# Source code for sympy.geometry.line

"""Line-like geometrical entities.

Contains
========
LinearEntity
Line
Ray
Segment
LinearEntity2D
Line2D
Ray2D
Segment2D
LinearEntity3D
Line3D
Ray3D
Segment3D

"""
from __future__ import division, print_function

from sympy.core import S, sympify
from sympy.core.relational import Eq
from sympy.functions.elementary.trigonometric import (_pi_coeff as pi_coeff, acos, tan)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.logic.boolalg import And
from sympy.simplify.simplify import simplify
from sympy.geometry.exceptions import GeometryError
from sympy.core.decorators import deprecated
from sympy.sets import Intersection
from sympy.matrices import Matrix

from .entity import GeometryEntity, GeometrySet
from .point import Point, Point3D
from .util import _symbol
from sympy.utilities.misc import Undecidable

[docs]class LinearEntity(GeometrySet):
"""A base class for all linear entities (Line, Ray and Segment)
in n-dimensional Euclidean space.

Attributes
==========

ambient_dimension
direction
length
p1
p2
points

Notes
=====

This is an abstract class and is not meant to be instantiated.

========

sympy.geometry.entity.GeometryEntity

"""
def __new__(cls, p1, p2=None, **kwargs):
p1, p2 = Point._normalize_dimension(p1, p2)
if p1 == p2:
# sometimes we return a single point if we are not given two unique
# points. This is done in the specific subclass
raise ValueError(
"%s.__new__ requires two unique Points." % cls.__name__)
if len(p1) != len(p2):
raise ValueError(
"%s.__new__ requires two Points of equal dimension." % cls.__name__)

return GeometryEntity.__new__(cls, p1, p2, **kwargs)

def __contains__(self, other):
"""Return a definitive answer or else raise an error if it cannot
be determined that other is on the boundaries of self."""
result = self.contains(other)

if result is not None:
return result
else:
raise Undecidable(
"can't decide whether '%s' contains '%s'" % (self, other))

def _span_test(self, other):
"""Test whether the point other lies in the positive span of self.
A point x is 'in front' of a point y if x.dot(y) >= 0.  Return
-1 if other is behind self.p1, 0 if other is self.p1 and
and 1 if other is in front of self.p1."""

if self.p1 == other:
return 0

rel_pos = other - self.p1
d = self.direction
if d.dot(rel_pos) > 0:
return 1
return -1

@property
def ambient_dimension(self):
return len(self.p1)

[docs]    def angle_between(l1, l2):
"""The angle formed between the two linear entities.

Parameters
==========

l1 : LinearEntity
l2 : LinearEntity

Returns
=======

Notes
=====

From the dot product of vectors v1 and v2 it is known that:

dot(v1, v2) = |v1|*|v2|*cos(A)

where A is the angle formed between the two vectors. We can
get the directional vectors of the two lines and readily
find the angle between the two using the above formula.

========

is_perpendicular

Examples
========

>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(0, 4), Point(2, 0)
>>> l1, l2 = Line(p1, p2), Line(p1, p3)
>>> l1.angle_between(l2)
pi/2
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(-1, 2, 0)
>>> l1, l2 = Line3D(p1, p2), Line3D(p2, p3)
>>> l1.angle_between(l2)
acos(-sqrt(2)/3)

"""
if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity):
raise TypeError('Must pass only LinearEntity objects')

v1, v2 = l1.direction, l2.direction
return acos(v1.dot(v2)/(abs(v1)*abs(v2)))

[docs]    def arbitrary_point(self, parameter='t'):
"""A parameterized point on the Line.

Parameters
==========

parameter : str, optional
The name of the parameter which will be used for the parametric
point. The default value is 't'. When this parameter is 0, the
first point used to define the line will be returned, and when
it is 1 the second point will be returned.

Returns
=======

point : Point

Raises
======

ValueError
When parameter already appears in the Line's definition.

========

sympy.geometry.point.Point

Examples
========

>>> from sympy import Point, Line
>>> p1, p2 = Point(1, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.arbitrary_point()
Point2D(4*t + 1, 3*t)
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(1, 0, 0), Point3D(5, 3, 1)
>>> l1 = Line3D(p1, p2)
>>> l1.arbitrary_point()
Point3D(4*t + 1, 3*t, t)

"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError('Symbol %s already appears in object '
'and cannot be used as a parameter.' % t.name)
# multiply on the right so the variable gets
# combined witht he coordinates of the point
return self.p1 + (self.p2 - self.p1)*t

@staticmethod
[docs]    def are_concurrent(*lines):
"""Is a sequence of linear entities concurrent?

Two or more linear entities are concurrent if they all
intersect at a single point.

Parameters
==========

lines : a sequence of linear entities.

Returns
=======

True : if the set of linear entities intersect in one point
False : otherwise.

========

sympy.geometry.util.intersection

Examples
========

>>> from sympy import Point, Line, Line3D
>>> p1, p2 = Point(0, 0), Point(3, 5)
>>> p3, p4 = Point(-2, -2), Point(0, 2)
>>> l1, l2, l3 = Line(p1, p2), Line(p1, p3), Line(p1, p4)
>>> Line.are_concurrent(l1, l2, l3)
True
>>> l4 = Line(p2, p3)
>>> Line.are_concurrent(l2, l3, l4)
False
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 5, 2)
>>> p3, p4 = Point3D(-2, -2, -2), Point3D(0, 2, 1)
>>> l1, l2, l3 = Line3D(p1, p2), Line3D(p1, p3), Line3D(p1, p4)
>>> Line3D.are_concurrent(l1, l2, l3)
True
>>> l4 = Line3D(p2, p3)
>>> Line3D.are_concurrent(l2, l3, l4)
False

"""

common_points = Intersection(*lines)
if common_points.is_FiniteSet and len(common_points) == 1:
return True
return False

[docs]    def contains(self, other):
"""Subclasses should implement this method and should return
True if other is on the boundaries of self;
False if not on the boundaries of self;
None if a determination cannot be made."""
raise NotImplementedError()

@property
def direction(self):
"""The direction vector of the LinearEntity.

Returns
=======

p : a Point; the ray from the origin to this point is the
direction of self

Examples
========

>>> from sympy.geometry import Line
>>> a, b = (1, 1), (1, 3)
>>> Line(a, b).direction
Point2D(0, 2)
>>> Line(b, a).direction
Point2D(0, -2)

This can be reported so the distance from the origin is 1:

>>> Line(b, a).direction.unit
Point2D(0, -1)

========

sympy.geometry.point.Point.unit

"""
return self.p2 - self.p1

[docs]    def intersection(self, other):
"""The intersection with another geometrical entity.

