# Source code for sympy.geometry.util

"""Utility functions for geometrical entities.

Contains
========
intersection
convex_hull
closest_points
farthest_points
are_coplanar
are_similar

"""
from __future__ import division, print_function

from sympy import Function, Symbol, solve
from sympy.core.compatibility import (
is_sequence, range, string_types, ordered)
from .point import Point, Point2D

def _ordered_points(p):
"""Return the tuple of points sorted numerically according to args"""
return tuple(sorted(p, key=lambda x: x.args))

def are_coplanar(*e):
""" Returns True if the given entities are coplanar otherwise False

Parameters
==========

e: entities to be checked for being coplanar

Returns
=======

Boolean

Examples
========

>>> from sympy import Point3D, Line3D
>>> from sympy.geometry.util import are_coplanar
>>> a = Line3D(Point3D(5, 0, 0), Point3D(1, -1, 1))
>>> b = Line3D(Point3D(0, -2, 0), Point3D(3, 1, 1))
>>> c = Line3D(Point3D(0, -1, 0), Point3D(5, -1, 9))
>>> are_coplanar(a, b, c)
False

"""
from sympy.geometry.line import LinearEntity3D
from sympy.geometry.point import Point3D
from sympy.geometry.plane import Plane
# XXX update tests for coverage

e = set(e)
# first work with a Plane if present
for i in list(e):
if isinstance(i, Plane):
e.remove(i)
return all(p.is_coplanar(i) for p in e)

if all(isinstance(i, Point3D) for i in e):
if len(e) < 3:
return False

# remove pts that are collinear with 2 pts
a, b = e.pop(), e.pop()
for i in list(e):
if Point3D.are_collinear(a, b, i):
e.remove(i)

if not e:
return False
else:
# define a plane
p = Plane(a, b, e.pop())
for i in e:
if i not in p:
return False
return True
else:
pt3d = []
for i in e:
if isinstance(i, Point3D):
pt3d.append(i)
elif isinstance(i, LinearEntity3D):
pt3d.extend(i.args)
elif isinstance(i, GeometryEntity):  # XXX we should have a GeometryEntity3D class so we can tell the difference between 2D and 3D -- here we just want to deal with 2D objects; if new 3D objects are encountered that we didn't hanlde above, an error should be raised
# all 2D objects have some Point that defines them; so convert those points to 3D pts by making z=0
for p in i.args:
if isinstance(p, Point):
pt3d.append(Point3D(*(p.args + (0,))))
return are_coplanar(*pt3d)

[docs]def are_similar(e1, e2):
"""Are two geometrical entities similar.

Can one geometrical entity be uniformly scaled to the other?

Parameters
==========

e1 : GeometryEntity
e2 : GeometryEntity

Returns
=======

are_similar : boolean

Raises
======

GeometryError
When e1 and e2 cannot be compared.

Notes
=====

If the two objects are equal then they are similar.

========

sympy.geometry.entity.GeometryEntity.is_similar

Examples
========

>>> from sympy import Point, Circle, Triangle, are_similar
>>> c1, c2 = Circle(Point(0, 0), 4), Circle(Point(1, 4), 3)
>>> t1 = Triangle(Point(0, 0), Point(1, 0), Point(0, 1))
>>> t2 = Triangle(Point(0, 0), Point(2, 0), Point(0, 2))
>>> t3 = Triangle(Point(0, 0), Point(3, 0), Point(0, 1))
>>> are_similar(t1, t2)
True
>>> are_similar(t1, t3)
False

"""
from .exceptions import GeometryError

if e1 == e2:
return True
try:
return e1.is_similar(e2)
except AttributeError:
try:
return e2.is_similar(e1)
except AttributeError:
n1 = e1.__class__.__name__
n2 = e2.__class__.__name__
raise GeometryError(
"Cannot test similarity between %s and %s" % (n1, n2))

[docs]def centroid(*args):
"""Find the centroid (center of mass) of the collection containing only Points,
Segments or Polygons. The centroid is the weighted average of the individual centroid
where the weights are the lengths (of segments) or areas (of polygons).
Overlapping regions will add to the weight of that region.

If there are no objects (or a mixture of objects) then None is returned.

