def reflect(self, line):
from sympy import atan, Line, Point, Dummy, oo
g = self
l = line
o = Point(0, 0)
if l.slope == 0:
y = l.args[0].y
if not y: # x-axis
return g.scale(y=-1)
reps = [(p, p.translate(y=2*(y - p.y))) for p in g.atoms(Point)]
elif l.slope == oo:
x = l.args[0].x
if not x: # y-axis
return g.scale(x=-1)
reps = [(p, p.translate(x=2*(x - p.x))) for p in g.atoms(Point)]
else:
if not hasattr(g, 'reflect') and not all(
isinstance(arg, Point) for arg in g.args):
raise NotImplementedError(
'reflect undefined or non-Point args in %s' % g)
a = atan(l.slope)
c = l.coefficients
d = -c[-1]/c[1] # y-intercept
# apply the transform to a single point
x, y = Dummy(), Dummy()
xf = Point(x, y)
xf = xf.translate(y=-d).rotate(-a, o).scale(y=-1
).rotate(a, o).translate(y=d)
# replace every point using that transform
reps = [(p, xf.xreplace({x: p.x, y: p.y})) for p in g.atoms(Point)]
return g.xreplace(dict(reps))
def reflect(self, line):
"""
Reflects an object across a line.
Parameters
==========
line: Line
Examples
========
>>> from sympy import pi, sqrt, Line, RegularPolygon
>>> l = Line((0, pi), slope=sqrt(2))
>>> pent = RegularPolygon((1, 2), 1, 5)
>>> rpent = pent.reflect(l)
>>> rpent
RegularPolygon(Point2D(-2*sqrt(2)*pi/3 - 1/3 + 4*sqrt(2)/3, 2/3 + 2*sqrt(2)/3 + 2*pi/3), -1, 5, -atan(2*sqrt(2)) + 3*pi/5)
>>> from sympy import pi, Line, Circle, Point
>>> l = Line((0, pi), slope=1)
>>> circ = Circle(Point(0, 0), 5)
>>> rcirc = circ.reflect(l)
>>> rcirc
Circle(Point2D(-pi, pi), -5)
"""
from sympy import atan, Point, Dummy, oo
g = self
l = line
o = Point(0, 0)
if l.slope == 0:
y = l.args[0].y
if not y: # x-axis
return g.scale(y=-1)
reps = [(p, p.translate(y=2*(y - p.y))) for p in g.atoms(Point)]
elif l.slope == oo:
x = l.args[0].x
if not x: # y-axis
return g.scale(x=-1)
reps = [(p, p.translate(x=2*(x - p.x))) for p in g.atoms(Point)]
else:
if not hasattr(g, 'reflect') and not all(
isinstance(arg, Point) for arg in g.args):
raise NotImplementedError(
'reflect undefined or non-Point args in %s' % g)
a = atan(l.slope)
c = l.coefficients
d = -c[-1]/c[1] # y-intercept
# apply the transform to a single point
x, y = Dummy(), Dummy()
xf = Point(x, y)
xf = xf.translate(y=-d).rotate(-a, o).scale(y=-1
).rotate(a, o).translate(y=d)
# replace every point using that transform
reps = [(p, xf.xreplace({x: p.x, y: p.y})) for p in g.atoms(Point)]
return g.xreplace(dict(reps))
def reflect(self, line):
from sympy import atan, Line, Point, Dummy, oo
g = self
l = line
o = Point(0, 0)
if l == Line(o, slope=0):
return g.scale(y=-1)
elif l == Line(o, slope=oo):
return g.scale(-1)
if not hasattr(g, 'reflect') and not all(
isinstance(arg, Point) for arg in g.args):
raise NotImplementedError
a = atan(l.slope)
c = l.coefficients
d = -c[-1]/c[1] # y-intercept
# apply the transform to a single point
x, y = Dummy(), Dummy()
xf = Point(x, y)
xf = xf.translate(y=-d).rotate(-a, o).scale(y=-1
).rotate(a, o).translate(y=d)
# replace every point using that transform
reps = [(p, xf.xreplace({x: p.x, y: p.y})) for p in g.atoms(Point)]
return g.xreplace(dict(reps))
def intersection(self, o):
""" The intersection with other geometrical entity.
