Source code for sympy.geometry.point
"""Geometrical Points.
Contains
========
Point
"""
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.functions.elementary.miscellaneous import sqrt
from .entity import GeometryEntity
from sympy.matrices import Matrix
from sympy.core.numbers import Float
[docs]class Point(GeometryEntity):
"""A point in a 2-dimensional Euclidean space.
Parameters
==========
coords : sequence of 2 coordinate values.
Attributes
==========
x
y
length
Raises
======
NotImplementedError
When trying to create a point with more than two 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 points are supported.
See Also
========
sympy.geometry.line.Segment : Connects two Points
Examples
========
>>> from sympy.geometry import Point
>>> from sympy.abc import x
>>> Point(1, 2)
Point(1, 2)
>>> Point([1, 2])
Point(1, 2)
>>> Point(0, x)
Point(0, x)
Floats are automatically converted to Rational unless the
evaluate flag is False:
>>> Point(0.5, 0.25)
Point(1/2, 1/4)
>>> Point(0.5, 0.25, evaluate=False)
Point(0.5, 0.25)
"""
def __new__(cls, *args, **kwargs):
if iterable(args[0]):
args = args[0]
elif isinstance(args[0], Point):
args = args[0].args
coords = Tuple(*args)
if len(coords) != 2:
raise NotImplementedError(
"Only two dimensional points currently supported")
if kwargs.get('evaluate', True):
coords = coords.xreplace(dict(
[(f, simplify(nsimplify(f, rational=True)))
for f in coords.atoms(Float)]))
return GeometryEntity.__new__(cls, *coords)
def __hash__(self):
return super(Point, self).__hash__()
def __eq__(self, other):
ts, to = type(self), type(other)
if ts is not to:
return False
return self.args == other.args
def __lt__(self, other):
return self.args < other.args
def __contains__(self, item):
return item == self
@property
[docs] def x(self):
"""
Returns the X coordinate of the Point.
Examples
========
>>> from sympy import Point
>>> p = Point(0, 1)
>>> p.x
0
"""
return self.args[0]
@property
[docs] def y(self):
"""
Returns the Y coordinate of the Point.
Examples
========
>>> from sympy import Point
>>> p = Point(0, 1)
>>> p.y
1
"""
return self.args[1]
@property
[docs] 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
[docs] def is_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
=======
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 and can confuse this algorithm.
# Use only unique points.
points = list(set(points))
if len(points) == 0:
return False
if len(points) <= 2:
return True # two points always form a line
points = [Point(a) for a in points]
# XXX Cross product is used now, but that only extends to three
# dimensions. If the concept needs to extend to greater
# dimensions then another method would have to be used
p1 = points[0]
p2 = points[1]
v1 = p2 - p1
x1, y1 = v1.args
rv = True
for p3 in points[2:]:
x2, y2 = (p3 - p1).args
test = simplify(x1*y2 - y1*x2).equals(0)
if test is False:
return False
if rv and not test:
rv = test
return rv
[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 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)
"""
p = Point(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
See Also
========
sympy.geometry.line.Segment.midpoint
Examples
========
>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(13, 5)
>>> p1.midpoint(p2)
Point(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
Point(1/2, 3/2)
>>> p1.evalf()
Point(0.5, 1.5)
"""
if prec is None:
coords = [x.evalf(**options) for x in self.args]
else:
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)
[Point(0, 0)]
"""
if isinstance(o, Point):
if self == o:
return [self]
return []
return o.intersection(self)
[docs] def rotate(self, angle, pt=None):
"""Rotate ``angle`` radians counterclockwise about Point ``pt``.
See Also
========
rotate, scale
Examples
========
>>> from sympy import Point, pi
>>> t = Point(1, 0)
>>> t.rotate(pi/2)
Point(0, 1)
>>> t.rotate(pi/2, (2, 0))
Point(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 Point
>>> t = Point(1, 1)
>>> t.scale(2)
Point(2, 1)
>>> t.scale(2, 2)
Point(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 Point
>>> t = Point(0, 1)
>>> t.translate(2)
Point(2, 1)
>>> t.translate(2, 2)
Point(2, 3)
>>> t + Point(2, 2)
Point(2, 3)
"""
return Point(self.x + x, self.y + y)
[docs] def transform(self, matrix):
"""Return the point after applying the transformation described
by the 3x3 Matrix, ``matrix``.
See Also
========
geometry.entity.rotate
geometry.entity.scale
geometry.entity.translate
"""
x, y = self.args
return Point(*(Matrix(1, 3, [x, y, 1])*matrix).tolist()[0][:2])
[docs] def dot(self, p2):
"""Return dot product of self with another Point."""
p2 = Point(p2)
x1, y1 = self.args
x2, y2 = p2.args
return x1*x2 + y1*y2
def __add__(self, other):
"""Add other to self by incrementing self's coordinates by those of other.
See Also
========
sympy.geometry.entity.translate
"""
if isinstance(other, Point):
if len(other.args) == len(self.args):
return Point(*[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."""
return self + (-other)
def __mul__(self, factor):
"""Multiply point's coordinates by a factor."""
factor = sympify(factor)
return Point([x*factor for x in self.args])
def __div__(self, divisor):
"""Divide point's coordinates by a factor."""
divisor = sympify(divisor)
return Point([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.args))
return Point.distance(origin, self)