========

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

Examples
========

>>> from sympy import Point, Segment, Polygon
>>> from sympy.geometry.util import centroid
>>> p = Polygon((0, 0), (10, 0), (10, 10))
>>> q = p.translate(0, 20)
>>> p.centroid, q.centroid
(Point2D(20/3, 10/3), Point2D(20/3, 70/3))
>>> centroid(p, q)
Point2D(20/3, 40/3)
>>> p, q = Segment((0, 0), (2, 0)), Segment((0, 0), (2, 2))
>>> centroid(p, q)
Point2D(1, -sqrt(2) + 2)
>>> centroid(Point(0, 0), Point(2, 0))
Point2D(1, 0)

Stacking 3 polygons on top of each other effectively triples the
weight of that polygon:

>>> p = Polygon((0, 0), (1, 0), (1, 1), (0, 1))
>>> q = Polygon((1, 0), (3, 0), (3, 1), (1, 1))
>>> centroid(p, q)
Point2D(3/2, 1/2)
>>> centroid(p, p, p, q) # centroid x-coord shifts left
Point2D(11/10, 1/2)

Stacking the squares vertically above and below p has the same
effect:

>>> centroid(p, p.translate(0, 1), p.translate(0, -1), q)
Point2D(11/10, 1/2)

"""

from sympy.geometry import Polygon, Segment, Point
if args:
if all(isinstance(g, Point) for g in args):
c = Point(0, 0)
for g in args:
c += g
den = len(args)
elif all(isinstance(g, Segment) for g in args):
c = Point(0, 0)
L = 0
for g in args:
l = g.length
c += g.midpoint*l
L += l
den = L
elif all(isinstance(g, Polygon) for g in args):
c = Point(0, 0)
A = 0
for g in args:
a = g.area
c += g.centroid*a
A += a
den = A
c /= den
return c.func(*[i.simplify() for i in c.args])

def closest_points(*args):
"""Return the subset of points from a set of points that were
the closest to each other in the 2D plane.

Parameters
==========

args : a collection of Points on 2D plane.

Notes
=====

This can only be performed on a set of points whose coordinates can
be ordered on the number line. If there are no ties then a single
pair of Points will be in the set.

References
==========

[1] http://www.cs.mcgill.ca/~cs251/ClosestPair/ClosestPairPS.html

[2] Sweep line algorithm
https://en.wikipedia.org/wiki/Sweep_line_algorithm

Examples
========

>>> from sympy.geometry import closest_points, Point2D, Triangle
>>> Triangle(sss=(3, 4, 5)).args
(Point2D(0, 0), Point2D(3, 0), Point2D(3, 4))
>>> closest_points(*_)
{(Point2D(0, 0), Point2D(3, 0))}

"""
from collections import deque
from math import hypot, sqrt as _sqrt
from sympy.functions.elementary.miscellaneous import sqrt

p = [Point2D(i) for i in set(args)]
if len(p) < 2:
raise ValueError('At least 2 distinct points must be given.')

try:
p.sort(key=lambda x: x.args)
except TypeError:
raise ValueError("The points could not be sorted.")

if any(not i.is_Rational for j in p for i in j.args):
def hypot(x, y):
arg = x*x + y*y
if arg.is_Rational:
return _sqrt(arg)
return sqrt(arg)

rv = [(0, 1)]
best_dist = hypot(p[1].x - p[0].x, p[1].y - p[0].y)
i = 2
left = 0
box = deque([0, 1])
while i < len(p):
while left < i and p[i][0] - p[left][0] > best_dist:
box.popleft()
left += 1

for j in box:
d = hypot(p[i].x - p[j].x, p[i].y - p[j].y)
if d < best_dist:
rv = [(j, i)]
elif d == best_dist:
rv.append((j, i))
else:
continue
best_dist = d
box.append(i)
i += 1

return {tuple([p[i] for i in pair]) for pair in rv}

[docs]def convex_hull(*args, **kwargs):
"""The convex hull surrounding the Points contained in the list of entities.

Parameters
==========

args : a collection of Points, Segments and/or Polygons

Returns
=======

convex_hull : Polygon if polygon is True else as a tuple (U, L) where L and U are the lower and upper hulls, respectively.

Notes
=====

This can only be performed on a set of points whose coordinates can
be ordered on the number line.