Parameters
==========

o : Point or LinearEntity

Returns
=======

intersection : list of geometrical entities

========

sympy.geometry.point.Point

Examples
========

>>> from sympy import Point, Line, Segment
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(7, 7)
>>> l1 = Line(p1, p2)
>>> l1.intersection(p3)
[Point2D(7, 7)]
>>> p4, p5 = Point(5, 0), Point(0, 3)
>>> l2 = Line(p4, p5)
>>> l1.intersection(l2)
[Point2D(15/8, 15/8)]
>>> p6, p7 = Point(0, 5), Point(2, 6)
>>> s1 = Segment(p6, p7)
>>> l1.intersection(s1)
[]
>>> from sympy import Point3D, Line3D, Segment3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(7, 7, 7)
>>> l1 = Line3D(p1, p2)
>>> l1.intersection(p3)
[Point3D(7, 7, 7)]
>>> l1 = Line3D(Point3D(4,19,12), Point3D(5,25,17))
>>> l2 = Line3D(Point3D(-3, -15, -19), direction_ratio=[2,8,8])
>>> l1.intersection(l2)
[Point3D(1, 1, -3)]
>>> p6, p7 = Point3D(0, 5, 2), Point3D(2, 6, 3)
>>> s1 = Segment3D(p6, p7)
>>> l1.intersection(s1)
[]

"""
def intersect_parallel_rays(ray1, ray2):
if ray1.direction.dot(ray2.direction) > 0:
# rays point in the same direction
# so return the one that is "in front"
return [ray2] if ray1._span_test(ray2.p1) >= 0 else [ray1]
else:
# rays point in opposite directions
st = ray1._span_test(ray2.p1)
if st < 0:
return []
elif st == 0:
return [ray2.p1]
return [Segment(ray1.p1, ray2.p1)]

def intersect_parallel_ray_and_segment(ray, seg):
st1, st2 = ray._span_test(seg.p1), ray._span_test(seg.p2)
if st1 < 0 and st2 < 0:
return []
elif st1 >= 0 and st2 >= 0:
return [seg]
elif st1 >= 0 and st2 < 0:
return [Segment(ray.p1, seg.p1)]
elif st1 <= 0 and st2 > 0:
return [Segment(ray.p1, seg.p2)]

def intersect_parallel_segments(seg1, seg2):
if seg1.contains(seg2):
return [seg2]
if seg2.contains(seg1):
return [seg1]

# direct the segments so they're oriented the same way
if seg1.direction.dot(seg2.direction) < 0:
seg2 = Segment(seg2.p1, seg2.p2)
# order the segments so seg1 is "behind" seg2
if seg1._span_test(seg2.p1) < 0:
seg1, seg2 = seg2, seg1
if seg2._span_test(seg1.p2) < 0:
return []
return [Segment(seg2.p1, seg1.p2)]

if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if other.is_Point:
if self.contains(other):
return [other]
else:
return []
elif isinstance(other, LinearEntity):
# break into cases based on whether
# the lines are parallel, non-parallel intersecting, or skew
pts = Point._normalize_dimension(self.p1, self.p2, other.p1, other.p2)
rank = Point.affine_rank(*pts)

if rank == 1:
# we're collinear
if isinstance(self, Line):
return [other]
if isinstance(other, Line):
return [self]

if isinstance(self, Ray) and isinstance(other, Ray):
return intersect_parallel_rays(self, other)
if isinstance(self, Ray) and isinstance(other, Segment):
return intersect_parallel_ray_and_segment(self, other)
if isinstance(self, Segment) and isinstance(other, Ray):
return intersect_parallel_ray_and_segment(other, self)
if isinstance(self, Segment) and isinstance(other, Segment):
return intersect_parallel_segments(self, other)
elif rank == 2:
# we're in the same plane
l1 = Line(*pts[:2])
l2 = Line(*pts[2:])

# check to see if we're parallel.  If we are, we can't
# be intersecting, since the collinear case was already
# handled
if l1.direction.is_scalar_multiple(l2.direction):
return []

# find the intersection as if everything were lines
# by solving the equation t*d + p1 == s*d' + p1'
m = Matrix([l1.direction, -l2.direction]).transpose()
v = Matrix([l2.p1 - l1.p1]).transpose()

# we cannot use m.solve(v) because that only works for square matrices
m_rref, pivots = m.col_insert(2, v).rref(simplify=True)
# rank == 2 ensures we have 2 pivots, but let's check anyway
if len(pivots) != 2:
raise GeometryError("Failed when solving Mx=b when M={} and b={}".format(m,v))
coeff = m_rref[0,2]
line_intersection = l1.direction*coeff + self.p1

# if we're both lines, we can skip a containment check
if isinstance(self, Line) and isinstance(other, Line):
return [line_intersection]

if self.contains(line_intersection) and other.contains(line_intersection):
return [line_intersection]
return []
else:
# we're skew
return []

return other.intersection(self)

[docs]    def is_parallel(l1, l2):
"""Are two linear entities parallel?

Parameters
==========

l1 : LinearEntity
l2 : LinearEntity

Returns
=======

True : if l1 and l2 are parallel,
False : otherwise.

========

coefficients

Examples
========

>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> p3, p4 = Point(3, 4), Point(6, 7)
>>> l1, l2 = Line(p1, p2), Line(p3, p4)
>>> Line.is_parallel(l1, l2)
True
>>> p5 = Point(6, 6)
>>> l3 = Line(p3, p5)
>>> Line.is_parallel(l1, l3)
False
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 4, 5)
>>> p3, p4 = Point3D(2, 1, 1), Point3D(8, 9, 11)
>>> l1, l2 = Line3D(p1, p2), Line3D(p3, p4)
>>> Line3D.is_parallel(l1, l2)
True
>>> p5 = Point3D(6, 6, 6)
>>> l3 = Line3D(p3, p5)
>>> Line3D.is_parallel(l1, l3)
False

"""

if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity):
raise TypeError('Must pass only LinearEntity objects')

return l1.direction.is_scalar_multiple(l2.direction)

[docs]    def is_perpendicular(l1, l2):
"""Are two linear entities perpendicular?

Parameters
==========

l1 : LinearEntity
l2 : LinearEntity

Returns
=======

True : if l1 and l2 are perpendicular,
False : otherwise.

========

coefficients

Examples
========

>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(-1, 1)
>>> l1, l2 = Line(p1, p2), Line(p1, p3)
>>> l1.is_perpendicular(l2)
True
>>> p4 = Point(5, 3)
>>> l3 = Line(p1, p4)
>>> l1.is_perpendicular(l3)
False
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(-1, 2, 0)
>>> l1, l2 = Line3D(p1, p2), Line3D(p2, p3)
>>> l1.is_perpendicular(l2)
False
>>> p4 = Point3D(5, 3, 7)
>>> l3 = Line3D(p1, p4)
>>> l1.is_perpendicular(l3)
False

"""
if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity):
raise TypeError('Must pass only LinearEntity objects')

return S.Zero.equals(l1.direction.dot(l2.direction))

[docs]    def is_similar(self, other):
"""
Return True if self and other are contained in the same line.

Examples
========

>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 1), Point(3, 4), Point(2, 3)
>>> l1 = Line(p1, p2)
>>> l2 = Line(p1, p3)
>>> l1.is_similar(l2)
True
"""
l = Line(self.p1, self.p2)
return l.contains(other)

@property
def length(self):
"""
The length of the line.

Examples
========

>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(3, 5)
>>> l1 = Line(p1, p2)
>>> l1.length
oo
"""
return S.Infinity

@property
def p1(self):
"""The first defining point of a linear entity.

