# Source code for sympy.geometry.polygon

from sympy.core import Expr, S, sympify, oo, pi, Symbol, zoo
from sympy.core.compatibility import as_int
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import cos, sin, tan, sqrt
from sympy.simplify import simplify, nsimplify
from sympy.geometry.exceptions import GeometryError
from sympy.matrices import Matrix
from sympy.solvers import solve
from sympy.utilities.iterables import has_variety, has_dups

from .entity import GeometryEntity
from .point import Point
from .ellipse import Circle
from .line import Line, Segment
from .util import _symbol

import warnings

[docs]class Polygon(GeometryEntity):
"""A two-dimensional polygon.

A simple polygon in space. Can be constructed from a sequence of points
or from a center, radius, number of sides and rotation angle.

Parameters
==========

vertices : sequence of Points

Attributes
==========

area
angles
perimeter
vertices
centroid
sides

Raises
======

GeometryError
If all parameters are not Points.

If the Polygon has intersecting sides.

========

sympy.geometry.point.Point, sympy.geometry.line.Segment, Triangle

Notes
=====

Polygons are treated as closed paths rather than 2D areas so
some calculations can be be negative or positive (e.g., area)
based on the orientation of the points.

Any consecutive identical points are reduced to a single point
and any points collinear and between two points will be removed
unless they are needed to define an explicit intersection (see examples).

A Triangle, Segment or Point will be returned when there are 3 or
fewer points provided.

Examples
========

>>> from sympy import Point, Polygon, pi
>>> p1, p2, p3, p4, p5 = [(0, 0), (1, 0), (5, 1), (0, 1), (3, 0)]
>>> Polygon(p1, p2, p3, p4)
Polygon(Point(0, 0), Point(1, 0), Point(5, 1), Point(0, 1))
>>> Polygon(p1, p2)
Segment(Point(0, 0), Point(1, 0))
>>> Polygon(p1, p2, p5)
Segment(Point(0, 0), Point(3, 0))

While the sides of a polygon are not allowed to cross implicitly, they
can do so explicitly. For example, a polygon shaped like a Z with the top
left connecting to the bottom right of the Z must have the point in the
middle of the Z explicitly given:

>>> mid = Point(1, 1)
>>> Polygon((0, 2), (2, 2), mid, (0, 0), (2, 0), mid).area
0
>>> Polygon((0, 2), (2, 2), mid, (2, 0), (0, 0), mid).area
-2

When the the keyword n is used to define the number of sides of the
Polygon then a RegularPolygon is created and the other arguments are
interpreted as center, radius and rotation. The unrotated RegularPolygon
will always have a vertex at Point(r, 0) where r is the radius of the
circle that circumscribes the RegularPolygon. Its method spin can be
used to increment that angle.

>>> p = Polygon((0,0), 1, n=3)
>>> p
RegularPolygon(Point(0, 0), 1, 3, 0)
>>> p.vertices[0]
Point(1, 0)
>>> p.args[0]
Point(0, 0)
>>> p.spin(pi/2)
>>> p.vertices[0]
Point(0, 1)

"""

def __new__(cls, *args, **kwargs):
if kwargs.get('n', 0):
n = kwargs.pop('n')
args = list(args)
# return a virtual polygon with n sides
if len(args) == 2: # center, radius
args.append(n)
elif len(args) == 3: # center, radius, rotation
args.insert(2, n)
return RegularPolygon(*args, **kwargs)

vertices = [Point(a) for a in args]

# remove consecutive duplicates
nodup = []
for p in vertices:
if nodup and p == nodup[-1]:
continue
nodup.append(p)
if len(nodup) > 1 and nodup[-1] == nodup[0]:
nodup.pop() # last point was same as first

# remove collinear points unless they are shared points
got = set()
shared = set()
for p in nodup:
if p in got:
else:
i = -3
while i < len(nodup) - 3 and len(nodup) > 2:
a, b, c = sorted([nodup[i], nodup[i + 1], nodup[i + 2]])
if b not in shared and Point.is_collinear(a, b, c):
nodup[i] = a
nodup[i + 1] = None
nodup.pop(i + 1)
i += 1

vertices = [x for x in nodup if x is not None]

if len(vertices) > 3:
rv = GeometryEntity.__new__(cls, *vertices, **kwargs)
elif len(vertices) == 3:
return Triangle(*vertices, **kwargs)
elif len(vertices) == 2:
return Segment(*vertices, **kwargs)
else:
return Point(*vertices, **kwargs)

# reject polygons that have intersecting sides unless the
# intersection is a shared point or a generalized intersection.
# A self-intersecting polygon is easier to detect than a
# random set of segments since only those sides that are not
# part of the convex hull can possibly intersect with other
# sides of the polygon...but for now we use the n**2 algorithm
# and check all sides with intersection with any preceding sides
hit = _symbol('hit')
if not rv.is_convex:
sides = rv.sides
for i, si in enumerate(sides):
pts = si[0], si[1]
ai = si.arbitrary_point(hit)
for j in range(i):
sj = sides[j]
if sj[0] not in pts and sj[1] not in pts:
aj = si.arbitrary_point(hit)
tx = (solve(ai[0] - aj[0]) or [S.Zero])[0]
if tx.is_number and 0 <= tx <= 1:
ty = (solve(ai[1] - aj[1]) or [S.Zero])[0]
if (tx or ty) and ty.is_number and 0 <= ty <= 1:
print(ai, aj)
raise GeometryError("Polygon has intersecting sides.")

return rv

@property
[docs]    def area(self):
"""
The area of the polygon.

