclockwise about Point ``pt``.
See Also
========
rotate, scale
Examples
========
>>> from sympy import Point2D, pi
>>> t = Point2D(1, 0)
>>> t.rotate(pi/2)
Point2D(0, 1)
>>> t.rotate(pi/2, (2, 0))
Point2D(2, -1)
"""
from sympy import cos, sin, Point
c = cos(angle)
s = sin(angle)
rv = self
if pt is not None:
pt = Point(pt)
rv -= pt
x, y = rv.args
rv = Point(c*x - s*y, s*x + c*y)
if pt is not None:
rv += pt
return rv
[docs] def scale(self, x=1, y=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).
See Also
========
rotate, translate
Examples
========
>>> from sympy import Point2D
>>> t = Point2D(1, 1)
>>> t.scale(2)
Point2D(2, 1)
>>> t.scale(2, 2)
Point2D(2, 2)
"""
if pt:
pt = Point(pt)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
return Point(self.x*x, self.y*y)
[docs] def translate(self, x=0, y=0):
"""Shift the Point by adding x and y to the coordinates of the Point.
See Also
========
rotate, scale
Examples
========
>>> from sympy import Point2D
>>> t = Point2D(0, 1)
>>> t.translate(2)
Point2D(2, 1)
>>> t.translate(2, 2)
Point2D(2, 3)
>>> t + Point2D(2, 2)
Point2D(2, 3)
"""
return Point(self.x + x, self.y + y)
[docs]class Point3D(Point):
"""A point in a 3-dimensional Euclidean space.
Parameters
==========
coords : sequence of 3 coordinate values.
Attributes
==========
x
y
z
length
Raises
======
TypeError
When trying to add or subtract points with different dimensions.
When `intersection` is called with object other than a Point.
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
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
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
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
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]
[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
See Also
========
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
"""
return Point.is_collinear(*points)
@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 ValueError:
pass
raise ValueError('At least 3 non-collinear points needed to define plane.')
[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).
See Also
========
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.
See Also
========
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)