Parameters
==========
Point, Point3D, LinearEntity, LinearEntity3D, Plane
Returns
=======
List
Examples
========
>>> from sympy import Point, Point3D, Line, Line3D, Plane
>>> a = Plane(Point3D(1, 2, 3), normal_vector=(1, 1, 1))
>>> b = Point3D(1, 2, 3)
>>> a.intersection(b)
[Point3D(1, 2, 3)]
>>> c = Line3D(Point3D(1, 4, 7), Point3D(2, 2, 2))
>>> a.intersection(c)
[Point3D(2, 2, 2)]
>>> d = Plane(Point3D(6, 0, 0), normal_vector=(2, -5, 3))
>>> e = Plane(Point3D(2, 0, 0), normal_vector=(3, 4, -3))
>>> d.intersection(e)
[Line3D(Point3D(78/23, -24/23, 0), Point3D(147/23, 321/23, 23))]
"""
from sympy.geometry.line import LinearEntity, LinearEntity3D
if not isinstance(o, GeometryEntity):
o = Point(o, dim=3)
if isinstance(o, Point):
if o in self:
return [o]
else:
return []
if isinstance(o, (LinearEntity, LinearEntity3D)):
if o in self:
p1, p2 = o.p1, o.p2
if isinstance(o, Segment):
o = Segment3D(p1, p2)
elif isinstance(o, Ray):
o = Ray3D(p1, p2)
elif isinstance(o, Line):
o = Line3D(p1, p2)
else:
raise ValueError('unhandled linear entity: %s' % o.func)
return [o]
else:
x, y, z = map(Dummy, 'xyz')
t = Dummy() # unnamed else it may clash with a symbol in o
a = Point3D(o.arbitrary_point(t))
b = self.equation(x, y, z)
# TODO: Replace solve with solveset, when this line is tested
c = solve(b.subs(list(zip((x, y, z), a.args))), t)
if not c:
return []
else:
p = a.subs(t, c[0])
if p not in self:
return [] # e.g. a segment might not intersect a plane
return [p]
if isinstance(o, Plane):
if self.equals(o):
return [self]
if self.is_parallel(o):
return []
else:
x, y, z = map(Dummy, 'xyz')
a, b = Matrix([self.normal_vector]), Matrix([o.normal_vector])
c = list(a.cross(b))
d = self.equation(x, y, z)
e = o.equation(x, y, z)
result = list(linsolve([d, e], x, y, z))[0]
for i in (x, y, z): result = result.subs(i, 0)
return [Line3D(Point3D(result), direction_ratio=c)]
def test_best_origin():
expr1 = y**2 * x**5 + y**5 * x**7 + 7 * x + x**12 + y**7 * x
l1 = Segment2D(Point(0, 3), Point(1, 1))
l2 = Segment2D(Point(S(3) / 2, 0), Point(S(3) / 2, 3))
l3 = Segment2D(Point(0, S(3) / 2), Point(3, S(3) / 2))
l4 = Segment2D(Point(0, 2), Point(2, 0))
l5 = Segment2D(Point(0, 2), Point(1, 1))
l6 = Segment2D(Point(2, 0), Point(1, 1))
assert best_origin((2, 1), 3, l1, expr1) == (0, 3)
assert best_origin((2, 0), 3, l2, x**7) == (S(3) / 2, 0)
assert best_origin((0, 2), 3, l3, x**7) == (0, S(3) / 2)
assert best_origin((1, 1), 2, l4, x**7 * y**3) == (0, 2)
assert best_origin((1, 1), 2, l4, x**3 * y**7) == (2, 0)
assert best_origin((1, 1), 2, l5, x**2 * y**9) == (0, 2)
assert best_origin((1, 1), 2, l6, x**9 * y**2) == (2, 0)
def test_polytopes_intersecting_sides():
fig5 = Polygon(Point(-4.165, -0.832), Point(-3.668, 1.568),
Point(-3.266, 1.279), Point(-1.090, -2.080),
Point(3.313, -0.683), Point(3.033, -4.845),
Point(-4.395, 4.840), Point(-1.007, -3.328))
assert polytope_integrate(fig5, x**2 + x*y + y**2) ==\
S(1633405224899363)/(24*10**12)
fig6 = Polygon(Point(-3.018, -4.473), Point(-0.103, 2.378),
Point(-1.605, -2.308), Point(4.516, -0.771),
Point(4.203, 0.478))
assert polytope_integrate(fig6, x**2 + x*y + y**2) ==\
S(88161333955921)/(3*10**12)
def intersection(self, o):
""" The intersection with other geometrical entity.
Parameters
==========
Point, Point3D, LinearEntity, LinearEntity3D, Plane
Returns
=======
List
Examples
========
>>> from sympy import Point, Point3D, Line, Line3D, Plane
>>> a = Plane(Point3D(1, 2, 3), normal_vector=(1, 1, 1))
>>> b = Point3D(1, 2, 3)
>>> a.intersection(b)
[Point3D(1, 2, 3)]
>>> c = Line3D(Point3D(1, 4, 7), Point3D(2, 2, 2))
>>> a.intersection(c)
[Point3D(2, 2, 2)]
>>> d = Plane(Point3D(6, 0, 0), normal_vector=(2, -5, 3))
>>> e = Plane(Point3D(2, 0, 0), normal_vector=(3, 4, -3))
>>> d.intersection(e)
[Line3D(Point3D(78/23, -24/23, 0), Point3D(147/23, 321/23, 23))]
"""
from sympy.geometry.line import LinearEntity, LinearEntity3D
if not isinstance(o, GeometryEntity):
o = Point(o, dim=3)
if isinstance(o, Point):
if o in self:
return [o]
else:
return []
if isinstance(o, (LinearEntity, LinearEntity3D)):
if o in self:
p1, p2 = o.p1, o.p2
if isinstance(o, Segment):
o = Segment3D(p1, p2)
elif isinstance(o, Ray):
o = Ray3D(p1, p2)
elif isinstance(o, Line):
o = Line3D(p1, p2)
else:
raise ValueError('unhandled linear entity: %s' % o.func)
return [o]
else:
x, y, z = map(Dummy, 'xyz')
t = Dummy() # unnamed else it may clash with a symbol in o
a = Point3D(o.arbitrary_point(t))
b = self.equation(x, y, z)
# TODO: Replace solve with solveset, when this line is tested
c = solve(b.subs(list(zip((x, y, z), a.args))), t)
if not c:
return []
else:
p = a.subs(t, c[0])
if p not in self:
return [] # e.g. a segment might not intersect a plane
return [p]
if isinstance(o, Plane):
if self.equals(o):
return [self]
if self.is_parallel(o):
return []
else:
x, y, z = map(Dummy, 'xyz')
a, b = Matrix([self.normal_vector]), Matrix([o.normal_vector])
c = list(a.cross(b))
d = self.equation(x, y, z)
e = o.equation(x, y, z)
result = list(linsolve([d, e], x, y, z))[0]
for i in (x, y, z):
result = result.subs(i, 0)
return [Line3D(Point3D(result), direction_ratio=c)]
def perpendicular_plane(self, *pts):
"""
Return a perpendicular passing through the given points. If the
direction ratio between the points is the same as the Plane's normal
vector then, to select from the infinite number of possible planes,
a third point will be chosen on the z-axis (or the y-axis
if the normal vector is already parallel to the z-axis). If less than
two points are given they will be supplied as follows: if no point is
given then pt1 will be self.p1; if a second point is not given it will
be a point through pt1 on a line parallel to the z-axis (if the normal
is not already the z-axis, otherwise on the line parallel to the
y-axis).