Notes
=====

The area calculation can be positive or negative based on the
orientation of the points.

========

sympy.geometry.ellipse.Ellipse.area

Examples
========

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.area
3

"""
area = 0
args = self.args
for i in range(len(args)):
x1, y1 = args[i - 1].args
x2, y2 = args[i].args
area += x1*y2 - x2*y1
return simplify(area) / 2

@property
[docs]    def angles(self):
"""The internal angle at each vertex.

Returns
=======

angles : dict
A dictionary where each key is a vertex and each value is the
internal angle at that vertex. The vertices are represented as
Points.

========

sympy.geometry.point.Point, sympy.geometry.line.LinearEntity.angle_between

Examples
========

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.angles[p1]
pi/2
>>> poly.angles[p2]
acos(-4*sqrt(17)/17)

"""

def _isright(a, b, c):
ba = b - a
ca = c - a
t_area = ba.x*ca.y - ca.x*ba.y
return bool(t_area <= 0)

# Determine orientation of points
args = self.vertices
cw = _isright(args[-1], args[0], args[1])

ret = {}
for i in range(len(args)):
a, b, c = args[i - 2], args[i - 1], args[i]
ang = Line.angle_between(Line(b, a), Line(b, c))
if cw ^ _isright(a, b, c):
ret[b] = 2*S.Pi - ang
else:
ret[b] = ang
return ret

@property
[docs]    def perimeter(self):
"""The perimeter of the polygon.

Returns
=======

perimeter : number or Basic instance

========

sympy.geometry.line.Segment.length

Examples
========

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.perimeter
sqrt(17) + 7
"""
p = 0
args = self.vertices
for i in range(len(args)):
p += args[i - 1].distance(args[i])
return simplify(p)

@property
[docs]    def vertices(self):
"""The vertices of the polygon.

Returns
=======

vertices : tuple of Points

Notes
=====

When iterating over the vertices, it is more efficient to index self
rather than to request the vertices and index them. Only use the
vertices when you want to process all of them at once. This is even
more important with RegularPolygons that calculate each vertex.

========

sympy.geometry.point.Point

Examples
========

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.vertices
(Point(0, 0), Point(1, 0), Point(5, 1), Point(0, 1))
>>> poly.args[0]
Point(0, 0)

"""
return self.args

@property
[docs]    def centroid(self):
"""The centroid of the polygon.

Returns
=======

centroid : Point

========

sympy.geometry.point.Point, sympy.geometry.util.centroid

Examples
========

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.centroid
Point(31/18, 11/18)

"""
A = 1/(6*self.area)
cx, cy = 0, 0
args = self.args
for i in range(len(args)):
x1, y1 = args[i - 1].args
x2, y2 = args[i].args
v = x1*y2 - x2*y1
cx += v*(x1 + x2)
cy += v*(y1 + y2)
return Point(simplify(A*cx), simplify(A*cy))

@property
[docs]    def sides(self):
"""The line segments that form the sides of the polygon.

Returns
=======

sides : list of sides
Each side is a Segment.

Notes
=====

The Segments that represent the sides are an undirected
line segment so cannot be used to tell the orientation of
the polygon.

========

sympy.geometry.point.Point, sympy.geometry.line.Segment

Examples
========

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.sides
[Segment(Point(0, 0), Point(1, 0)),
Segment(Point(1, 0), Point(5, 1)),
Segment(Point(0, 1), Point(5, 1)), Segment(Point(0, 0), Point(0, 1))]

"""
res = []
args = self.vertices
for i in range(-len(args), 0):
res.append(Segment(args[i], args[i + 1]))
return res

[docs]    def is_convex(self):
"""Is the polygon convex?

A polygon is convex if all its interior angles are less than 180
degrees.

Returns
=======

is_convex : boolean
True if this polygon is convex, False otherwise.

========

sympy.geometry.util.convex_hull

Examples
========

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.is_convex()
True

"""

def _isright(a, b, c):
ba = b - a
ca = c - a
t_area = simplify(ba.x*ca.y - ca.x*ba.y)
return bool(t_area <= 0)

# Determine orientation of points
args = self.vertices
cw = _isright(args[-2], args[-1], args[0])
for i in range(1, len(args)):
if cw ^ _isright(args[i - 2], args[i - 1], args[i]):
return False

return True

[docs]    def encloses_point(self, p):
"""
Return True if p is enclosed by (is inside of) self.

Notes
=====

Being on the border of self is considered False.

Parameters
==========

p : Point

Returns
=======

encloses_point : True, False or None

========

sympy.geometry.point.Point, sympy.geometry.ellipse.Ellipse.encloses_point

Examples
========

>>> from sympy import Polygon, Point
>>> from sympy.abc import t
>>> p = Polygon((0, 0), (4, 0), (4, 4))
>>> p.encloses_point(Point(2, 1))
True
>>> p.encloses_point(Point(2, 2))
False
>>> p.encloses_point(Point(5, 5))
False

References
==========

[1] http://www.ariel.com.au/a/python-point-int-poly.html
[2] http://local.wasp.uwa.edu.au/~pbourke/geometry/insidepoly/

"""
p = Point(p)
if p in self.vertices or any(p in s for s in self.sides):
return False

# move to p, checking that the result is numeric
lit = []
for v in self.vertices:
lit.append(v - p) # the difference is simplified
if lit[-1].free_symbols:
return None
self = Polygon(*lit)