========

sympy.geometry.point.Point

Examples
========

>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.p1
Point2D(0, 0)

"""
return self.args[0]

@property
def p2(self):
"""The second defining point of a linear entity.

========

sympy.geometry.point.Point

Examples
========

>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.p2
Point2D(5, 3)

"""
return self.args[1]

[docs]    def parallel_line(self, p):
"""Create a new Line parallel to this linear entity which passes
through the point p.

Parameters
==========

p : Point

Returns
=======

line : Line

========

is_parallel

Examples
========

>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> l1 = Line(p1, p2)
>>> l2 = l1.parallel_line(p3)
>>> p3 in l2
True
>>> l1.is_parallel(l2)
True
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(2, 3, 4), Point3D(-2, 2, 0)
>>> l1 = Line3D(p1, p2)
>>> l2 = l1.parallel_line(p3)
>>> p3 in l2
True
>>> l1.is_parallel(l2)
True

"""
p = Point(p, dim=self.ambient_dimension)
return Line(p, p + self.direction)

[docs]    def perpendicular_line(self, p):
"""Create a new Line perpendicular to this linear entity which passes
through the point p.

Parameters
==========

p : Point

Returns
=======

line : Line

========

is_perpendicular, perpendicular_segment

Examples
========

>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> l1 = Line(p1, p2)
>>> l2 = l1.perpendicular_line(p3)
>>> p3 in l2
True
>>> l1.is_perpendicular(l2)
True
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(2, 3, 4), Point3D(-2, 2, 0)
>>> l1 = Line3D(p1, p2)
>>> l2 = l1.perpendicular_line(p3)
>>> p3 in l2
True
>>> l1.is_perpendicular(l2)
True

"""
p = Point(p, dim=self.ambient_dimension)
if p in self:
p = p + self.direction.orthogonal_direction
return Line(p, self.projection(p))

[docs]    def perpendicular_segment(self, p):
"""Create a perpendicular line segment from p to this line.

The enpoints of the segment are p and the closest point in
the line containing self. (If self is not a line, the point might
not be in self.)

Parameters
==========

p : Point

Returns
=======

segment : Segment

Notes
=====

Returns p itself if p is on this linear entity.

========

perpendicular_line

Examples
========

>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 2)
>>> l1 = Line(p1, p2)
>>> s1 = l1.perpendicular_segment(p3)
>>> l1.is_perpendicular(s1)
True
>>> p3 in s1
True
>>> l1.perpendicular_segment(Point(4, 0))
Segment2D(Point2D(2, 2), Point2D(4, 0))
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 2, 0)
>>> l1 = Line3D(p1, p2)
>>> s1 = l1.perpendicular_segment(p3)
>>> l1.is_perpendicular(s1)
True
>>> p3 in s1
True
>>> l1.perpendicular_segment(Point3D(4, 0, 0))
Segment3D(Point3D(4/3, 4/3, 4/3), Point3D(4, 0, 0))

"""
p = Point(p, dim=self.ambient_dimension)
if p in self:
return p
l = self.perpendicular_line(p)
# The intersection should be unique, so unpack the singleton
p2, = Intersection(Line(self.p1, self.p2), l)

return Segment(p, p2)

@property
def points(self):
"""The two points used to define this linear entity.

Returns
=======

points : tuple of Points

========

sympy.geometry.point.Point

Examples
========

>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 11)
>>> l1 = Line(p1, p2)
>>> l1.points
(Point2D(0, 0), Point2D(5, 11))

"""
return (self.p1, self.p2)

[docs]    def projection(self, other):
"""Project a point, line, ray, or segment onto this linear entity.

Parameters
==========

other : Point or LinearEntity (Line, Ray, Segment)

Returns
=======

projection : Point or LinearEntity (Line, Ray, Segment)
The return type matches the type of the parameter other.

Raises
======

GeometryError
When method is unable to perform projection.

Notes
=====

A projection involves taking the two points that define
the linear entity and projecting those points onto a
Line and then reforming the linear entity using these
projections.
A point P is projected onto a line L by finding the point
on L that is closest to P. This point is the intersection
of L and the line perpendicular to L that passes through P.

========

sympy.geometry.point.Point, perpendicular_line

Examples
========

>>> from sympy import Point, Line, Segment, Rational
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(Rational(1, 2), 0)
>>> l1 = Line(p1, p2)
>>> l1.projection(p3)
Point2D(1/4, 1/4)
>>> p4, p5 = Point(10, 0), Point(12, 1)
>>> s1 = Segment(p4, p5)
>>> l1.projection(s1)
Segment2D(Point2D(5, 5), Point2D(13/2, 13/2))
>>> p1, p2, p3 = Point(0, 0, 1), Point(1, 1, 2), Point(2, 0, 1)
>>> l1 = Line(p1, p2)
>>> l1.projection(p3)
Point3D(2/3, 2/3, 5/3)
>>> p4, p5 = Point(10, 0, 1), Point(12, 1, 3)
>>> s1 = Segment(p4, p5)
>>> l1.projection(s1)
Segment3D(Point3D(10/3, 10/3, 13/3), Point3D(5, 5, 6))

"""

if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)

def proj_point(p):
return Point.project(p - self.p1, self.direction) + self.p1

if isinstance(other, Point):
return proj_point(other)
elif isinstance(other, LinearEntity):
p1, p2 = proj_point(other.p1), proj_point(other.p2)
# test to see if we're degenerate
if p1 == p2:
return p1
projected = other.__class__(p1, p2)
projected = Intersection(self, projected)
# if we happen to have intersected in only a point, return that
if projected.is_FiniteSet and len(projected) == 1:
# projected is a set of size 1, so unpack it in a
a, = projected
return a
return projected

raise GeometryError(
"Do not know how to project %s onto %s" % (other, self))

[docs]    def random_point(self):
"""A random point on a LinearEntity.

Returns
=======

point : Point

========

sympy.geometry.point.Point

Examples
========

>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> p3 = l1.random_point()
>>> # random point - don't know its coords in advance
>>> p3 # doctest: +ELLIPSIS
Point2D(...)
>>> # point should belong to the line
>>> p3 in l1
True

"""
from random import randint
from sympy.functions import floor

# The lower and upper
lower, upper = -2**32 - 1, 2**32

if isinstance(self, Ray):
lower = 0
if isinstance(self, Segment):
lower = 0
upper = floor(self.length)
t = randint(lower, upper)

return self.direction*t/abs(self.direction) + self.p1

[docs]class Line(LinearEntity):
"""An infinite line in space.

A line is declared with two distinct points.
A 2D line may be declared with a point and slope
and a 3D line may be defined with a point and a direction ratio.

Parameters
==========

p1 : Point
p2 : Point
slope : sympy expression
direction_ratio : list

Notes
=====

Line will automatically subclass to Line2D or Line3D based
on the dimension of p1.  The slope argument is only relevant
for Line2D and the direction_ratio argument is only relevant
for Line3D.