Parameters
==========
pts: 0, 1 or 2 Point3D
Returns
=======
Plane
Examples
========
>>> from sympy import Plane, Point3D, Line3D
>>> a, b = Point3D(0, 0, 0), Point3D(0, 1, 0)
>>> Z = (0, 0, 1)
>>> p = Plane(a, normal_vector=Z)
>>> p.perpendicular_plane(a, b)
Plane(Point3D(0, 0, 0), (1, 0, 0))
"""
if len(pts) > 2:
raise ValueError('No more than 2 pts should be provided.')
pts = list(pts)
if len(pts) == 0:
pts.append(self.p1)
if len(pts) == 1:
x, y, z = self.normal_vector
if x == y == 0:
dir = (0, 1, 0)
else:
dir = (0, 0, 1)
pts.append(pts[0] + Point3D(*dir))
p1, p2 = [Point(i, dim=3) for i in pts]
l = Line3D(p1, p2)
n = Line3D(p1, direction_ratio=self.normal_vector)
if l in n: # XXX should an error be raised instead?
# there are infinitely many perpendicular planes;
x, y, z = self.normal_vector
if x == y == 0:
# the z axis is the normal so pick a pt on the y-axis
p3 = Point3D(0, 1, 0) # case 1
else:
# else pick a pt on the z axis
p3 = Point3D(0, 0, 1) # case 2
# in case that point is already given, move it a bit
if p3 in l:
p3 *= 2 # case 3
else:
p3 = p1 + Point3D(*self.normal_vector) # case 4
return Plane(p1, p2, p3)
def _eval_subs(self, old, new):
if old == self.parameter:
return Point(*[f.subs(old, new) for f in self.functions])
def test_polytope_integrate():
# Convex 2-Polytopes
# Vertex representation
assert polytope_integrate(Polygon(Point(0, 0), Point(0, 2), Point(4, 0)),
1,
dims=(x, y)) == 4
assert polytope_integrate(Polygon(Point(0, 0), Point(0, 1),
Point(1, 1), Point(1, 0)), x * y) ==\
S(1)/4
assert polytope_integrate(Polygon(Point(0, 3), Point(5, 3), Point(1, 1)),
6 * x**2 - 40 * y) == S(-935) / 3
assert polytope_integrate(
Polygon(Point(0, 0), Point(0, sqrt(3)), Point(sqrt(3), sqrt(3)),
Point(sqrt(3), 0)), 1) == 3
hexagon = Polygon(Point(0, 0), Point(-sqrt(3) / 2,
S(1) / 2), Point(-sqrt(3) / 2, 3 / 2),
Point(0, 2), Point(sqrt(3) / 2, 3 / 2),
Point(sqrt(3) / 2,
S(1) / 2))
assert polytope_integrate(hexagon, 1) == S(3 * sqrt(3)) / 2
# Hyperplane representation
assert polytope_integrate([((-1, 0), 0), ((1, 2), 4), ((0, -1), 0)],
1,
dims=(x, y)) == 4
assert polytope_integrate([((-1, 0), 0), ((0, 1), 1), ((1, 0), 1),
((0, -1), 0)], x * y) == S(1) / 4
assert polytope_integrate([((0, 1), 3), ((1, -2), -1), ((-2, -1), -3)],
6 * x**2 - 40 * y) == S(-935) / 3
assert polytope_integrate([((-1, 0), 0), ((0, sqrt(3)), 3),
((sqrt(3), 0), 3), ((0, -1), 0)], 1) == 3
hexagon = [((-1 / 2, -sqrt(3) / 2), 0), ((-1, 0), sqrt(3) / 2),
((-1 / 2, sqrt(3) / 2), sqrt(3)),
((1 / 2, sqrt(3) / 2), sqrt(3)), ((1, 0), sqrt(3) / 2),
((1 / 2, -sqrt(3) / 2), 0)]
assert polytope_integrate(hexagon, 1) == S(3 * sqrt(3)) / 2
# Non-convex polytopes
# Vertex representation
assert polytope_integrate(
Polygon(Point(-1, -1), Point(-1, 1), Point(1, 1), Point(0, 0),
Point(1, -1)), 1) == 3
assert polytope_integrate(
Polygon(Point(-1, -1), Point(-1, 1), Point(0, 0), Point(1, 1),
Point(1, -1), Point(0, 0)), 1) == 2
# Hyperplane representation
assert polytope_integrate([((-1, 0), 1), ((0, 1), 1), ((1, -1), 0),
((1, 1), 0), ((0, -1), 1)], 1) == 3
assert polytope_integrate([((-1, 0), 1), ((1, 1), 0), ((-1, 1), 0),
((1, 0), 1), ((-1, -1), 0), ((1, -1), 0)],
1) == 2
# Tests for 2D polytopes mentioned in Chin et al(Page 10):
# http://dilbert.engr.ucdavis.edu/~suku/quadrature/cls-integration.pdf
fig1 = Polygon(Point(1.220, -0.827), Point(-1.490, -4.503),