# polygon closure is assumed in the following test but Polygon removes duplicate pts so
# the last point has to be added so all sides are computed. Using Polygon.sides is
# not good since Segments are unordered.
args = self.args
indices = list(range(-len(args), 1))

if self.is_convex():
orientation = None
for i in indices:
a = args[i]
b = args[i + 1]
test = ((-a.y)*(b.x - a.x) - (-a.x)*(b.y - a.y)).is_negative
if orientation is None:
orientation = test
elif test is not orientation:
return False
return True

hit_odd = False
p1x, p1y = args[0].args
for i in indices[1:]:
p2x, p2y = args[i].args
if 0 > min(p1y, p2y):
if 0 <= max(p1y, p2y):
if 0 <= max(p1x, p2x):
if p1y != p2y:
xinters = (-p1y)*(p2x - p1x)/(p2y - p1y) + p1x
if p1x == p2x or 0 <= xinters:
hit_odd = not hit_odd
p1x, p1y = p2x, p2y
return hit_odd

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

The parameter, varying from 0 to 1, assigns points to the position on
the perimeter that is that fraction of the total perimeter. So the
point evaluated at t=1/2 would return the point from the first vertex
that is 1/2 way around the polygon.

Parameters
==========

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

Returns
=======

arbitrary_point : Point

Raises
======

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

========

sympy.geometry.point.Point

Examples
========

>>> from sympy import Polygon, S, Symbol
>>> t = Symbol('t', real=True)
>>> tri = Polygon((0, 0), (1, 0), (1, 1))
>>> p = tri.arbitrary_point('t')
>>> perimeter = tri.perimeter
>>> s1, s2 = [s.length for s in tri.sides[:2]]
>>> p.subs(t, (s1 + s2/2)/perimeter)
Point(1, 1/2)

"""
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)
sides = []
perimeter = self.perimeter
perim_fraction_start = 0
for s in self.sides:
side_perim_fraction = s.length/perimeter
perim_fraction_end = perim_fraction_start + side_perim_fraction
pt  = s.arbitrary_point(parameter).subs(
t, (t - perim_fraction_start)/side_perim_fraction)
sides.append((pt, (perim_fraction_start <= t < perim_fraction_end)))
perim_fraction_start = perim_fraction_end
return Piecewise(*sides)

[docs]    def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the polygon.

Parameters
==========

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

Returns
=======

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

Examples
========

>>> from sympy import Polygon
>>> p = Polygon((0, 0), (1, 0), (1, 1))
>>> p.plot_interval()
[t, 0, 1]

"""
t = Symbol(parameter, real=True)
return [t, 0, 1]

[docs]    def intersection(self, o):
"""The intersection of two polygons.

The intersection may be empty and can contain individual Points and
complete Line Segments.

Parameters
==========

other: Polygon

Returns
=======

intersection : list
The list of Segments and Points

========

sympy.geometry.point.Point, sympy.geometry.line.Segment

Examples
========

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))
>>> poly1 = Polygon(p1, p2, p3, p4)
>>> p5, p6, p7 = list(map(Point, [(3, 2), (1, -1), (0, 2)]))
>>> poly2 = Polygon(p5, p6, p7)
>>> poly1.intersection(poly2)
[Point(2/3, 0), Point(9/5, 1/5), Point(7/3, 1), Point(1/3, 1)]

"""
res = []
for side in self.sides:
inter = side.intersection(o)
if inter is not None:
res.extend(inter)
return res

[docs]    def distance(self, o):
"""
Returns the shortest distance between self and o.

If o is a point, then self does not need to be convex.
If o is another polygon self and o must be complex.

Examples
========

>>> from sympy import Point, Polygon, RegularPolygon
>>> p1, p2 = list(map(Point, [(0, 0), (7, 5)]))
>>> poly = Polygon(*RegularPolygon(p1, 1, 3).vertices)
>>> poly.distance(p2)
sqrt(61)
"""
if isinstance(o, Point):
dist = oo
for side in self.sides:
current = side.distance(o)
if current == 0:
return S.Zero
elif current < dist:
dist = current
return dist
elif isinstance(o, Polygon) and self.is_convex() and o.is_convex():
return self._do_poly_distance(o)
raise NotImplementedError()

def _do_poly_distance(self, e2):
"""
Calculates the least distance between the exteriors of two
convex polygons e1 and e2. Does not check for the convexity
of the polygons as this is checked by Polygon.distance.

Notes
=====

- Prints a warning if the two polygons possibly intersect as the return
value will not be valid in such a case. For a more through test of
intersection use intersection().

========

sympy.geometry.point.Point.distance

Examples
=======

>>> from sympy.geometry import Point, Polygon
>>> square = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0))
>>> triangle = Polygon(Point(1, 2), Point(2, 2), Point(2, 1))
>>> square._do_poly_distance(triangle)
sqrt(2)/2

Description of method used
==========================

Method:
[1] http://cgm.cs.mcgill.ca/~orm/mind2p.html
Uses rotating calipers:
[2] http://en.wikipedia.org/wiki/Rotating_calipers
and antipodal points:
[3] http://en.wikipedia.org/wiki/Antipodal_point
"""
e1 = self

'''Tests for a possible intersection between the polygons and outputs a warning'''
e1_center = e1.centroid
e2_center = e2.centroid
for vertex in e1.vertices:
r = Point.distance(e1_center, vertex)
for vertex in e2.vertices:
r = Point.distance(e2_center, vertex)
center_dist = Point.distance(e1_center, e2_center)
warnings.warn("Polygons may intersect producing erroneous output")

