# Source code for sympy.geometry.line

"""Line-like geometrical entities.

Contains
========
LinearEntity
Line
Ray
Segment

"""
from __future__ import print_function, division

from sympy.core import S, C, sympify, Dummy
from sympy.functions.elementary.trigonometric import _pi_coeff as pi_coeff
from sympy.core.logic import fuzzy_and
from sympy.simplify.simplify import simplify
from sympy.solvers import solve
from sympy.geometry.exceptions import GeometryError
from .entity import GeometryEntity
from .point import Point
from .util import _symbol

# TODO: this should be placed elsewhere and reused in other modules

class Undecidable(ValueError):
pass

[docs]class LinearEntity(GeometryEntity):
"""An abstract 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.
Subclasses should implement the following methods:

* __eq__
* contains

========

sympy.geometry.entity.GeometryEntity

"""

def __new__(cls, p1, p2, **kwargs):
p1 = Point(p1)
p2 = Point(p2)
if p1 == p2:
# Rolygon returns lower priority classes...should LinearEntity, too?
return p1  # raise ValueError("%s.__new__ requires two unique Points." % cls.__name__)

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

@property
[docs]    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
Point(0, 0)

"""
return self.args[0]

@property
[docs]    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
Point(5, 3)

"""
return self.args[1]

@property
[docs]    def coefficients(self):
"""The coefficients (a, b, c) for the linear equation 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 is_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 are concurrent,
False : otherwise.

Notes
=====

Simply take the first two lines and find their intersection.
If there is no intersection, then the first two lines were
parallel and had no intersection so concurrency is impossible
amongst the whole set. Otherwise, check to see if the
intersection point of the first two lines is a member on
the rest of the lines. If so, the lines are concurrent.

========

sympy.geometry.util.intersection

Examples
========

>>> from sympy import Point, Line
>>> 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)
>>> l1.is_concurrent(l2, l3)
True

>>> l4 = Line(p2, p3)
>>> l4.is_concurrent(l2, l3)
False

"""

# Concurrency requires intersection at a single point; One linear
# entity cannot be concurrent.
if len(lines) <= 1:
return False

try:
# Get the intersection (if parallel)
p = lines[0].intersection(lines[1])
if len(p) == 0:
return False

# Make sure the intersection is on every linear entity
for line in lines[2:]:
if p[0] not in line:
return False
return True
except AttributeError:
return False

[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

"""
try:
a1, b1, c1 = l1.coefficients
a2, b2, c2 = l2.coefficients
return bool(simplify(a1*b2 - b1*a2) == 0)
except AttributeError:
return False

[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

"""
try:
a1, b1, c1 = l1.coefficients
a2, b2, c2 = l2.coefficients
return bool(simplify(a1*a2 + b1*b2) == 0)
except AttributeError:
return False

[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

"""
v1 = l1.p2 - l1.p1
v2 = l2.p2 - l2.p1
return C.acos(v1.dot(v2)/(abs(v1)*abs(v2)))

[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

"""
d = self.p1 - self.p2
return Line(p, p + d)

[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

"""
d1, d2 = (self.p1 - self.p2).args
if d2 == 0:  # If a horizontal line
if p.y == self.p1.y:  # if p is on this linear entity
return Line(p, p + Point(0, 1))
else:
p2 = Point(p.x, self.p1.y)
return Line(p, p2)
else:
p2 = Point(p.x - d2, p.y + d1)
return Line(p, p2)

[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))
Segment(Point(2, 2), Point(4, 0))

"""
if p in self:
return p
pl = self.perpendicular_line(p)
p2 = Line(self).intersection(pl)[0]
return Segment(p, p2)

@property
[docs]    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
[docs]    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)

@property
[docs]    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
(Point(0, 0), Point(5, 11))

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

[docs]    def projection(self, o):
"""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 is done by creating a
perpendicular line through P and L and finding its
intersection with L.