Point(-3.766, -1.622), Point(-4.240, -0.091),
Point(-3.160, 4), Point(-0.981, 4.447), Point(0.132, 4.027))
assert polytope_integrate(fig1, x**2 + x*y + y**2) ==\
S(2031627344735367)/(8*10**12)
fig2 = Polygon(Point(4.561, 2.317), Point(1.491, -1.315),
Point(-3.310, -3.164), Point(-4.845, -3.110),
Point(-4.569, 1.867))
assert polytope_integrate(fig2, x**2 + x*y + y**2) ==\
S(517091313866043)/(16*10**11)
fig3 = Polygon(Point(-2.740, -1.888), Point(-3.292, 4.233),
Point(-2.723, -0.697), Point(-0.643, -3.151))
assert polytope_integrate(fig3, x**2 + x*y + y**2) ==\
S(147449361647041)/(8*10**12)
fig4 = Polygon(Point(0.211, -4.622), Point(-2.684, 3.851),
Point(0.468, 4.879), Point(4.630, -1.325),
Point(-0.411, -1.044))
assert polytope_integrate(fig4, x**2 + x*y + y**2) ==\
S(180742845225803)/(10**12)
# Tests for many polynomials with maximum degree given.
tri = Polygon(Point(0, 3), Point(5, 3), Point(1, 1))
polys = []
expr1 = x**9 * y + x**7 * y**3 + 2 * x**2 * y**8
expr2 = x**6 * y**4 + x**5 * y**5 + 2 * y**10
expr3 = x**10 + x**9 * y + x**8 * y**2 + x**5 * y**5
polys.extend((expr1, expr2, expr3))
result_dict = polytope_integrate(tri, polys, max_degree=10)
assert result_dict[expr1] == S(615780107) / 594
assert result_dict[expr2] == S(13062161) / 27
assert result_dict[expr3] == S(1946257153) / 924
# Tests for 3D polytopes
cube1 = [[(0, 0, 0), (0, 6, 6), (6, 6, 6), (3, 6, 0), (0, 6, 0), (6, 0, 6),
(3, 0, 0), (0, 0, 6)], [1, 2, 3, 4], [3, 2, 5, 6], [1, 7, 5, 2],
[0, 6, 5, 7], [1, 4, 0, 7], [0, 4, 3, 6]]
assert polytope_integrate(cube1, 1) == S(162)
# 3D Test cases in Chin et al(2015)
cube2 = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0), (5, 0, 5),
(5, 5, 0), (5, 5, 5)], [3, 7, 6, 2], [1, 5, 7, 3], [5, 4, 6, 7],
[0, 4, 5, 1], [2, 0, 1, 3], [2, 6, 4, 0]]
cube3 = [[(0, 0, 0), (5, 0, 0), (5, 4, 0), (3, 2, 0), (3, 5, 0), (0, 5, 0),
(0, 0, 5), (5, 0, 5), (5, 4, 5), (3, 2, 5), (3, 5, 5),
(0, 5, 5)], [6, 11, 5, 0], [1, 7, 6, 0], [5, 4, 3, 2, 1, 0],
[11, 10, 4, 5], [10, 9, 3, 4], [9, 8, 2, 3], [8, 7, 1, 2],
[7, 8, 9, 10, 11, 6]]
cube4 = [[(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1),
(S(1) / 4, S(1) / 4, S(1) / 4)], [0, 2, 1], [1, 3, 0], [4, 2, 3],
[4, 3, 1], [0, 1, 2], [2, 4, 1], [0, 3, 2]]
assert polytope_integrate(cube2, x ** 2 + y ** 2 + x * y + z ** 2) ==\
S(15625)/4
assert polytope_integrate(cube3, x ** 2 + y ** 2 + x * y + z ** 2) ==\
S(33835) / 12
assert polytope_integrate(cube4, x ** 2 + y ** 2 + x * y + z ** 2) ==\
S(37) / 960
def test_polytopes_intersecting_sides():
# Intersecting polygons not implemented yet in SymPy. Will be implemented
# soon. As of now, the intersection point will have to be manually
# supplied by user.
fig5 = Polygon(Point(-4.165, -0.832), Point(-3.668, 1.568),
Point(-3.266, 1.279), Point(-1.090, -2.080),
Point(3.313, -0.683), Point(3.033, -4.845),
Point(-4.395, 4.840), Point(-1.007, -3.328))
assert polytope_integrate(fig5, x**2 + x*y + y**2) ==\
S(1633405224899363)/(24*10**12)
fig6 = Polygon(Point(-3.018, -4.473), Point(-0.103, 2.378),
Point(-1.605, -2.308), Point(4.516, -0.771),
Point(4.203, 0.478))
assert polytope_integrate(fig6, x**2 + x*y + y**2) ==\
S(88161333955921)/(3*10**12)
def intersection(self, o):
"""The intersection of the parabola and another geometrical entity `o`.