'''
Find the upper rightmost vertex of e1 and the lowest leftmost vertex of e2
'''
e1_ymax = Point(0, -oo)
e2_ymin = Point(0, oo)

for vertex in e1.vertices:
if vertex.y > e1_ymax.y or (vertex.y == e1_ymax.y and vertex.x > e1_ymax.x):
e1_ymax = vertex
for vertex in e2.vertices:
if vertex.y < e2_ymin.y or (vertex.y == e2_ymin.y and vertex.x < e2_ymin.x):
e2_ymin = vertex
min_dist = Point.distance(e1_ymax, e2_ymin)

'''
Produce a dictionary with vertices of e1 as the keys and, for each vertex, the points
to which the vertex is connected as its value. The same is then done for e2.
'''
e1_connections = {}
e2_connections = {}

for side in e1.sides:
if side.p1 in e1_connections:
e1_connections[side.p1].append(side.p2)
else:
e1_connections[side.p1] = [side.p2]

if side.p2 in e1_connections:
e1_connections[side.p2].append(side.p1)
else:
e1_connections[side.p2] = [side.p1]

for side in e2.sides:
if side.p1 in e2_connections:
e2_connections[side.p1].append(side.p2)
else:
e2_connections[side.p1] = [side.p2]

if side.p2 in e2_connections:
e2_connections[side.p2].append(side.p1)
else:
e2_connections[side.p2] = [side.p1]

e1_current = e1_ymax
e2_current = e2_ymin
support_line = Line(Point(S.Zero, S.Zero), Point(S.One, S.Zero))

'''
Determine which point in e1 and e2 will be selected after e2_ymin and e1_ymax,
this information combined with the above produced dictionaries determines the
path that will be taken around the polygons
'''
point1 = e1_connections[e1_ymax][0]
point2 = e1_connections[e1_ymax][1]
angle1 = support_line.angle_between(Line(e1_ymax, point1))
angle2 = support_line.angle_between(Line(e1_ymax, point2))
if angle1 < angle2: e1_next = point1
elif angle2 < angle1: e1_next = point2
elif Point.distance(e1_ymax, point1) > Point.distance(e1_ymax, point2):
e1_next = point2
else: e1_next = point1

point1 = e2_connections[e2_ymin][0]
point2 = e2_connections[e2_ymin][1]
angle1 = support_line.angle_between(Line(e2_ymin, point1))
angle2 = support_line.angle_between(Line(e2_ymin, point2))
if angle1 > angle2: e2_next = point1
elif angle2 > angle1: e2_next = point2
elif Point.distance(e2_ymin, point1) > Point.distance(e2_ymin, point2):
e2_next = point2
else: e2_next = point1

'''
Loop which determins the distance between anti-podal pairs and updates the
minimum distance accordingly. It repeats until it reaches the starting position.
'''
while True:
e1_angle = support_line.angle_between(Line(e1_current, e1_next))
e2_angle = pi - support_line.angle_between(Line(e2_current, e2_next))

if e1_angle < e2_angle:
support_line = Line(e1_current, e1_next)
e1_segment = Segment(e1_current, e1_next)
min_dist_current = e1_segment.distance(e2_current)

if min_dist_current.evalf() < min_dist.evalf(): min_dist = min_dist_current

if e1_connections[e1_next][0] != e1_current:
e1_current = e1_next
e1_next = e1_connections[e1_next][0]
else:
e1_current = e1_next
e1_next = e1_connections[e1_next][1]
elif e1_angle > e2_angle:
support_line = Line(e2_next, e2_current)
e2_segment = Segment(e2_current, e2_next)
min_dist_current = e2_segment.distance(e1_current)

if min_dist_current.evalf() < min_dist.evalf(): min_dist = min_dist_current

if e2_connections[e2_next][0] != e2_current:
e2_current = e2_next
e2_next = e2_connections[e2_next][0]
else:
e2_current = e2_next
e2_next = e2_connections[e2_next][1]
else:
support_line = Line(e1_current, e1_next)
e1_segment = Segment(e1_current, e1_next)
e2_segment = Segment(e2_current, e2_next)
min1 = e1_segment.distance(e2_next)
min2 = e2_segment.distance(e1_next)

min_dist_current = min(min1, min2)
if min_dist_current.evalf() < min_dist.evalf(): min_dist = min_dist_current

if e1_connections[e1_next][0] != e1_current:
e1_current = e1_next
e1_next = e1_connections[e1_next][0]
else:
e1_current = e1_next
e1_next = e1_connections[e1_next][1]

if e2_connections[e2_next][0] != e2_current:
e2_current = e2_next
e2_next = e2_connections[e2_next][0]
else:
e2_current = e2_next
e2_next = e2_connections[e2_next][1]
if e1_current == e1_ymax and e2_current == e2_ymin: break
return min_dist

def __eq__(self, o):
if not isinstance(o, Polygon) or len(self.args) != len(o.args):
return False

# See if self can ever be traversed (cw or ccw) from any of its
# vertices to match all points of o
args = self.args
oargs = o.args
n = len(args)
o0 = oargs[0]
for i0 in range(n):
if args[i0] == o0:
if all(args[(i0 + i) % n] == oargs[i] for i in range(1, n)):
return True
if all(args[(i0 - i) % n] == oargs[i] for i in range(1, n)):
return True
return False

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

def __contains__(self, o):
"""
Return True if o is contained within the boundary lines of self.altitudes

Parameters
==========

other : GeometryEntity

Returns
=======

contained in : bool
The points (and sides, if applicable) are contained in self.