========

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)
Point(1/4, 1/4)

>>> p4, p5 = Point(10, 0), Point(12, 1)
>>> s1 = Segment(p4, p5)
>>> l1.projection(s1)
Segment(Point(5, 5), Point(13/2, 13/2))

"""
tline = Line(self.p1, self.p2)

def _project(p):
"""Project a point onto the line representing self."""
if p in tline:
return p
l1 = tline.perpendicular_line(p)
return tline.intersection(l1)[0]

projected = None
if isinstance(o, Point):
return _project(o)
elif isinstance(o, LinearEntity):
n_p1 = _project(o.p1)
n_p2 = _project(o.p2)
if n_p1 == n_p2:
projected = n_p1
else:
projected = o.__class__(n_p1, n_p2)

# Didn't know how to project so raise an error
if projected is None:
n1 = self.__class__.__name__
n2 = o.__class__.__name__
raise GeometryError(
"Do not know how to project %s onto %s" % (n2, n1))

return self.intersection(projected)[0]

[docs]    def intersection(self, o):
"""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)
[Point(7, 7)]

>>> p4, p5 = Point(5, 0), Point(0, 3)
>>> l2 = Line(p4, p5)
>>> l1.intersection(l2)
[Point(15/8, 15/8)]

>>> p6, p7 = Point(0, 5), Point(2, 6)
>>> s1 = Segment(p6, p7)
>>> l1.intersection(s1)
[]

"""
if isinstance(o, Point):
if o in self:
return [o]
else:
return []
elif isinstance(o, LinearEntity):
a1, b1, c1 = self.coefficients
a2, b2, c2 = o.coefficients
t = simplify(a1*b2 - a2*b1)
if t.equals(0) is not False:  # assume they are parallel
if isinstance(self, Line):
if o.p1 in self:
return [o]
return []
elif isinstance(o, Line):
if self.p1 in o:
return [self]
return []
elif isinstance(self, Ray):
if isinstance(o, Ray):
# case 1, rays in the same direction
if self.xdirection == o.xdirection:
if self.source.x < o.source.x:
return [o]
return [self]
# case 2, rays in the opposite directions
else:
if o.source in self:
if self.source == o.source:
return [self.source]
return [Segment(o.source, self.source)]
return []
elif isinstance(o, Segment):
if o.p1 in self:
if o.p2 in self:
return [o]
return [Segment(o.p1, self.source)]
elif o.p2 in self:
return [Segment(o.p2, self.source)]
return []
elif isinstance(self, Segment):
if isinstance(o, Ray):
return o.intersection(self)
elif isinstance(o, Segment):
# A reminder that the points of Segments are ordered
# in such a way that the following works. See
# Segment.__new__ for details on the ordering.
if self.p1 not in o:
if self.p2 not in o:
# Neither of the endpoints are in o so either
# o is contained in this segment or it isn't
if o in self:
return [self]
return []
else:
# p1 not in o but p2 is. Either there is a
# segment as an intersection, or they only
# intersect at an endpoint
if self.p2 == o.p1:
return [o.p1]
return [Segment(o.p1, self.p2)]
elif self.p2 not in o:
# p2 not in o but p1 is. Either there is a
# segment as an intersection, or they only
# intersect at an endpoint
if self.p1 == o.p2:
return [o.p2]
return [Segment(o.p2, self.p1)]

# Both points of self in o so the whole segment
# is in o
return [self]

# Unknown linear entity
return []

# Not parallel, so find the point of intersection
px = simplify((b1*c2 - c1*b2) / t)
py = simplify((a2*c1 - a1*c2) / t)
inter = Point(px, py)
# we do not use a simplistic 'inter in self and inter in o'
# because that requires an equality test that is fragile;
# instead we employ some diagnostics to see if the intersection
# is valid

def inseg(self):
def _between(a, b, c):
return c >= a and c <= b or c <= a and c >= b
if _between(self.p1.x, self.p2.x, inter.x) and \
_between(self.p1.y, self.p2.y, inter.y):
return True

def inray(self):
sray = Ray(self.p1, inter)
if sray.xdirection == self.xdirection and \
sray.ydirection == self.ydirection:
return True
for i in range(2):
if isinstance(self, Line):
if isinstance(o, Line):
return [inter]
elif isinstance(o, Ray) and inray(o):
return [inter]
elif isinstance(o, Segment) and inseg(o):
return [inter]
elif isinstance(self, Ray) and inray(self):
if isinstance(o, Ray) and inray(o):
return [inter]
elif isinstance(o, Segment) and inseg(o):
return [inter]
elif isinstance(self, Segment) and inseg(self):
if isinstance(o, Segment) and inseg(o):
return [inter]
self, o = o, self
return []

return o.intersection(self)