========

sympy.geometry.point.Point
sympy.geometry.line.Line2D
sympy.geometry.line.Line3D

Examples
========

>>> import sympy
>>> from sympy import Point
>>> from sympy.geometry import Line, Segment
>>> L = Line(Point(2,3), Point(3,5))
>>> L
Line2D(Point2D(2, 3), Point2D(3, 5))
>>> L.points
(Point2D(2, 3), Point2D(3, 5))
>>> L.equation()
-2*x + y + 1
>>> L.coefficients
(-2, 1, 1)

Instantiate with keyword slope:

>>> Line(Point(0, 0), slope=0)
Line2D(Point2D(0, 0), Point2D(1, 0))

Instantiate with another linear object

>>> s = Segment((0, 0), (0, 1))
>>> Line(s).equation()
x
"""

def __new__(cls, p1, p2=None, **kwargs):
if isinstance(p1, LinearEntity):
if p2:
raise ValueError('If p1 is a LinearEntity, p2 must be None.')
dim = len(p1.p1)
else:
p1 = Point(p1)
dim = len(p1)
if p2 is not None or isinstance(p2, Point) and p2.ambient_dimension != dim:
p2 = Point(p2)

if dim == 2:
return Line2D(p1, p2, **kwargs)
elif dim == 3:
return Line3D(p1, p2, **kwargs)
return LinearEntity.__new__(cls, p1, p2, **kwargs)

[docs]    def contains(self, other):
"""
Return True if other is on this Line, or False otherwise.

Examples
========

>>> from sympy import Line,Point
>>> p1, p2 = Point(0, 1), Point(3, 4)
>>> l = Line(p1, p2)
>>> l.contains(p1)
True
>>> l.contains((0, 1))
True
>>> l.contains((0, 0))
False
>>> a = (0, 0, 0)
>>> b = (1, 1, 1)
>>> c = (2, 2, 2)
>>> l1 = Line(a, b)
>>> l2 = Line(b, a)
>>> l1 == l2
False
>>> l1 in l2
True

"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if isinstance(other, Point):
return Point.is_collinear(other, self.p1, self.p2)
if isinstance(other, LinearEntity):
return Point.is_collinear(self.p1, self.p2, other.p1, other.p2)
return False

[docs]    def distance(self, other):
"""
Finds the shortest distance between a line and a point.

Raises
======

NotImplementedError is raised if other is not a Point

Examples
========

>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> s = Line(p1, p2)
>>> s.distance(Point(-1, 1))
sqrt(2)
>>> s.distance((-1, 2))
3*sqrt(2)/2
>>> p1, p2 = Point(0, 0, 0), Point(1, 1, 1)
>>> s = Line(p1, p2)
>>> s.distance(Point(-1, 1, 1))
2*sqrt(6)/3
>>> s.distance((-1, 1, 1))
2*sqrt(6)/3

"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if self.contains(other):
return S.Zero
return self.perpendicular_segment(other).length

def equal(self, other):
return self.equals(other)

[docs]    def equals(self, other):
"""Returns True if self and other are the same mathematical entities"""
if not isinstance(other, Line):
return False
return Point.is_collinear(self.p1, other.p1, self.p2, other.p2)

[docs]    def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of line. Gives
values that will produce a line that is +/- 5 units long (where a
unit is the distance between the two points that define the line).

Parameters
==========

parameter : str, optional
Default value is 't'.

Returns
=======

plot_interval : list (plot interval)
[parameter, lower_bound, upper_bound]

Examples
========

>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.plot_interval()
[t, -5, 5]

"""
t = _symbol(parameter)
return [t, -5, 5]

[docs]class Ray(LinearEntity):
"""A Ray is a semi-line in the space with a source point and a direction.

Parameters
==========

p1 : Point
The source of the Ray
p2 : Point or radian value
This point determines the direction in which the Ray propagates.
If given as an angle it is interpreted in radians with the positive
direction being ccw.

Attributes
==========

source

========

sympy.geometry.line.Ray2D
sympy.geometry.line.Ray3D
sympy.geometry.point.Point
sympy.geometry.line.Line

Notes
=====

Ray will automatically subclass to Ray2D or Ray3D based on the
dimension of p1.

Examples
========

>>> import sympy
>>> from sympy import Point, pi
>>> from sympy.geometry import Ray
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r
Ray2D(Point2D(2, 3), Point2D(3, 5))
>>> r.points
(Point2D(2, 3), Point2D(3, 5))
>>> r.source
Point2D(2, 3)
>>> r.xdirection
oo
>>> r.ydirection
oo
>>> r.slope
2
>>> Ray(Point(0, 0), angle=pi/4).slope
1

"""
def __new__(cls, p1, p2=None, **kwargs):
p1 = Point(p1)
if p2 is not None:
p1, p2 = Point._normalize_dimension(p1, Point(p2))
dim = len(p1)

if dim == 2:
return Ray2D(p1, p2, **kwargs)
elif dim == 3:
return Ray3D(p1, p2, **kwargs)
return LinearEntity.__new__(cls, p1, *pts, **kwargs)

def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG path element for the LinearEntity.

Parameters
==========

scale_factor : float
Multiplication factor for the SVG stroke-width.  Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""

from sympy.core.evalf import N

verts = (N(self.p1), N(self.p2))
coords = ["{0},{1}".format(p.x, p.y) for p in verts]
path = "M {0} L {1}".format(coords[0], " L ".join(coords[1:]))

return (
'<path fill-rule="evenodd" fill="{2}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" d="{1}" '
'marker-start="url(#markerCircle)" marker-end="url(#markerArrow)"/>'
).format(2. * scale_factor, path, fill_color)

[docs]    def contains(self, other):
"""
Is other GeometryEntity contained in this Ray?

Examples
========

>>> from sympy import Ray,Point,Segment
>>> p1, p2 = Point(0, 0), Point(4, 4)
>>> r = Ray(p1, p2)
>>> r.contains(p1)
True
>>> r.contains((1, 1))
True
>>> r.contains((1, 3))
False
>>> s = Segment((1, 1), (2, 2))
>>> r.contains(s)
True
>>> s = Segment((1, 2), (2, 5))
>>> r.contains(s)
False
>>> r1 = Ray((2, 2), (3, 3))
>>> r.contains(r1)
True
>>> r1 = Ray((2, 2), (3, 5))
>>> r.contains(r1)
False
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if isinstance(other, Point):
if Point.is_collinear(self.p1, self.p2, other):
# if we're in the direction of the ray, our
# direction vector dot the ray's direction vector
# should be non-negative
return bool( (self.p2 - self.p1).dot(other - self.p1) >= S.Zero )
return False
elif isinstance(other, Ray):
if Point.is_collinear(self.p1, self.p2, other.p1, other.p2):
return bool( (self.p2 - self.p1).dot(other.p2 - other.p1) > S.Zero )
return False
elif isinstance(other, Segment):
return other.p1 in self and other.p2 in self

# No other known entity can be contained in a Ray
return False

[docs]    def distance(self, other):
"""
Finds the shortest distance between the ray and a point.