Parameters
==========
o : GeometryEntity, LinearEntity
Returns
=======
intersection : list of GeometryEntity objects
Examples
========
>>> from sympy import Parabola, Point, Ellipse, Line, Segment
>>> p1 = Point(0,0)
>>> l1 = Line(Point(1, -2), Point(-1,-2))
>>> parabola1 = Parabola(p1, l1)
>>> parabola1.intersection(Ellipse(Point(0, 0), 2, 5))
[Point2D(-2, 0), Point2D(2, 0)]
>>> parabola1.intersection(Line(Point(-7, 3), Point(12, 3)))
[Point2D(-4, 3), Point2D(4, 3)]
>>> parabola1.intersection(Segment((-12, -65), (14, -68)))
[]
"""
x, y = symbols('x y', real=True)
parabola_eq = self.equation()
if isinstance(o, Parabola):
if o in self:
return [o]
else:
return list(
ordered([
Point(i)
for i in solve([parabola_eq, o.equation()], [x, y])
]))
elif isinstance(o, Point2D):
if simplify(parabola_eq.subs(([(x, o._args[0]),
(y, o._args[1])]))) == 0:
return [o]
else:
return []
elif isinstance(o, (Segment2D, Ray2D)):
result = solve(
[parabola_eq,
Line2D(o.points[0], o.points[1]).equation()], [x, y])
return list(ordered([Point2D(i) for i in result if i in o]))
elif isinstance(o, (Line2D, Ellipse)):
return list(
ordered([
Point2D(i)
for i in solve([parabola_eq, o.equation()], [x, y])
]))
elif isinstance(o, LinearEntity3D):
raise TypeError(
'Entity must be two dimensional, not three dimensional')
else:
raise TypeError('Wrong type of argument were put')
def test_polytope_integrate():
# Convex 2-Polytopes
# Vertex representation
assert polytope_integrate(Polygon(Point(0, 0), Point(0, 2), Point(4, 0)),
1,
dims=(x, y)) == 4
assert polytope_integrate(Polygon(Point(0, 0), Point(0, 1),
Point(1, 1), Point(1, 0)), x * y) ==\
S(1)/4
assert polytope_integrate(Polygon(Point(0, 3), Point(5, 3), Point(1, 1)),
6 * x**2 - 40 * y) == S(-935) / 3
assert polytope_integrate(
Polygon(Point(0, 0), Point(0, sqrt(3)), Point(sqrt(3), sqrt(3)),
Point(sqrt(3), 0)), 1) == 3
hexagon = Polygon(Point(0, 0), Point(-sqrt(3) / 2,
S(1) / 2),
Point(-sqrt(3) / 2,
S(3) / 2), Point(0, 2),
Point(sqrt(3) / 2,
S(3) / 2), Point(sqrt(3) / 2,
S(1) / 2))
assert polytope_integrate(hexagon, 1) == S(3 * sqrt(3)) / 2
# Hyperplane representation
assert polytope_integrate([((-1, 0), 0), ((1, 2), 4), ((0, -1), 0)],
1,
dims=(x, y)) == 4
assert polytope_integrate([((-1, 0), 0), ((0, 1), 1), ((1, 0), 1),
((0, -1), 0)], x * y) == S(1) / 4
assert polytope_integrate([((0, 1), 3), ((1, -2), -1), ((-2, -1), -3)],
6 * x**2 - 40 * y) == S(-935) / 3
assert polytope_integrate([((-1, 0), 0), ((0, sqrt(3)), 3),
((sqrt(3), 0), 3), ((0, -1), 0)], 1) == 3
hexagon = [((-S(1) / 2, -sqrt(3) / 2), 0), ((-1, 0), sqrt(3) / 2),
((-S(1) / 2, sqrt(3) / 2), sqrt(3)),
((S(1) / 2, sqrt(3) / 2), sqrt(3)), ((1, 0), sqrt(3) / 2),
((S(1) / 2, -sqrt(3) / 2), 0)]
assert polytope_integrate(hexagon, 1) == S(3 * sqrt(3)) / 2
# Non-convex polytopes
# Vertex representation
assert polytope_integrate(
Polygon(Point(-1, -1), Point(-1, 1), Point(1, 1), Point(0, 0),
Point(1, -1)), 1) == 3
assert polytope_integrate(
Polygon(Point(-1, -1), Point(-1, 1), Point(0, 0), Point(1, 1),
Point(1, -1), Point(0, 0)), 1) == 2
# Hyperplane representation
assert polytope_integrate([((-1, 0), 1), ((0, 1), 1), ((1, -1), 0),
((1, 1), 0), ((0, -1), 1)], 1) == 3
assert polytope_integrate([((-1, 0), 1), ((1, 1), 0), ((-1, 1), 0),
((1, 0), 1), ((-1, -1), 0), ((1, -1), 0)],
1) == 2
# Tests for 2D polytopes mentioned in Chin et al(Page 10):
# http://dilbert.engr.ucdavis.edu/~suku/quadrature/cls-integration.pdf
fig1 = Polygon(Point(1.220, -0.827), Point(-1.490, -4.503),
Point(-3.766, -1.622), Point(-4.240, -0.091),
Point(-3.160, 4), Point(-0.981, 4.447), Point(0.132, 4.027))
assert polytope_integrate(fig1, x**2 + x*y + y**2) ==\
S(2031627344735367)/(8*10**12)
fig2 = Polygon(Point(4.561, 2.317), Point(1.491, -1.315),
Point(-3.310, -3.164), Point(-4.845, -3.110),
Point(-4.569, 1.867))
assert polytope_integrate(fig2, x**2 + x*y + y**2) ==\
S(517091313866043)/(16*10**11)
fig3 = Polygon(Point(-2.740, -1.888), Point(-3.292, 4.233),
Point(-2.723, -0.697), Point(-0.643, -3.151))
assert polytope_integrate(fig3, x**2 + x*y + y**2) ==\
S(147449361647041)/(8*10**12)
fig4 = Polygon(Point(0.211, -4.622), Point(-2.684, 3.851),
Point(0.468, 4.879), Point(4.630, -1.325),
Point(-0.411, -1.044))
assert polytope_integrate(fig4, x**2 + x*y + y**2) ==\
S(180742845225803)/(10**12)