========

sympy.geometry.entity.GeometryEntity.encloses

Examples
========

>>> from sympy import Line, Segment, Point
>>> p = Point(0, 0)
>>> q = Point(1, 1)
>>> s = Segment(p, q*2)
>>> l = Line(p, q)
>>> p in q
False
>>> p in s
True
>>> q*3 in s
False
>>> s in l
True

"""

if isinstance(o, Polygon):
return self == o
elif isinstance(o, Segment):
return any(o in s for s in self.sides)
elif isinstance(o, Point):
if o in self.vertices:
return True
for side in self.sides:
if o in side:
return True

return False

[docs]class RegularPolygon(Polygon):
"""
A regular polygon.

Such a polygon has all internal angles equal and all sides the same length.

Parameters
==========

center : Point
radius : number or Basic instance
The distance from the center to a vertex
n : int
The number of sides

Attributes
==========

vertices
center
rotation
apothem
interior_angle
exterior_angle
circumcircle
incircle
angles

Raises
======

GeometryError
If the center is not a Point, or the radius is not a number or Basic
instance, or the number of sides, n, is less than three.

Notes
=====

A RegularPolygon can be instantiated with Polygon with the kwarg n.

Regular polygons are instantiated with a center, radius, number of sides
and a rotation angle. Whereas the arguments of a Polygon are vertices, the
vertices of the RegularPolygon must be obtained with the vertices method.

========

sympy.geometry.point.Point, Polygon

Examples
========

>>> from sympy.geometry import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r
RegularPolygon(Point(0, 0), 5, 3, 0)
>>> r.vertices[0]
Point(5, 0)

"""

__slots__ = ['_n', '_center', '_radius', '_rot']

def __new__(self, c, r, n, rot=0, **kwargs):
r, n, rot = sympify([r, n, rot])
c = Point(c)
if not isinstance(r, Expr):
raise GeometryError("r must be an Expr object, not %s" % r)
if n.is_Number:
as_int(n) # let an error raise if necessary
if n < 3:
raise GeometryError("n must be a >= 3, not %s" % n)

obj = GeometryEntity.__new__(self, c, r, n, **kwargs)
obj._n = n
obj._center = c
obj._rot = rot
return obj

@property
[docs]    def args(self):
"""
Returns the center point, the radius,
the number of sides, and the orientation angle.

Examples
========

>>> from sympy import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r.args
(Point(0, 0), 5, 3, 0)
"""

def __str__(self):
return 'RegularPolygon(%s, %s, %s, %s)' % tuple(self.args)

def __repr__(self):
return 'RegularPolygon(%s, %s, %s, %s)' % tuple(self.args)

@property
[docs]    def area(self):
"""Returns the area.

Examples
========
>>> from sympy.geometry import RegularPolygon
>>> square = RegularPolygon((0, 0), 1, 4)
>>> square.area
2
>>> _ == square.length**2
True
"""
c, r, n, rot = self.args
return n*self.length**2/(4*tan(pi/n))

@property
[docs]    def length(self):
"""Returns the length of the sides.

The half-length of the side and the apothem form two legs
of a right triangle whose hypotenuse is the radius of the
regular polygon.

Examples
========
>>> from sympy.geometry import RegularPolygon
>>> from sympy import sqrt
>>> s = square_in_unit_circle = RegularPolygon((0, 0), 1, 4)
>>> s.length
sqrt(2)
>>> sqrt((_/2)**2 + s.apothem**2) == s.radius
True

"""

@property
[docs]    def center(self):
"""The center of the RegularPolygon

This is also the center of the circumscribing circle.

Returns
=======

center : Point

========

sympy.geometry.point.Point, sympy.geometry.ellipse.Ellipse.center

Examples
========

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.center
Point(0, 0)
"""
return self._center

@property
[docs]    def circumcenter(self):
"""
Alias for center.

Examples
========

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.circumcenter
Point(0, 0)
"""
return self.center

@property

This is also the radius of the circumscribing circle.

Returns
=======

radius : number or instance of Basic

========

Examples
========

>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
r

"""

@property
"""

Examples
========

>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
r
"""

@property
[docs]    def rotation(self):
"""CCW angle by which the RegularPolygon is rotated

Returns
=======

rotation : number or instance of Basic

Examples
========

>>> from sympy import pi
>>> from sympy.geometry import RegularPolygon, Point
>>> RegularPolygon(Point(0, 0), 3, 4, pi).rotation
pi

"""
return self._rot

@property
[docs]    def apothem(self):

Returns
=======

apothem : number or instance of Basic

========

Examples
========

>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.apothem
sqrt(2)*r/2

"""

@property
"""
Alias for apothem.

Examples
========

>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
sqrt(2)*r/2
"""
return self.apothem

@property
[docs]    def interior_angle(self):
"""Measure of the interior angles.

Returns
=======

interior_angle : number

========

sympy.geometry.line.LinearEntity.angle_between

Examples
========

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.interior_angle
3*pi/4

"""
return (self._n - 2)*S.Pi/self._n

@property
[docs]    def exterior_angle(self):
"""Measure of the exterior angles.

Returns
=======

exterior_angle : number

========

sympy.geometry.line.LinearEntity.angle_between

Examples
========

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.exterior_angle
pi/4

"""
return 2*S.Pi/self._n

@property
[docs]    def circumcircle(self):
"""The circumcircle of the RegularPolygon.

Returns
=======

circumcircle : Circle

========

circumcenter, sympy.geometry.ellipse.Circle

Examples
========

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.circumcircle
Circle(Point(0, 0), 4)

"""

@property
[docs]    def incircle(self):
"""The incircle of the RegularPolygon.

Returns
=======

incircle : Circle

========

Examples
========

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.incircle
Circle(Point(0, 0), 4*cos(pi/8))

"""
return Circle(self.center, self.apothem)

@property
[docs]    def angles(self):
"""
Returns a dictionary with keys, the vertices of the Polygon,
and values, the interior angle at each vertex.