Raises
======

NotImplementedError is raised if other is not a Point

Examples
========

>>> from sympy import Point, Ray
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> s = Ray(p1, p2)
>>> s.distance(Point(-1, -1))
sqrt(2)
>>> s.distance((-1, 2))
3*sqrt(2)/2
>>> p1, p2 = Point(0, 0, 0), Point(1, 1, 2)
>>> s = Ray(p1, p2)
>>> s
Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 2))
>>> s.distance(Point(-1, -1, 2))
4*sqrt(3)/3
>>> s.distance((-1, -1, 2))
4*sqrt(3)/3

"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if self.contains(other):
return S.Zero

proj = Line(self.p1, self.p2).projection(other)
if self.contains(proj):
return abs(other - proj)
else:
return abs(other - self.source)

[docs]    def equals(self, other):
"""Returns True if self and other are the same mathematical entities"""
if not isinstance(other, Ray):
return False
return self.source == other.source and other.p2 in self

[docs]    def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Ray. Gives
values that will produce a ray that is 10 units long (where a unit is
the distance between the two points that define the ray).

Parameters
==========

parameter : str, optional
Default value is 't'.

Returns
=======

plot_interval : list
[parameter, lower_bound, upper_bound]

Examples
========

>>> from sympy import Point, Ray, pi
>>> r = Ray((0, 0), angle=pi/4)
>>> r.plot_interval()
[t, 0, 10]

"""
t = _symbol(parameter)
return [t, 0, 10]

@property
def source(self):
"""The point from which the ray emanates.

========

sympy.geometry.point.Point

Examples
========

>>> from sympy import Point, Ray
>>> p1, p2 = Point(0, 0), Point(4, 1)
>>> r1 = Ray(p1, p2)
>>> r1.source
Point2D(0, 0)
>>> p1, p2 = Point(0, 0, 0), Point(4, 1, 5)
>>> r1 = Ray(p2, p1)
>>> r1.source
Point3D(4, 1, 5)

"""
return self.p1

[docs]class Segment(LinearEntity):
"""An undirected line segment in space.

Parameters
==========

p1 : Point
p2 : Point

Attributes
==========

length : number or sympy expression
midpoint : Point

========

sympy.geometry.line.Segment2D
sympy.geometry.line.Segment3D
sympy.geometry.point.Point
sympy.geometry.line.Line

Notes
=====

If 2D or 3D points are used to define Segment, it will
be automatically subclassed to Segment2D or Segment3D.

Examples
========

>>> import sympy
>>> from sympy import Point
>>> from sympy.geometry import Segment
>>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts
Segment2D(Point2D(1, 0), Point2D(1, 1))
>>> s = Segment(Point(4, 3), Point(1, 1))
>>> s
Segment2D(Point2D(1, 1), Point2D(4, 3))
>>> s.points
(Point2D(1, 1), Point2D(4, 3))
>>> s.slope
2/3
>>> s.length
sqrt(13)
>>> s.midpoint
Point2D(5/2, 2)
>>> Segment((1, 0, 0), (1, 1, 1)) # tuples are interpreted as pts
Segment3D(Point3D(1, 0, 0), Point3D(1, 1, 1))
>>> s = Segment(Point(4, 3, 9), Point(1, 1, 7))
>>> s
Segment3D(Point3D(1, 1, 7), Point3D(4, 3, 9))
>>> s.points
(Point3D(1, 1, 7), Point3D(4, 3, 9))
>>> s.length
sqrt(17)
>>> s.midpoint
Point3D(5/2, 2, 8)

"""
def __new__(cls, p1, p2, **kwargs):
p1, p2 = Point._normalize_dimension(Point(p1), Point(p2))
dim = len(p1)

if dim == 2:
return Segment2D(p1, p2, **kwargs)
elif dim == 3:
return Segment3D(p1, p2, **kwargs)
return LinearEntity.__new__(cls, p1, p2, **kwargs)

[docs]    def contains(self, other):
"""
Is the other GeometryEntity contained within this Segment?

Examples
========

>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 1), Point(3, 4)
>>> s = Segment(p1, p2)
>>> s2 = Segment(p2, p1)
>>> s.contains(s2)
True
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 1, 1), Point3D(3, 4, 5)
>>> s = Segment3D(p1, p2)
>>> s2 = Segment3D(p2, p1)
>>> s.contains(s2)
True
>>> s.contains((p1 + p2) / 2)
True
"""

if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if isinstance(other, Point):
if Point.is_collinear(other, self.p1, self.p2):
d1, d2 = other - self.p1, other - self.p2
d = self.p2 - self.p1
# without the call to simplify, sympy cannot tell that an expression
# like (a+b)*(a/2+b/2) is always non-negative.  If it cannot be
# determined, raise an Undecidable error
try:
# the triangle inequality says that |d1|+|d2| >= |d| and is strict
# only if other lies in the line segment
return bool(Eq(simplify(abs(d1) + abs(d2) - abs(d)), 0))
except TypeError:
raise Undecidable("Cannot determine if {} is in {}".format(other, self))
if isinstance(other, Segment):
return other.p1 in self and other.p2 in self

return False

[docs]    def distance(self, other):
"""
Finds the shortest distance between a line segment and a point.

Raises
======

NotImplementedError is raised if other is not a Point

Examples
========

>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 1), Point(3, 4)
>>> s = Segment(p1, p2)
>>> s.distance(Point(10, 15))
sqrt(170)
>>> s.distance((0, 12))
sqrt(73)
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 0, 3), Point3D(1, 1, 4)
>>> s = Segment3D(p1, p2)
>>> s.distance(Point3D(10, 15, 12))
sqrt(341)
>>> s.distance((10, 15, 12))
sqrt(341)
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if isinstance(other, Point):
vp1 = other - self.p1
vp2 = other - self.p2

dot_prod_sign_1 = self.direction.dot(vp1) >= 0
dot_prod_sign_2 = self.direction.dot(vp2) <= 0
if dot_prod_sign_1 and dot_prod_sign_2:
return Line(self.p1, self.p2).distance(other)
if dot_prod_sign_1 and not dot_prod_sign_2:
return abs(vp2)
if not dot_prod_sign_1 and dot_prod_sign_2:
return abs(vp1)
raise NotImplementedError()

@property
def length(self):
"""The length of the line segment.

========

sympy.geometry.point.Point.distance

Examples
========

>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(4, 3)
>>> s1 = Segment(p1, p2)
>>> s1.length
5
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3)
>>> s1 = Segment3D(p1, p2)
>>> s1.length
sqrt(34)

"""
return Point.distance(self.p1, self.p2)

@property
def midpoint(self):
"""The midpoint of the line segment.

========

sympy.geometry.point.Point.midpoint

Examples
========

>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(4, 3)
>>> s1 = Segment(p1, p2)
>>> s1.midpoint
Point2D(2, 3/2)
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3)
>>> s1 = Segment3D(p1, p2)
>>> s1.midpoint
Point3D(2, 3/2, 3/2)

"""
return Point.midpoint(self.p1, self.p2)

[docs]    def perpendicular_bisector(self, p=None):
"""The perpendicular bisector of this segment.

If no point is specified or the point specified is not on the
bisector then the bisector is returned as a Line. Otherwise a
Segment is returned that joins the point specified and the
intersection of the bisector and the segment.

