"""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)
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 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
See Also
========
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)
def _symbol(s, matching_symbol=None):
"""Return s if s is a Symbol, else return either a new Symbol (real=True)
with the same name s or the matching_symbol if s is a string and it matches
the name of the matching_symbol.
>>> from sympy import Symbol
>>> from sympy.geometry.util import _symbol
>>> x = Symbol('x')
>>> _symbol('y')
y
>>> _.is_real
True
>>> _symbol(x)
x
>>> _.is_real is None
True
>>> arb = Symbol('foo')
>>> _symbol('arb', arb) # arb's name is foo so foo will not be returned
arb
>>> _symbol('foo', arb) # now it will
foo
NB: the symbol here may not be the same as a symbol with the same
name defined elsewhere as a result of different assumptions.
See Also
========
sympy.core.symbol.Symbol
"""
if isinstance(s, string_types):
if matching_symbol and matching_symbol.name == s:
return matching_symbol
return Symbol(s, real=True)
elif isinstance(s, Symbol):
return s
else:
raise ValueError('symbol must be string for symbol name or Symbol')
def _uniquely_named_symbol(xname, *exprs):
"""Return a symbol which, when printed, will have a name unique
from any other already in the expressions given. The name is made
unique by prepending underscores.
"""
prefix = '%s'
x = prefix % xname
syms = set().union(*[e.free_symbols for e in exprs])
while any(x == str(s) for s in syms):
prefix = '_' + prefix
x = prefix % xname
return _symbol(x)
[docs]def intersection(*entities):
"""The intersection of a collection of GeometryEntity instances.
Parameters
==========
entities : sequence of GeometryEntity
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.
See Also
========
sympy.geometry.entity.GeometryEntity.intersection
Examples
========
>>> from sympy.geometry import Point, Line, Circle, intersection
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(-1, 5)
>>> l1, l2 = Line(p1, p2), Line(p3, p2)
>>> c = Circle(p2, 1)
>>> intersection(l1, p2)
[Point2D(1, 1)]
>>> intersection(l1, l2)
[Point2D(1, 1)]
>>> intersection(c, p2)
[]
>>> intersection(c, Point(1, 0))
[Point2D(1, 0)]
>>> intersection(c, l2)
[Point2D(-sqrt(5)/5 + 1, 2*sqrt(5)/5 + 1),
Point2D(sqrt(5)/5 + 1, -2*sqrt(5)/5 + 1)]
"""
from .entity import GeometryEntity
from .point import Point
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):
try:
entities[i] = Point(e)
except NotImplementedError:
raise ValueError('%s is not a GeometryEntity and cannot be made into Point' % str(e))
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
[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
See Also
========
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):
p.add(e)
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 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(*_)
set([(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}
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(*_)
set([(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 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.line3d 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.
See Also
========
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.
See Also
========
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])