Examples
========

>>> from sympy import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r.angles
{Point(-5/2, -5*sqrt(3)/2): pi/3, Point(-5/2, 5*sqrt(3)/2): pi/3, Point(5, 0): pi/3}
"""
ret = {}
ang = self.interior_angle
for v in self.vertices:
ret[v] = ang
return ret

[docs]    def encloses_point(self, p):
"""
Return True if p is enclosed by (is inside of) self.

Notes
=====

Being on the border of self is considered False.

The general Polygon.encloses_point method is called only if
a point is not within or beyond the incircle or circumcircle,
respectively.

Parameters
==========

p : Point

Returns
=======

encloses_point : True, False or None

========

sympy.geometry.ellipse.Ellipse.encloses_point

Examples
========

>>> from sympy import RegularPolygon, S, Point, Symbol
>>> p = RegularPolygon((0, 0), 3, 4)
>>> p.encloses_point(Point(0, 0))
True
>>> p.encloses_point(Point((r + R)/2, 0))
True
>>> p.encloses_point(Point(R/2, R/2 + (R - r)/10))
False
>>> t = Symbol('t', real=True)
>>> p.encloses_point(p.arbitrary_point().subs(t, S.Half))
False
>>> p.encloses_point(Point(5, 5))
False

"""

c = self.center
d = Segment(c, p).length
return False
return True
else:
# now enumerate the RegularPolygon like a general polygon.
return Polygon.encloses_point(self, p)

[docs]    def spin(self, angle):
"""Increment *in place* the virtual Polygon's rotation by ccw angle.

>>> from sympy import Polygon, Point, pi
>>> r = Polygon(Point(0,0), 1, n=3)
>>> r.vertices[0]
Point(1, 0)
>>> r.spin(pi/6)
>>> r.vertices[0]
Point(sqrt(3)/2, 1/2)

========

rotation
rotate : Creates a copy of the RegularPolygon rotated about a Point

"""
self._rot += angle

[docs]    def rotate(self, angle, pt=None):
"""Override GeometryEntity.rotate to first rotate the RegularPolygon

>>> from sympy import Point, RegularPolygon, Polygon, pi
>>> t = RegularPolygon(Point(1, 0), 1, 3)
>>> t.vertices[0] # vertex on x-axis
Point(2, 0)
>>> t.rotate(pi/2).vertices[0] # vertex on y axis now
Point(0, 2)

========

rotation
spin : Rotates a RegularPolygon in place

"""

r = type(self)(*self.args) # need a copy or else changes are in-place
r._rot += angle
return GeometryEntity.rotate(r, angle, pt)

[docs]    def scale(self, x=1, y=1, pt=None):
"""Override GeometryEntity.scale since it is the radius that must be
scaled (if x == y) or else a new Polygon must be returned.

>>> from sympy import RegularPolygon

Symmetric scaling returns a RegularPolygon:

>>> RegularPolygon((0, 0), 1, 4).scale(2, 2)
RegularPolygon(Point(0, 0), 2, 4, 0)

Asymmetric scaling returns a kite as a Polygon:

>>> RegularPolygon((0, 0), 1, 4).scale(2, 1)
Polygon(Point(2, 0), Point(0, 1), Point(-2, 0), Point(0, -1))

"""
if pt:
pt = Point(pt)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
if x != y:
return Polygon(*self.vertices).scale(x, y)
c, r, n, rot = self.args
r *= x
return self.func(c, r, n, rot)

@property
[docs]    def vertices(self):
"""The vertices of the RegularPolygon.

Returns
=======

vertices : list
Each vertex is a Point.

========

sympy.geometry.point.Point

Examples
========

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.vertices
[Point(5, 0), Point(0, 5), Point(-5, 0), Point(0, -5)]

"""
c = self._center
rot = self._rot
v = 2*S.Pi/self._n

return [Point(c.x + r*cos(k*v + rot), c.y + r*sin(k*v + rot))
for k in range(self._n)]

def __eq__(self, o):
if not isinstance(o, Polygon):
return False
elif not isinstance(o, RegularPolygon):
return Polygon.__eq__(o, self)
return self.args == o.args

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

[docs]class Triangle(Polygon):
"""
A polygon with three vertices and three sides.

Parameters
==========

points : sequence of Points
keyword: asa, sas, or sss to specify sides/angles of the triangle

Attributes
==========

vertices
altitudes
orthocenter
circumcenter
circumcircle
incircle
medians
medial

Raises
======

GeometryError
If the number of vertices is not equal to three, or one of the vertices
is not a Point, or a valid keyword is not given.

========

sympy.geometry.point.Point, Polygon

Examples
========

>>> from sympy.geometry import Triangle, Point
>>> Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
Triangle(Point(0, 0), Point(4, 0), Point(4, 3))

Keywords sss, sas, or asa can be used to give the desired
side lengths (in order) and interior angles (in degrees) that
define the triangle:

>>> Triangle(sss=(3, 4, 5))
Triangle(Point(0, 0), Point(3, 0), Point(3, 4))
>>> Triangle(asa=(30, 1, 30))
Triangle(Point(0, 0), Point(1, 0), Point(1/2, sqrt(3)/6))
>>> Triangle(sas=(1, 45, 2))
Triangle(Point(0, 0), Point(2, 0), Point(sqrt(2)/2, sqrt(2)/2))