Parameters
==========

p : Point

Returns
=======

bisector : Line or Segment

========

LinearEntity.perpendicular_segment

Examples
========

>>> from sympy import Point, Segment
>>> p1, p2, p3 = Point(0, 0), Point(6, 6), Point(5, 1)
>>> s1 = Segment(p1, p2)
>>> s1.perpendicular_bisector()
Line2D(Point2D(3, 3), Point2D(-3, 9))

>>> s1.perpendicular_bisector(p3)
Segment2D(Point2D(3, 3), Point2D(5, 1))

"""
l = self.perpendicular_line(self.midpoint)
if p is not None:
p2 = Point(p, dim=self.ambient_dimension)
if p2 in l:
return Segment(self.midpoint, p2)
return l

[docs]    def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Segment gives
values that will produce the full segment in a plot.

Parameters
==========

parameter : str, optional
Default value is 't'.

Returns
=======

plot_interval : list
[parameter, lower_bound, upper_bound]

Examples
========

>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> s1 = Segment(p1, p2)
>>> s1.plot_interval()
[t, 0, 1]

"""
t = _symbol(parameter)
return [t, 0, 1]

[docs]class LinearEntity2D(LinearEntity):
"""A base class for all linear entities (line, ray and segment)
in a 2-dimensional Euclidean space.

Attributes
==========

p1
p2
coefficients
slope
points

Notes
=====

This is an abstract class and is not meant to be instantiated.

========

sympy.geometry.entity.GeometryEntity

"""
@property
def bounds(self):
"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
rectangle for the geometric figure.

"""

verts = self.points
xs = [p.x for p in verts]
ys = [p.y for p in verts]
return (min(xs), min(ys), max(xs), max(ys))

[docs]    def perpendicular_line(self, p):
"""Create a new Line perpendicular to this linear entity which passes
through the point p.

Parameters
==========

p : Point

Returns
=======

line : Line

========

is_perpendicular, perpendicular_segment

Examples
========

>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> l1 = Line(p1, p2)
>>> l2 = l1.perpendicular_line(p3)
>>> p3 in l2
True
>>> l1.is_perpendicular(l2)
True

"""
p = Point(p, dim=self.ambient_dimension)
# any two lines in R^2 intersect, so blindly making
# a line through p in an orthogonal direction will work
return Line(p, p + self.direction.orthogonal_direction)

@property
def slope(self):
"""The slope of this linear entity, or infinity if vertical.

Returns
=======

slope : number or sympy expression

========

coefficients

Examples
========

>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(3, 5)
>>> l1 = Line(p1, p2)
>>> l1.slope
5/3

>>> p3 = Point(0, 4)
>>> l2 = Line(p1, p3)
>>> l2.slope
oo

"""
d1, d2 = (self.p1 - self.p2).args
if d1 == 0:
return S.Infinity
return simplify(d2/d1)

[docs]class Line2D(LinearEntity2D, Line):
"""An infinite line in space 2D.

A line is declared with two distinct points or a point and slope
as defined using keyword slope.

Parameters
==========

p1 : Point
pt : Point
slope : sympy expression

========

sympy.geometry.point.Point

Examples
========

>>> import sympy
>>> from sympy import Point
>>> from sympy.abc import L
>>> from sympy.geometry import Line, Segment
>>> L = Line(Point(2,3), Point(3,5))
>>> L
Line2D(Point2D(2, 3), Point2D(3, 5))
>>> L.points
(Point2D(2, 3), Point2D(3, 5))
>>> L.equation()
-2*x + y + 1
>>> L.coefficients
(-2, 1, 1)

Instantiate with keyword slope:

>>> Line(Point(0, 0), slope=0)
Line2D(Point2D(0, 0), Point2D(1, 0))

Instantiate with another linear object

>>> s = Segment((0, 0), (0, 1))
>>> Line(s).equation()
x
"""
def __new__(cls, p1, pt=None, slope=None, **kwargs):
if isinstance(p1, LinearEntity):
if pt is not None:
raise ValueError('When p1 is a LinearEntity, pt should be None')
p1, pt = Point._normalize_dimension(*p1.args, dim=2)
else:
p1 = Point(p1, dim=2)
if pt is not None and slope is None:
try:
p2 = Point(pt, dim=2)
except (NotImplementedError, TypeError, ValueError):
raise ValueError('The 2nd argument was not a valid Point. '
'If it was a slope, enter it with keyword "slope".')
elif slope is not None and pt is None:
slope = sympify(slope)
if slope.is_finite is False:
# when infinite slope, don't change x
dx = 0
dy = 1
else:
# go over 1 up slope
dx = 1
dy = slope
# XXX avoiding simplification by adding to coords directly
p2 = Point(p1.x + dx, p1.y + dy, evaluate=False)
else:
raise ValueError('A 2nd Point or keyword "slope" must be used.')
return LinearEntity2D.__new__(cls, p1, p2, **kwargs)

def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG path element for the LinearEntity.

Parameters
==========

scale_factor : float
Multiplication factor for the SVG stroke-width.  Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""

from sympy.core.evalf import N

verts = (N(self.p1), N(self.p2))
coords = ["{0},{1}".format(p.x, p.y) for p in verts]
path = "M {0} L {1}".format(coords[0], " L ".join(coords[1:]))

return (
'<path fill-rule="evenodd" fill="{2}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" d="{1}" '
'marker-start="url(#markerReverseArrow)" marker-end="url(#markerArrow)"/>'
).format(2. * scale_factor, path, fill_color)

@property
def coefficients(self):
"""The coefficients (a, b, c) for ax + by + c = 0.

========

sympy.geometry.line.Line.equation

Examples
========

>>> from sympy import Point, Line
>>> from sympy.abc import x, y
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.coefficients
(-3, 5, 0)

>>> p3 = Point(x, y)
>>> l2 = Line(p1, p3)
>>> l2.coefficients
(-y, x, 0)

"""
p1, p2 = self.points
if p1.x == p2.x:
return (S.One, S.Zero, -p1.x)
elif p1.y == p2.y:
return (S.Zero, S.One, -p1.y)
return tuple([simplify(i) for i in
(self.p1.y - self.p2.y,
self.p2.x - self.p1.x,
self.p1.x*self.p2.y - self.p1.y*self.p2.x)])

[docs]    def equation(self, x='x', y='y'):
"""The equation of the line: ax + by + c.

Parameters
==========

x : str, optional
The name to use for the x-axis, default value is 'x'.
y : str, optional
The name to use for the y-axis, default value is 'y'.

Returns
=======

equation : sympy expression

========

LinearEntity.coefficients

Examples
========

>>> from sympy import Point, Line
>>> p1, p2 = Point(1, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.equation()
-3*x + 4*y + 3

"""
x, y = _symbol(x), _symbol(y)
p1, p2 = self.points
if p1.x == p2.x:
return x - p1.x
elif p1.y == p2.y:
return y - p1.y

a, b, c = self.coefficients
return a*x + b*y + c

[docs]class Ray2D(LinearEntity2D, Ray):
"""
A Ray is a semi-line in the space with a source point and a direction.