[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()
Point(4*t + 1, 3*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)
x = simplify(self.p1.x + t*(self.p2.x - self.p1.x))
y = simplify(self.p1.y + t*(self.p2.y - self.p1.y))
return Point(x, y)

[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
Point(...)
>>> # point should belong to the line
>>> p3 in l1
True

"""
from random import randint

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

if self.slope is S.Infinity:
if isinstance(self, Ray):
if self.ydirection is S.Infinity:
lower = self.p1.y
else:
upper = self.p1.y
elif isinstance(self, Segment):
lower = self.p1.y
upper = self.p2.y

x = self.p1.x
y = randint(lower, upper)
else:
if isinstance(self, Ray):
if self.xdirection is S.Infinity:
lower = self.p1.x
else:
upper = self.p1.x
elif isinstance(self, Segment):
lower = self.p1.x
upper = self.p2.x

a, b, c = self.coefficients
x = randint(lower, upper)
y = (-c - a*x) / b
return Point(x, y)

[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
"""
def _norm(a, b, c):
if a != 0:
return 1, b/a, c/a
elif b != 0:
return a/b, 1, c/b
else:
return c
return _norm(*self.coefficients) == _norm(*other.coefficients)

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))

[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()

def __eq__(self, other):
"""Subclasses should implement this method."""
raise NotImplementedError()

def __hash__(self):
return super(LinearEntity, self).__hash__()

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

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

Notes
=====

At the moment only lines in a 2D space can be declared, because
Points can be defined only for 2D spaces.

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
Line(Point(2, 3), Point(3, 5))
>>> L.points
(Point(2, 3), Point(3, 5))
>>> L.equation()
-2*x + y + 1
>>> L.coefficients
(-2, 1, 1)

Instantiate with keyword slope:

>>> Line(Point(0, 0), slope=0)
Line(Point(0, 0), Point(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):
p1, pt = p1.args
else:
p1 = Point(p1)
if pt is not None and slope is None:
try:
p2 = Point(pt)
except NotImplementedError:
raise ValueError('The 2nd argument was not a valid Point. '
'If it was a slope, enter it with keyword "slope".')
if p1 == p2:
raise ValueError('A line requires two distinct points.')
elif slope is not None and pt is None:
slope = sympify(slope)
if slope.is_bounded is False:
# when unbounded slope, don't change x
p2 = p1 + Point(0, 1)
else:
# go over 1 up slope
p2 = p1 + Point(1, slope)
else:
raise ValueError('A 2nd Point or keyword "slope" must be used.')

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

[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]    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 simplify(a*x + b*y + c)

[docs]    def contains(self, o):
"""Return True if o is on this Line, or False otherwise."""
if isinstance(o, Point):
x, y = Dummy(), Dummy()
eq = self.equation(x, y)
if not eq.has(y):
return (solve(eq, x)[0] - o.x).equals(0)
if not eq.has(x):
return (solve(eq, y)[0] - o.y).equals(0)
return (solve(eq.subs(x, o.x), y)[0] - o.y).equals(0)
elif not isinstance(o, LinearEntity):
return False
elif isinstance(o, Line):
return self.__eq__(o)
elif not self.is_similar(o):
return False
else:
return o.p1 in self and o.p2 in self

def __eq__(self, other):
"""Return True if other is equal to this Line, or False otherwise."""
if not isinstance(other, Line):
return False
return Point.is_collinear(self.p1, self.p2, other.p1, other.p2)

def __hash__(self):
return super(Line, self).__hash__()

[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
xdirection
ydirection

========

sympy.geometry.point.Point, Line

Notes
=====

At the moment only rays in a 2D space can be declared, because
Points can be defined only for 2D spaces.

Examples
========

>>> import sympy
>>> from sympy import Point, pi
>>> from sympy.abc import r
>>> from sympy.geometry import Ray
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r
Ray(Point(2, 3), Point(3, 5))
>>> r.points
(Point(2, 3), Point(3, 5))
>>> r.source
Point(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)
if pt is not None and angle is None:
try:
p2 = Point(pt)
except NotImplementedError:
raise ValueError(
'The 2nd argument was not a valid Point;\nif '
'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
if not p2:
p2 = p1 + Point(1, C.tan(c))
else:
raise ValueError('A 2nd point or keyword "angle" must be used.')