"""

def __new__(cls, *args, **kwargs):
if len(args) != 3:
if 'sss' in kwargs:
return _sss(*[nsimplify(a) for a in kwargs['sss']])
if 'asa' in kwargs:
return _asa(*[nsimplify(a) for a in kwargs['asa']])
if 'sas' in kwargs:
return _sas(*[nsimplify(a) for a in kwargs['sas']])
msg = "Triangle instantiates with three points or a valid keyword."
raise GeometryError(msg)

vertices = [Point(a) for a in args]

# remove consecutive duplicates
nodup = []
for p in vertices:
if nodup and p == nodup[-1]:
continue
nodup.append(p)
if len(nodup) > 1 and nodup[-1] == nodup[0]:
nodup.pop() # last point was same as first

# remove collinear points
i = -3
while i < len(nodup) - 3 and len(nodup) > 2:
a, b, c = sorted([nodup[i], nodup[i + 1], nodup[i + 2]])
if Point.is_collinear(a, b, c):
nodup[i] = a
nodup[i + 1] = None
nodup.pop(i + 1)
i += 1

vertices = [x for x in nodup if x is not None]

if len(vertices) == 3:
return GeometryEntity.__new__(cls, *vertices, **kwargs)
elif len(vertices) == 2:
return Segment(*vertices, **kwargs)
else:
return Point(*vertices, **kwargs)

@property
[docs]    def vertices(self):
"""The triangle's vertices

Returns
=======

vertices : tuple
Each element in the tuple is a Point

========

sympy.geometry.point.Point

Examples
========

>>> from sympy.geometry import Triangle, Point
>>> t = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t.vertices
(Point(0, 0), Point(4, 0), Point(4, 3))

"""
return self.args

[docs]    def is_similar(t1, t2):
"""Is another triangle similar to this one.

Two triangles are similar if one can be uniformly scaled to the other.

Parameters
==========

other: Triangle

Returns
=======

is_similar : boolean

========

sympy.geometry.entity.GeometryEntity.is_similar

Examples
========

>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -3))
>>> t1.is_similar(t2)
True

>>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -4))
>>> t1.is_similar(t2)
False

"""
if not isinstance(t2, Polygon):
return False

s1_1, s1_2, s1_3 = [side.length for side in t1.sides]
s2 = [side.length for side in t2.sides]
def _are_similar(u1, u2, u3, v1, v2, v3):
e1 = simplify(u1/v1)
e2 = simplify(u2/v2)
e3 = simplify(u3/v3)
return bool(e1 == e2) and bool(e2 == e3)

# There's only 6 permutations, so write them out
return _are_similar(s1_1, s1_2, s1_3, *s2) or \
_are_similar(s1_1, s1_3, s1_2, *s2) or \
_are_similar(s1_2, s1_1, s1_3, *s2) or \
_are_similar(s1_2, s1_3, s1_1, *s2) or \
_are_similar(s1_3, s1_1, s1_2, *s2) or \
_are_similar(s1_3, s1_2, s1_1, *s2)

[docs]    def is_equilateral(self):
"""Are all the sides the same length?

Returns
=======

is_equilateral : boolean

========

sympy.geometry.entity.GeometryEntity.is_similar, RegularPolygon
is_isosceles, is_right, is_scalene

Examples
========

>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t1.is_equilateral()
False

>>> from sympy import sqrt
>>> t2 = Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3)))
>>> t2.is_equilateral()
True

"""
return not has_variety(s.length for s in self.sides)

[docs]    def is_isosceles(self):
"""Are two or more of the sides the same length?

Returns
=======

is_isosceles : boolean

========

is_equilateral, is_right, is_scalene

Examples
========

>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(2, 4))
>>> t1.is_isosceles()
True

"""
return has_dups(s.length for s in self.sides)

[docs]    def is_scalene(self):
"""Are all the sides of the triangle of different lengths?

Returns
=======

is_scalene : boolean

========

is_equilateral, is_isosceles, is_right

Examples
========

>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(1, 4))
>>> t1.is_scalene()
True

"""
return not has_dups(s.length for s in self.sides)

[docs]    def is_right(self):
"""Is the triangle right-angled.

Returns
=======

is_right : boolean

========

sympy.geometry.line.LinearEntity.is_perpendicular
is_equilateral, is_isosceles, is_scalene

Examples
========

>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t1.is_right()
True

"""
s = self.sides
return Segment.is_perpendicular(s[0], s[1]) or \
Segment.is_perpendicular(s[1], s[2]) or \
Segment.is_perpendicular(s[0], s[2])

@property
[docs]    def altitudes(self):
"""The altitudes of the triangle.

An altitude of a triangle is a segment through a vertex,
perpendicular to the opposite side, with length being the
height of the vertex measured from the line containing the side.

Returns
=======

altitudes : dict
The dictionary consists of keys which are vertices and values
which are Segments.

========

sympy.geometry.point.Point, sympy.geometry.line.Segment.length

Examples
========

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.altitudes[p1]
Segment(Point(0, 0), Point(1/2, 1/2))

"""
s = self.sides
v = self.vertices
return {v[0]: s[1].perpendicular_segment(v[0]),
v[1]: s[2].perpendicular_segment(v[1]),
v[2]: s[0].perpendicular_segment(v[2])}

@property
[docs]    def orthocenter(self):
"""The orthocenter of the triangle.

The orthocenter is the intersection of the altitudes of a triangle.
It may lie inside, outside or on the triangle.

Returns
=======

orthocenter : Point

========

sympy.geometry.point.Point

Examples
========

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.orthocenter
Point(0, 0)

"""
a = self.altitudes
v = self.vertices
return Line(a[v[0]]).intersection(Line(a[v[1]]))[0]

@property
[docs]    def circumcenter(self):
"""The circumcenter of the triangle

The circumcenter is the center of the circumcircle.