# Tests for many polynomials with maximum degree given(2D case).
tri = Polygon(Point(0, 3), Point(5, 3), Point(1, 1))
polys = []
expr1 = x**9 * y + x**7 * y**3 + 2 * x**2 * y**8
expr2 = x**6 * y**4 + x**5 * y**5 + 2 * y**10
expr3 = x**10 + x**9 * y + x**8 * y**2 + x**5 * y**5
polys.extend((expr1, expr2, expr3))
result_dict = polytope_integrate(tri, polys, max_degree=10)
assert result_dict[expr1] == S(615780107) / 594
assert result_dict[expr2] == S(13062161) / 27
assert result_dict[expr3] == S(1946257153) / 924
# Tests when all integral of all monomials up to a max_degree is to be
# calculated.
assert polytope_integrate(Polygon(Point(0, 0), Point(0, 1), Point(1, 1),
Point(1, 0)),
max_degree=4) == {
0: 0,
1: 1,
x: S(1) / 2,
x**2 * y**2: S(1) / 9,
x**4: S(1) / 5,
y**4: S(1) / 5,
y: S(1) / 2,
x * y**2: S(1) / 6,
y**2: S(1) / 3,
x**3: S(1) / 4,
x**2 * y: S(1) / 6,
x**3 * y: S(1) / 8,
x * y: S(1) / 4,
y**3: S(1) / 4,
x**2: S(1) / 3,
x * y**3: S(1) / 8
}
# Tests for 3D polytopes
cube1 = [[(0, 0, 0), (0, 6, 6), (6, 6, 6), (3, 6, 0), (0, 6, 0), (6, 0, 6),
(3, 0, 0), (0, 0, 6)], [1, 2, 3, 4], [3, 2, 5, 6], [1, 7, 5, 2],
[0, 6, 5, 7], [1, 4, 0, 7], [0, 4, 3, 6]]
assert polytope_integrate(cube1, 1) == S(162)
# 3D Test cases in Chin et al(2015)
cube2 = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0), (5, 0, 5),
(5, 5, 0), (5, 5, 5)], [3, 7, 6, 2], [1, 5, 7, 3], [5, 4, 6, 7],
[0, 4, 5, 1], [2, 0, 1, 3], [2, 6, 4, 0]]
cube3 = [[(0, 0, 0), (5, 0, 0), (5, 4, 0), (3, 2, 0), (3, 5, 0), (0, 5, 0),
(0, 0, 5), (5, 0, 5), (5, 4, 5), (3, 2, 5), (3, 5, 5),
(0, 5, 5)], [6, 11, 5, 0], [1, 7, 6, 0], [5, 4, 3, 2, 1, 0],
[11, 10, 4, 5], [10, 9, 3, 4], [9, 8, 2, 3], [8, 7, 1, 2],
[7, 8, 9, 10, 11, 6]]
cube4 = [[(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1),
(S(1) / 4, S(1) / 4, S(1) / 4)], [0, 2, 1], [1, 3, 0], [4, 2, 3],
[4, 3, 1], [0, 1, 2], [2, 4, 1], [0, 3, 2]]
assert polytope_integrate(cube2, x ** 2 + y ** 2 + x * y + z ** 2) ==\
S(15625)/4
assert polytope_integrate(cube3, x ** 2 + y ** 2 + x * y + z ** 2) ==\
S(33835) / 12
assert polytope_integrate(cube4, x ** 2 + y ** 2 + x * y + z ** 2) ==\
S(37) / 960
# Test cases from Mathematica's PolyhedronData library
octahedron = [[(S(-1) / sqrt(2), 0, 0), (0, S(1) / sqrt(2), 0),
(0, 0, S(-1) / sqrt(2)), (0, 0, S(1) / sqrt(2)),
(0, S(-1) / sqrt(2), 0), (S(1) / sqrt(2), 0, 0)], [3, 4, 5],
[3, 5, 1], [3, 1, 0], [3, 0, 4], [4, 0, 2], [4, 2, 5],
[2, 0, 1], [5, 2, 1]]
assert polytope_integrate(octahedron, 1) == sqrt(2) / 3
great_stellated_dodecahedron =\
[[(-0.32491969623290634095, 0, 0.42532540417601993887),
(0.32491969623290634095, 0, -0.42532540417601993887),
(-0.52573111211913359231, 0, 0.10040570794311363956),
(0.52573111211913359231, 0, -0.10040570794311363956),
(-0.10040570794311363956, -0.3090169943749474241, 0.42532540417601993887),
(-0.10040570794311363956, 0.30901699437494742410, 0.42532540417601993887),
(0.10040570794311363956, -0.3090169943749474241, -0.42532540417601993887),
(0.10040570794311363956, 0.30901699437494742410, -0.42532540417601993887),
(-0.16245984811645317047, -0.5, 0.10040570794311363956),
(-0.16245984811645317047, 0.5, 0.10040570794311363956),
(0.16245984811645317047, -0.5, -0.10040570794311363956),
(0.16245984811645317047, 0.5, -0.10040570794311363956),
(-0.42532540417601993887, -0.3090169943749474241, -0.10040570794311363956),
(-0.42532540417601993887, 0.30901699437494742410, -0.10040570794311363956),
(-0.26286555605956679615, 0.1909830056250525759, -0.42532540417601993887),
(-0.26286555605956679615, -0.1909830056250525759, -0.42532540417601993887),
(0.26286555605956679615, 0.1909830056250525759, 0.42532540417601993887),