Parameters
==========

p1 : Point
The source of the Ray
p2 : Point or radian value
This point determines the direction in which the Ray propagates.
If given as an angle it is interpreted in radians with the positive
direction being ccw.

Attributes
==========

source
xdirection
ydirection

========

sympy.geometry.point.Point, Line

Examples
========

>>> import sympy
>>> from sympy import Point, pi
>>> from sympy.geometry import Ray
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r
Ray2D(Point2D(2, 3), Point2D(3, 5))
>>> r.points
(Point2D(2, 3), Point2D(3, 5))
>>> r.source
Point2D(2, 3)
>>> r.xdirection
oo
>>> r.ydirection
oo
>>> r.slope
2
>>> Ray(Point(0, 0), angle=pi/4).slope
1

"""
def __new__(cls, p1, pt=None, angle=None, **kwargs):
p1 = Point(p1, dim=2)
if pt is not None and angle is None:
try:
p2 = Point(pt, dim=2)
except (NotImplementedError, TypeError, ValueError):
from sympy.utilities.misc import filldedent
raise ValueError(filldedent('''
The 2nd argument was not a valid Point; if
it was meant to be an angle it should be
given with keyword "angle".'''))
if p1 == p2:
raise ValueError('A Ray requires two distinct points.')
elif angle is not None and pt is None:
# we need to know if the angle is an odd multiple of pi/2
c = pi_coeff(sympify(angle))
p2 = None
if c is not None:
if c.is_Rational:
if c.q == 2:
if c.p == 1:
p2 = p1 + Point(0, 1)
elif c.p == 3:
p2 = p1 + Point(0, -1)
elif c.q == 1:
if c.p == 0:
p2 = p1 + Point(1, 0)
elif c.p == 1:
p2 = p1 + Point(-1, 0)
if p2 is None:
c *= S.Pi
else:
c = angle % (2*S.Pi)
if not p2:
m = 2*c/S.Pi
left = And(1 < m, m < 3)  # is it in quadrant 2 or 3?
x = Piecewise((-1, left), (Piecewise((0, Eq(m % 1, 0)), (1, True)), True))
y = Piecewise((-tan(c), left), (Piecewise((1, Eq(m, 1)), (-1, Eq(m, 3)), (tan(c), True)), True))
p2 = p1 + Point(x, y)
else:
raise ValueError('A 2nd point or keyword "angle" must be used.')

return LinearEntity2D.__new__(cls, p1, p2, **kwargs)

@property
def xdirection(self):
"""The x direction of the ray.

Positive infinity if the ray points in the positive x direction,
negative infinity if the ray points in the negative x direction,
or 0 if the ray is vertical.

========

ydirection

Examples
========

>>> from sympy import Point, Ray
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, -1)
>>> r1, r2 = Ray(p1, p2), Ray(p1, p3)
>>> r1.xdirection
oo
>>> r2.xdirection
0

"""
if self.p1.x < self.p2.x:
return S.Infinity
elif self.p1.x == self.p2.x:
return S.Zero
else:
return S.NegativeInfinity

@property
def ydirection(self):
"""The y direction of the ray.

Positive infinity if the ray points in the positive y direction,
negative infinity if the ray points in the negative y direction,
or 0 if the ray is horizontal.

========

xdirection

Examples
========

>>> from sympy import Point, Ray
>>> p1, p2, p3 = Point(0, 0), Point(-1, -1), Point(-1, 0)
>>> r1, r2 = Ray(p1, p2), Ray(p1, p3)
>>> r1.ydirection
-oo
>>> r2.ydirection
0

"""
if self.p1.y < self.p2.y:
return S.Infinity
elif self.p1.y == self.p2.y:
return S.Zero
else:
return S.NegativeInfinity

[docs]class Segment2D(LinearEntity2D, Segment):
"""An undirected line segment in 2D space.

Parameters
==========

p1 : Point
p2 : Point

Attributes
==========

length : number or sympy expression
midpoint : Point

========

sympy.geometry.point.Point, Line

Examples
========

>>> import sympy
>>> from sympy import Point
>>> from sympy.geometry import Segment
>>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts
Segment2D(Point2D(1, 0), Point2D(1, 1))
>>> s = Segment(Point(4, 3), Point(1, 1))
>>> s
Segment2D(Point2D(1, 1), Point2D(4, 3))
>>> s.points
(Point2D(1, 1), Point2D(4, 3))
>>> s.slope
2/3
>>> s.length
sqrt(13)
>>> s.midpoint
Point2D(5/2, 2)

"""
def __new__(cls, p1, p2, **kwargs):
# Reorder the two points under the following ordering:
#   if p1.x != p2.x then p1.x < p2.x
#   if p1.x == p2.x then p1.y < p2.y
p1 = Point(p1, dim=2)
p2 = Point(p2, dim=2)
if p1 == p2:
return p1
if (p1.x > p2.x) == True:
p1, p2 = p2, p1
elif (p1.x == p2.x) == True and (p1.y > p2.y) == True:
p1, p2 = p2, p1
return LinearEntity2D.__new__(cls, p1, p2, **kwargs)

def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG path element for the LinearEntity.

Parameters
==========

scale_factor : float
Multiplication factor for the SVG stroke-width.  Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""

from sympy.core.evalf import N

verts = (N(self.p1), N(self.p2))
coords = ["{0},{1}".format(p.x, p.y) for p in verts]
path = "M {0} L {1}".format(coords[0], " L ".join(coords[1:]))
return (
'<path fill-rule="evenodd" fill="{2}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" d="{1}" />'
).format(2. * scale_factor, path, fill_color)

[docs]class LinearEntity3D(LinearEntity):
"""An base class for all linear entities (line, ray and segment)
in a 3-dimensional Euclidean space.

Attributes
==========

p1
p2
direction_ratio
direction_cosine
points

Notes
=====

This is a base class and is not meant to be instantiated.
"""

def __new__(cls, p1, p2, **kwargs):
p1 = Point3D(p1, dim=3)
p2 = Point3D(p2, dim=3)
if p1 == p2:
# if it makes sense to return a Point, handle in subclass
raise ValueError(
"%s.__new__ requires two unique Points." % cls.__name__)

return GeometryEntity.__new__(cls, p1, p2, **kwargs)

ambient_dimension = 3

@property
def direction_ratio(self):
"""The direction ratio of a given line in 3D.

========

sympy.geometry.line.Line.equation

Examples
========

>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l = Line3D(p1, p2)
>>> l.direction_ratio
[5, 3, 1]
"""
p1, p2 = self.points
return p1.direction_ratio(p2)

@property
def direction_cosine(self):
"""The normalized direction ratio of a given line in 3D.

========

sympy.geometry.line.Line.equation

Examples
========

>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l = Line3D(p1, p2)
>>> l.direction_cosine
[sqrt(35)/7, 3*sqrt(35)/35, sqrt(35)/35]
>>> sum(i**2 for i in _)
1
"""
p1, p2 = self.points
return p1.direction_cosine(p2)

[docs]class Line3D(LinearEntity3D, Line):
"""An infinite 3D line in space.

A line is declared with two distinct points or a point and direction_ratio
as defined using keyword direction_ratio.