References
==========

[1] http://en.wikipedia.org/wiki/Graham_scan

[2] Andrew's Monotone Chain Algorithm
(A.M. Andrew,
"Another Efficient Algorithm for Convex Hulls in Two Dimensions", 1979)
http://geomalgorithms.com/a10-_hull-1.html

========

sympy.geometry.point.Point, sympy.geometry.polygon.Polygon

Examples
========

>>> from sympy.geometry import Point, convex_hull
>>> points = [(1, 1), (1, 2), (3, 1), (-5, 2), (15, 4)]
>>> convex_hull(*points)
Polygon(Point2D(-5, 2), Point2D(1, 1), Point2D(3, 1), Point2D(15, 4))
>>> convex_hull(*points, **dict(polygon=False))
([Point2D(-5, 2), Point2D(15, 4)],
[Point2D(-5, 2), Point2D(1, 1), Point2D(3, 1), Point2D(15, 4)])

"""
from .entity import GeometryEntity
from .point import Point
from .line import Segment
from .polygon import Polygon

polygon = kwargs.get('polygon', True)
p = set()
for e in args:
if not isinstance(e, GeometryEntity):
try:
e = Point(e)
except NotImplementedError:
raise ValueError('%s is not a GeometryEntity and cannot be made into Point' % str(e))
if isinstance(e, Point):
elif isinstance(e, Segment):
p.update(e.points)
elif isinstance(e, Polygon):
p.update(e.vertices)
else:
raise NotImplementedError(
'Convex hull for %s not implemented.' % type(e))

# make sure all our points are of the same dimension
if any(len(x) != 2 for x in p):
raise ValueError('Can only compute the convex hull in two dimensions')

p = list(p)
if len(p) == 1:
return p[0] if polygon else (p[0], None)
elif len(p) == 2:
s = Segment(p[0], p[1])
return s if polygon else (s, None)

def _orientation(p, q, r):
'''Return positive if p-q-r are clockwise, neg if ccw, zero if
collinear.'''
return (q.y - p.y)*(r.x - p.x) - (q.x - p.x)*(r.y - p.y)

# scan to find upper and lower convex hulls of a set of 2d points.
U = []
L = []
try:
p.sort(key=lambda x: x.args)
except TypeError:
raise ValueError("The points could not be sorted.")
for p_i in p:
while len(U) > 1 and _orientation(U[-2], U[-1], p_i) <= 0:
U.pop()
while len(L) > 1 and _orientation(L[-2], L[-1], p_i) >= 0:
L.pop()
U.append(p_i)
L.append(p_i)
U.reverse()
convexHull = tuple(L + U[1:-1])

if len(convexHull) == 2:
s = Segment(convexHull[0], convexHull[1])
return s if polygon else (s, None)
if polygon:
return Polygon(*convexHull)
else:
U.reverse()
return (U, L)

def farthest_points(*args):
"""Return the subset of points from a set of points that were
the furthest apart from each other in the 2D plane.

Parameters
==========

args : a collection of Points on 2D plane.

Notes
=====

This can only be performed on a set of points whose coordinates can
be ordered on the number line. If there are no ties then a single
pair of Points will be in the set.

References
==========

[1] http://code.activestate.com/recipes/117225-convex-hull-and-diameter-of-2d-point-sets/

[2] Rotating Callipers Technique
https://en.wikipedia.org/wiki/Rotating_calipers

Examples
========

>>> from sympy.geometry import farthest_points, Point2D, Triangle
>>> Triangle(sss=(3, 4, 5)).args
(Point2D(0, 0), Point2D(3, 0), Point2D(3, 4))
>>> farthest_points(*_)
{(Point2D(0, 0), Point2D(3, 4))}

"""
from math import hypot, sqrt as _sqrt

def rotatingCalipers(Points):
U, L = convex_hull(*Points, **dict(polygon=False))

if L is None:
if isinstance(U, Point):
raise ValueError('At least two distinct points must be given.')
yield U.args
else:
i = 0
j = len(L) - 1
while i < len(U) - 1 or j > 0:
yield U[i], L[j]
# if all the way through one side of hull, advance the other side
if i == len(U) - 1:
j -= 1
elif j == 0:
i += 1
# still points left on both lists, compare slopes of next hull edges
# being careful to avoid divide-by-zero in slope calculation
elif (U[i+1].y - U[i].y) * (L[j].x - L[j-1].x) > \
(L[j].y - L[j-1].y) * (U[i+1].x - U[i].x):
i += 1
else:
j -= 1

p = [Point2D(i) for i in set(args)]

if any(not i.is_Rational for j in p for i in j.args):
def hypot(x, y):
arg = x*x + y*y
if arg.is_Rational:
return _sqrt(arg)
return sqrt(arg)

rv = []
diam = 0
for pair in rotatingCalipers(args):
h, q = _ordered_points(pair)
d = hypot(h.x - q.x, h.y - q.y)
if d > diam:
rv = [(h, q)]
elif d == diam:
rv.append((h, q))
else:
continue
diam = d

return set(rv)

def idiff(eq, y, x, n=1):
"""Return dy/dx assuming that eq == 0.