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

@property
[docs]    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
Point(0, 0)

"""
return self.p1

@property
[docs]    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
[docs]    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]    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]

def __eq__(self, other):
"""Is the other GeometryEntity equal to this Ray?"""
if not isinstance(other, Ray):
return False
return (self.source == other.source) and (other.p2 in self)

def __hash__(self):
return super(Ray, self).__hash__()

[docs]    def contains(self, o):
"""Is other GeometryEntity contained in this Ray?"""
if isinstance(o, Ray):
return (Point.is_collinear(self.p1, self.p2, o.p1, o.p2) and
self.xdirection == o.xdirection and
self.ydirection == o.ydirection)
elif isinstance(o, Segment):
return o.p1 in self and o.p2 in self
elif isinstance(o, Point):
if Point.is_collinear(self.p1, self.p2, o):
if self.xdirection is S.Infinity:
rv = o.x >= self.source.x
elif self.xdirection is S.NegativeInfinity:
rv = o.x <= self.source.x
elif self.ydirection is S.Infinity:
rv = o.y >= self.source.y
else:
rv = o.y <= self.source.y
if isinstance(rv, bool):
return rv
raise Undecidable(
'Cannot determine if %s is in %s' % (o, self))
else:
# Points are not collinear, so the rays are not parallel
# and hence it is impossible for self to contain o
return False

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

[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.point.Point, Line

Notes
=====

At the moment only segments in a 2D space can be declared, because
Points can be defined only for 2D spaces.

Examples
========

>>> import sympy
>>> from sympy import Point
>>> from sympy.abc import s
>>> from sympy.geometry import Segment
>>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts
Segment(Point(1, 0), Point(1, 1))
>>> s = Segment(Point(4, 3), Point(1, 1))
>>> s
Segment(Point(1, 1), Point(4, 3))
>>> s.points
(Point(1, 1), Point(4, 3))
>>> s.slope
2/3
>>> s.length
sqrt(13)
>>> s.midpoint
Point(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)
p2 = Point(p2)
if p1 == p2:
return Point(p1)
if (p1.x > p2.x) is True:
p1, p2 = p2, p1
elif (p1.x == p2.x) is (p1.y > p2.y) is True:
p1, p2 = p2, p1
return LinearEntity.__new__(cls, p1, p2, **kwargs)

[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]    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()
Line(Point(3, 3), Point(9, -3))

>>> s1.perpendicular_bisector(p3)
Segment(Point(3, 3), Point(5, 1))

"""
l = LinearEntity.perpendicular_line(self, self.midpoint)
if p is None or p not in l:
return l
else:
return Segment(self.midpoint, p)

@property
[docs]    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

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

@property
[docs]    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
Point(2, 3/2)

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

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

Raises
======

NotImplementedError is raised if o 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)
"""
if isinstance(o, Point):
return self._do_point_distance(o)
raise NotImplementedError()

def _do_point_distance(self, pt):
"""Calculates the distance between a point and a line segment."""

seg_vector = self.p2 - self.p1
pt_vector = pt - self.p1
t = seg_vector.dot(pt_vector)/self.length**2
if t >= 1:
distance = Point.distance(self.p2, pt)
elif t <= 0:
distance = Point.distance(self.p1, pt)
else:
distance = Point.distance(
self.p1 + Point(t*seg_vector.x, t*seg_vector.y), pt)
return distance

def __eq__(self, other):
"""Is the other GeometryEntity equal to this Ray?"""
if not isinstance(other, Segment):
return False
return (self.p1 == other.p1) and (self.p2 == other.p2)

def __hash__(self):
return super(Segment, self).__hash__()

[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
"""
if isinstance(other, Segment):
return other.p1 in self and other.p2 in self
elif isinstance(other, Point):
if Point.is_collinear(self.p1, self.p2, other):
t = Dummy('t')
x, y = self.arbitrary_point(t).args
if self.p1.x != self.p2.x:
ti = solve(x - other.x, t)[0]
else:
ti = solve(y - other.y, t)[0]
if ti.is_number:
return 0 <= ti <= 1
return None

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