(0.26286555605956679615, -0.1909830056250525759, 0.42532540417601993887),
(0.42532540417601993887, -0.3090169943749474241, 0.10040570794311363956),
(0.42532540417601993887, 0.30901699437494742410, 0.10040570794311363956)],
[12, 3, 0, 6, 16], [17, 7, 0, 3, 13],
[9, 6, 0, 7, 8], [18, 2, 1, 4, 14],
[15, 5, 1, 2, 19], [11, 4, 1, 5, 10],
[8, 19, 2, 18, 9], [10, 13, 3, 12, 11],
[16, 14, 4, 11, 12], [13, 10, 5, 15, 17],
[14, 16, 6, 9, 18], [19, 8, 7, 17, 15]]
# Actual volume is : 0.163118960624632
assert Abs(polytope_integrate(great_stellated_dodecahedron, 1) -\
0.163118960624632) < 1e-12
expr = x**2 + y**2 + z**2
octahedron_five_compound = [[
(0, -0.7071067811865475244, 0), (0, 0.70710678118654752440, 0),
(0.1148764602736805918, -0.35355339059327376220,
-0.60150095500754567366),
(0.1148764602736805918, 0.35355339059327376220,
-0.60150095500754567366),
(0.18587401723009224507, -0.57206140281768429760,
0.37174803446018449013),
(0.18587401723009224507, 0.57206140281768429760,
0.37174803446018449013),
(0.30075047750377283683, -0.21850801222441053540,
0.60150095500754567366),
(0.30075047750377283683, 0.21850801222441053540,
0.60150095500754567366),
(0.48662449473386508189, -0.35355339059327376220,
-0.37174803446018449013),
(0.48662449473386508189, 0.35355339059327376220,
-0.37174803446018449013),
(-0.60150095500754567366, 0, -0.37174803446018449013),
(-0.30075047750377283683, -0.21850801222441053540,
-0.60150095500754567366),
(-0.30075047750377283683, 0.21850801222441053540,
-0.60150095500754567366),
(0.60150095500754567366, 0, 0.37174803446018449013),
(0.4156269377774534286, -0.57206140281768429760, 0),
(0.4156269377774534286, 0.57206140281768429760, 0),
(0.37174803446018449013, 0, -0.60150095500754567366),
(-0.4156269377774534286, -0.57206140281768429760, 0),
(-0.4156269377774534286, 0.57206140281768429760, 0),
(-0.67249851196395732696, -0.21850801222441053540, 0),
(-0.67249851196395732696, 0.21850801222441053540, 0),
(0.67249851196395732696, -0.21850801222441053540, 0),
(0.67249851196395732696, 0.21850801222441053540, 0),
(-0.37174803446018449013, 0, 0.60150095500754567366),
(-0.48662449473386508189, -0.35355339059327376220,
0.37174803446018449013),
(-0.48662449473386508189, 0.35355339059327376220,
0.37174803446018449013),
(-0.18587401723009224507, -0.57206140281768429760,
-0.37174803446018449013),
(-0.18587401723009224507, 0.57206140281768429760,
-0.37174803446018449013),
(-0.11487646027368059176, -0.35355339059327376220,
0.60150095500754567366),
(-0.11487646027368059176, 0.35355339059327376220,
0.60150095500754567366)
], [0, 10, 16], [23, 10, 0], [16, 13, 0], [0, 13, 23], [16, 10, 1],
[1, 10, 23], [1, 13, 16], [23, 13, 1],
[2, 4, 19], [22, 4, 2], [2, 19,
27], [27, 22, 2],
[20, 5, 3], [3, 5, 21],
[26, 20, 3], [3, 21, 26], [29, 19, 4],
[4, 22, 29], [5, 20, 28], [28, 21, 5],
[6, 8, 15], [17, 8, 6], [6, 15,
25], [25, 17, 6],
[14, 9, 7], [7, 9, 18], [24, 14,
7], [7, 18, 24],
[8, 12, 15], [17, 12, 8], [14, 11, 9],
[9, 11, 18], [11, 14, 24], [24, 18, 11],
[25, 15, 12], [12, 17, 25], [29, 27, 19],
[20, 26, 28], [28, 26, 21], [22, 27, 29]]
assert Abs(polytope_integrate(octahedron_five_compound, expr)) - 0.353553\
< 1e-6
cube_five_compound = [[
(-0.1624598481164531631, -0.5, -0.6881909602355867691),
(-0.1624598481164531631, 0.5, -0.6881909602355867691),
(0.1624598481164531631, -0.5, 0.68819096023558676910),
(0.1624598481164531631, 0.5, 0.68819096023558676910),
(-0.52573111211913359231, 0, -0.6881909602355867691),
(0.52573111211913359231, 0, 0.68819096023558676910),
(-0.26286555605956679615, -0.8090169943749474241,
-0.1624598481164531631),