Parameters
==========

p1 : Point3D
pt : Point3D
direction_ratio : list

========

sympy.geometry.point.Point3D
sympy.geometry.line.Line
sympy.geometry.line.Line2D

Examples
========

>>> import sympy
>>> from sympy import Point3D
>>> from sympy.geometry import Line3D, Segment3D
>>> L = Line3D(Point3D(2, 3, 4), Point3D(3, 5, 1))
>>> L
Line3D(Point3D(2, 3, 4), Point3D(3, 5, 1))
>>> L.points
(Point3D(2, 3, 4), Point3D(3, 5, 1))
"""

def __new__(cls, p1, pt=None, direction_ratio=[], **kwargs):
if isinstance(p1, LinearEntity3D):
if pt is not None:
raise ValueError('if p1 is a LinearEntity, pt must be None.')
p1, pt = p1.args
else:
p1 = Point(p1, dim=3)
if pt is not None and len(direction_ratio) == 0:
pt = Point(pt, dim=3)
elif len(direction_ratio) == 3 and pt is None:
pt = Point3D(p1.x + direction_ratio[0], p1.y + direction_ratio[1],
p1.z + direction_ratio[2])
else:
raise ValueError('A 2nd Point or keyword "direction_ratio" must '
'be used.')

return LinearEntity3D.__new__(cls, p1, pt, **kwargs)

[docs]    def equation(self, x='x', y='y', z='z', k='k'):
"""The equation of the line in 3D

Parameters
==========

x : str, optional
The name to use for the x-axis, default value is 'x'.
y : str, optional
The name to use for the y-axis, default value is 'y'.
z : str, optional
The name to use for the x-axis, default value is 'z'.

Returns
=======

equation : tuple

Examples
========

>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(1, 0, 0), Point3D(5, 3, 0)
>>> l1 = Line3D(p1, p2)
>>> l1.equation()
(x/4 - 1/4, y/3, zoo*z, k)

"""
x, y, z, k = _symbol(x), _symbol(y), _symbol(z), _symbol(k)
p1, p2 = self.points
a = p1.direction_ratio(p2)
return (((x - p1.x)/a[0]), ((y - p1.y)/a[1]),
((z - p1.z)/a[2]), k)

[docs]class Ray3D(LinearEntity3D, Ray):
"""
A Ray is a semi-line in the space with a source point and a direction.

Parameters
==========

p1 : Point3D
The source of the Ray
p2 : Point or a direction vector
direction_ratio: Determines the direction in which the Ray propagates.

Attributes
==========

source
xdirection
ydirection
zdirection

========

sympy.geometry.point.Point3D, Line3D

Examples
========

>>> import sympy
>>> from sympy import Point3D, pi
>>> from sympy.geometry import Ray3D
>>> r = Ray3D(Point3D(2, 3, 4), Point3D(3, 5, 0))
>>> r
Ray3D(Point3D(2, 3, 4), Point3D(3, 5, 0))
>>> r.points
(Point3D(2, 3, 4), Point3D(3, 5, 0))
>>> r.source
Point3D(2, 3, 4)
>>> r.xdirection
oo
>>> r.ydirection
oo
>>> r.direction_ratio
[1, 2, -4]

"""

def __new__(cls, p1, pt=None, direction_ratio=[], **kwargs):
if isinstance(p1, LinearEntity3D):
if pt is not None:
raise ValueError('If p1 is a LinearEntity, pt must be None')
p1, pt = p1.args
else:
p1 = Point(p1, dim=3)
if pt is not None and len(direction_ratio) == 0:
pt = Point(pt, dim=3)
elif len(direction_ratio) == 3 and pt is None:
pt = Point3D(p1.x + direction_ratio[0], p1.y + direction_ratio[1],
p1.z + direction_ratio[2])
else:
raise ValueError('A 2nd Point or keyword "direction_ratio" must'
'be used.')

return LinearEntity3D.__new__(cls, p1, pt, **kwargs)

@property
def xdirection(self):
"""The x direction of the ray.

Positive infinity if the ray points in the positive x direction,
negative infinity if the ray points in the negative x direction,
or 0 if the ray is vertical.

========

ydirection

Examples
========

>>> from sympy import Point3D, Ray3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, -1, 0)
>>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3)
>>> r1.xdirection
oo
>>> r2.xdirection
0

"""
if self.p1.x < self.p2.x:
return S.Infinity
elif self.p1.x == self.p2.x:
return S.Zero
else:
return S.NegativeInfinity

@property
def ydirection(self):
"""The y direction of the ray.

Positive infinity if the ray points in the positive y direction,
negative infinity if the ray points in the negative y direction,
or 0 if the ray is horizontal.

========

xdirection

Examples
========

>>> from sympy import Point3D, Ray3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(-1, -1, -1), Point3D(-1, 0, 0)
>>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3)
>>> r1.ydirection
-oo
>>> r2.ydirection
0

"""
if self.p1.y < self.p2.y:
return S.Infinity
elif self.p1.y == self.p2.y:
return S.Zero
else:
return S.NegativeInfinity

@property
def zdirection(self):
"""The z direction of the ray.

Positive infinity if the ray points in the positive z direction,
negative infinity if the ray points in the negative z direction,
or 0 if the ray is horizontal.

========

xdirection

Examples
========

>>> from sympy import Point3D, Ray3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(-1, -1, -1), Point3D(-1, 0, 0)
>>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3)
>>> r1.ydirection
-oo
>>> r2.ydirection
0
>>> r2.zdirection
0

"""
if self.p1.z < self.p2.z:
return S.Infinity
elif self.p1.z == self.p2.z:
return S.Zero
else:
return S.NegativeInfinity

[docs]class Segment3D(LinearEntity3D, Segment):
"""A undirected line segment in a 3D space.

Parameters
==========

p1 : Point3D
p2 : Point3D

Attributes
==========

length : number or sympy expression
midpoint : Point3D

========

sympy.geometry.point.Point3D, Line3D

Examples
========

>>> import sympy
>>> from sympy import Point3D
>>> from sympy.geometry import Segment3D
>>> Segment3D((1, 0, 0), (1, 1, 1)) # tuples are interpreted as pts
Segment3D(Point3D(1, 0, 0), Point3D(1, 1, 1))
>>> s = Segment3D(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s
Segment3D(Point3D(1, 1, 7), Point3D(4, 3, 9))
>>> s.points
(Point3D(1, 1, 7), Point3D(4, 3, 9))
>>> s.length
sqrt(17)
>>> s.midpoint
Point3D(5/2, 2, 8)

"""

def __new__(cls, p1, p2, **kwargs):
# Reorder the two points under the following ordering:
#   if p1.x != p2.x then p1.x < p2.x
#   if p1.x == p2.x then p1.y < p2.y
#   The z-coordinate will not come into picture while ordering
p1 = Point(p1, dim=3)
p2 = Point(p2, dim=3)

if p1 == p2:
return p1
if (p1.x > p2.x) == True:
p1, p2 = p2, p1
elif (p1.x == p2.x) == True and (p1.y > p2.y) == True:
p1, p2 = p2, p1
return LinearEntity3D.__new__(cls, p1, p2, **kwargs)