Returns
=======

circumcenter : Point

========

sympy.geometry.point.Point

Examples
========

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.circumcenter
Point(1/2, 1/2)
"""
a,b,c = [x.perpendicular_bisector() for x in self.sides]
return a.intersection(b)[0]

@property
"""The radius of the circumcircle of the triangle.

Returns
=======

circumradius : number of Basic instance

========

Examples
========

>>> from sympy import Symbol
>>> from sympy.geometry import Point, Triangle
>>> a = Symbol('a')
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, a)
>>> t = Triangle(p1, p2, p3)
sqrt(a**2/4 + 1/4)
"""
return Point.distance(self.circumcenter, self.vertices[0])

@property
[docs]    def circumcircle(self):
"""The circle which passes through the three vertices of the triangle.

Returns
=======

circumcircle : Circle

========

sympy.geometry.ellipse.Circle

Examples
========

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.circumcircle
Circle(Point(1/2, 1/2), sqrt(2)/2)

"""

[docs]    def bisectors(self):
"""The angle bisectors of the triangle.

An angle bisector of a triangle is a straight line through a vertex
which cuts the corresponding angle in half.

Returns
=======

bisectors : dict
Each key is a vertex (Point) and each value is the corresponding
bisector (Segment).

========

sympy.geometry.point.Point, sympy.geometry.line.Segment

Examples
========

>>> from sympy.geometry import Point, Triangle, Segment
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> from sympy import sqrt
>>> t.bisectors()[p2] == Segment(Point(0, sqrt(2) - 1), Point(1, 0))
True

"""
s = self.sides
v = self.vertices
c = self.incenter
l1 = Segment(v[0], Line(v[0], c).intersection(s[1])[0])
l2 = Segment(v[1], Line(v[1], c).intersection(s[2])[0])
l3 = Segment(v[2], Line(v[2], c).intersection(s[0])[0])
return {v[0]: l1, v[1]: l2, v[2]: l3}

@property
[docs]    def incenter(self):
"""The center of the incircle.

The incircle is the circle which lies inside the triangle and touches
all three sides.

Returns
=======

incenter : Point

========

incircle, sympy.geometry.point.Point

Examples
========

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.incenter
Point(-sqrt(2)/2 + 1, -sqrt(2)/2 + 1)

"""
s = self.sides
l = Matrix([s[i].length for i in [1, 2, 0]])
p = sum(l)
v = self.vertices
x = simplify(l.dot(Matrix([vi.x for vi in v]))/p)
y = simplify(l.dot(Matrix([vi.y for vi in v]))/p)
return Point(x, y)

@property

Returns
=======

inradius : number of Basic instance

========

Examples
========

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(4, 0), Point(0, 3)
>>> t = Triangle(p1, p2, p3)
1

"""
return simplify(2 * self.area / self.perimeter)

@property
[docs]    def incircle(self):
"""The incircle of the triangle.

The incircle is the circle which lies inside the triangle and touches
all three sides.

Returns
=======

incircle : Circle

========

sympy.geometry.ellipse.Circle

Examples
========

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(2, 0), Point(0, 2)
>>> t = Triangle(p1, p2, p3)
>>> t.incircle
Circle(Point(-sqrt(2) + 2, -sqrt(2) + 2), -sqrt(2) + 2)

"""

@property
[docs]    def medians(self):
"""The medians of the triangle.

A median of a triangle is a straight line through a vertex and the
midpoint of the opposite side, and divides the triangle into two
equal areas.

Returns
=======

medians : dict
Each key is a vertex (Point) and each value is the median (Segment)
at that point.

========

sympy.geometry.point.Point.midpoint, sympy.geometry.line.Segment.midpoint

Examples
========

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.medians[p1]
Segment(Point(0, 0), Point(1/2, 1/2))

"""
s = self.sides
v = self.vertices
return {v[0]: Segment(s[1].midpoint, v[0]),
v[1]: Segment(s[2].midpoint, v[1]),
v[2]: Segment(s[0].midpoint, v[2])}

@property
[docs]    def medial(self):
"""The medial triangle of the triangle.

The triangle which is formed from the midpoints of the three sides.

Returns
=======

medial : Triangle

========

sympy.geometry.line.Segment.midpoint

Examples
========

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.medial
Triangle(Point(1/2, 0), Point(1/2, 1/2), Point(0, 1/2))

"""
s = self.sides
return Triangle(s[0].midpoint, s[1].midpoint, s[2].midpoint)

#@property
#def excircles(self):
#    """Returns a list of the three excircles for this triangle."""
#    pass

"""Return the radian value for the given degrees (pi = 180 degrees)."""
return d*pi/180

def deg(r):
"""Return the degree value for the given radians (pi = 180 degrees)."""
return r/pi*180

def _slope(d):
return rv

def _asa(d1, l, d2):
"""Return triangle having side with length l on the x-axis."""
xy = Line((0, 0), slope=_slope(d1)).intersection(
Line((l, 0), slope=_slope(180 - d2)))[0]
return Triangle((0,0),(l,0),xy)

def _sss(l1, l2, l3):
"""Return triangle having side of length l1 on the x-axis."""
c1 = Circle((0,0), l3)
c2 = Circle((l1, 0), l2)
inter = [a for a in c1.intersection(c2) if a.y.is_nonnegative]
if not inter:
return None
pt = inter[0]
return Triangle((0,0), (l1,0), pt)

def _sas(l1, d, l2):
"""Return triangle having side with length l2 on the x-axis."""
p1 = Point(0, 0)
p2 = Point(l2, 0)