Parameters
==========

y : the dependent variable or a list of dependent variables (with y first)
x : the variable that the derivative is being taken with respect to
n : the order of the derivative (default is 1)

Examples
========

>>> from sympy.abc import x, y, a
>>> from sympy.geometry.util import idiff

>>> circ = x**2 + y**2 - 4
>>> idiff(circ, y, x)
-x/y
>>> idiff(circ, y, x, 2).simplify()
-(x**2 + y**2)/y**3

Here, a is assumed to be independent of x:

>>> idiff(x + a + y, y, x)
-1

Now the x-dependence of a is made explicit by listing a after
y in a list.

>>> idiff(x + a + y, [y, a], x)
-Derivative(a, x) - 1

========

sympy.core.function.Derivative: represents unevaluated derivatives
sympy.core.function.diff: explicitly differentiates wrt symbols

"""
if is_sequence(y):
dep = set(y)
y = y[0]
elif isinstance(y, Symbol):
dep = {y}
else:
raise ValueError("expecting x-dependent symbol(s) but got: %s" % y)

f = dict([(s, Function(
s.name)(x)) for s in eq.free_symbols if s != x and s in dep])
dydx = Function(y.name)(x).diff(x)
eq = eq.subs(f)
derivs = {}
for i in range(n):
yp = solve(eq.diff(x), dydx)[0].subs(derivs)
if i == n - 1:
return yp.subs([(v, k) for k, v in f.items()])
derivs[dydx] = yp
eq = dydx - yp
dydx = dydx.diff(x)

[docs]def intersection(*entities, **kwargs):
"""The intersection of a collection of GeometryEntity instances.

Parameters
==========
entities : sequence of GeometryEntity
pairwise (keyword argument) : Can be either True or False

Returns
=======
intersection : list of GeometryEntity

Raises
======
NotImplementedError
When unable to calculate intersection.

Notes
=====
The intersection of any geometrical entity with itself should return
a list with one item: the entity in question.
An intersection requires two or more entities. If only a single
entity is given then the function will return an empty list.
It is possible for intersection to miss intersections that one
knows exists because the required quantities were not fully
simplified internally.
Reals should be converted to Rationals, e.g. Rational(str(real_num))
or else failures due to floating point issues may result.

Case 1: When the keyword argument 'pairwise' is False (default value):
In this case, the functon returns a list of intersections common to
all entities.

Case 2: When the keyword argument 'pairwise' is True:
In this case, the functions returns a list intersections that occur
between any pair of entities.

========

sympy.geometry.entity.GeometryEntity.intersection

Examples
========

>>> from sympy.geometry import Ray, Circle, intersection
>>> c = Circle((0, 1), 1)
>>> intersection(c, c.center)
[]
>>> right = Ray((0, 0), (1, 0))
>>> up = Ray((0, 0), (0, 1))
>>> intersection(c, right, up)
[Point2D(0, 0)]
>>> intersection(c, right, up, pairwise=True)
[Point2D(0, 0), Point2D(0, 2)]
>>> left = Ray((1, 0), (0, 0))
>>> intersection(right, left)
[Segment2D(Point2D(0, 0), Point2D(1, 0))]

"""

from .entity import GeometryEntity
from .point import Point

pairwise = kwargs.pop('pairwise', False)

if len(entities) <= 1:
return []

# entities may be an immutable tuple
entities = list(entities)
for i, e in enumerate(entities):
if not isinstance(e, GeometryEntity):
entities[i] = Point(e)

if not pairwise:
# find the intersection common to all objects
res = entities[0].intersection(entities[1])
for entity in entities[2:]:
newres = []
for x in res:
newres.extend(x.intersection(entity))
res = newres
return res

# find all pairwise intersections
ans = []
for j in range(0, len(entities)):
for k in range(j + 1, len(entities)):
ans.extend(intersection(entities[j], entities[k]))
return list(ordered(set(ans)))