(-0.26286555605956679615, 0.8090169943749474241,
-0.1624598481164531631),
(0.26286555605956680301, -0.8090169943749474241,
0.1624598481164531631),
(0.26286555605956680301, 0.8090169943749474241, 0.1624598481164531631),
(-0.42532540417601993887, -0.3090169943749474241,
0.68819096023558676910),
(-0.42532540417601993887, 0.30901699437494742410,
0.68819096023558676910),
(0.42532540417601996609, -0.3090169943749474241,
-0.6881909602355867691),
(0.42532540417601996609, 0.30901699437494742410,
-0.6881909602355867691),
(-0.6881909602355867691, -0.5, 0.1624598481164531631),
(-0.6881909602355867691, 0.5, 0.1624598481164531631),
(0.68819096023558676910, -0.5, -0.1624598481164531631),
(0.68819096023558676910, 0.5, -0.1624598481164531631),
(-0.85065080835203998877, 0, -0.1624598481164531631),
(0.85065080835203993218, 0, 0.1624598481164531631)
], [18, 10, 3, 7], [13, 19, 8, 0], [18, 0, 8, 10], [3, 19, 13, 7],
[18, 7, 13, 0], [8, 19, 3, 10], [6, 2, 11, 18],
[1, 9, 19, 12], [11, 9, 1, 18], [6, 12, 19, 2],
[1, 12, 6, 18], [11, 2, 19, 9], [4, 14, 11, 7],
[17, 5, 8, 12], [4, 12, 8, 14], [11, 5, 17, 7],
[4, 7, 17, 12], [8, 5, 11, 14], [6, 10, 15, 4],
[13, 9, 5, 16], [15, 9, 13, 4], [6, 16, 5, 10],
[13, 16, 6, 4], [15, 10, 5, 9], [14, 15, 1, 0],
[16, 17, 3, 2], [14, 2, 3, 15], [1, 17, 16, 0],
[14, 0, 16, 2], [3, 17, 1, 15]]
assert Abs(polytope_integrate(cube_five_compound, expr) - 1.25) < 1e-12
echidnahedron = [[
(0, 0, -2.4898982848827801995), (0, 0, 2.4898982848827802734),
(0, -4.2360679774997896964, -2.4898982848827801995),
(0, -4.2360679774997896964, 2.4898982848827802734),
(0, 4.2360679774997896964, -2.4898982848827801995),
(0, 4.2360679774997896964, 2.4898982848827802734),
(-4.0287400534704067567, -1.3090169943749474241,
-2.4898982848827801995),
(-4.0287400534704067567, -1.3090169943749474241,
2.4898982848827802734),
(-4.0287400534704067567, 1.3090169943749474241,
-2.4898982848827801995),
(-4.0287400534704067567, 1.3090169943749474241, 2.4898982848827802734),
(4.0287400534704069747, -1.3090169943749474241,
-2.4898982848827801995),
(4.0287400534704069747, -1.3090169943749474241, 2.4898982848827802734),
(4.0287400534704069747, 1.3090169943749474241, -2.4898982848827801995),
(4.0287400534704069747, 1.3090169943749474241, 2.4898982848827802734),
(-2.4898982848827801995, -3.4270509831248422723,
-2.4898982848827801995),
(-2.4898982848827801995, -3.4270509831248422723,
2.4898982848827802734),
(-2.4898982848827801995, 3.4270509831248422723,
-2.4898982848827801995),
(-2.4898982848827801995, 3.4270509831248422723, 2.4898982848827802734),
(2.4898982848827802734, -3.4270509831248422723,
-2.4898982848827801995),
(2.4898982848827802734, -3.4270509831248422723, 2.4898982848827802734),
(2.4898982848827802734, 3.4270509831248422723, -2.4898982848827801995),
(2.4898982848827802734, 3.4270509831248422723, 2.4898982848827802734),
(-4.7169310137059934362, -0.8090169943749474241,
-1.1135163644116066184),
(-4.7169310137059934362, 0.8090169943749474241,
-1.1135163644116066184),
(4.7169310137059937438, -0.8090169943749474241,
1.11351636441160673519),
(4.7169310137059937438, 0.8090169943749474241, 1.11351636441160673519),
(-4.2916056095299737777, -2.1180339887498948482,
1.11351636441160673519),
(-4.2916056095299737777, 2.1180339887498948482,
1.11351636441160673519),
(4.2916056095299737777, -2.1180339887498948482,
-1.1135163644116066184),
(4.2916056095299737777, 2.1180339887498948482, -1.1135163644116066184),
(-3.6034146492943870399, 0, -3.3405490932348205213),
(3.6034146492943870399, 0, 3.3405490932348202056),
(-3.3405490932348205213, -3.4270509831248422723,
1.11351636441160673519),
(-3.3405490932348205213, 3.4270509831248422723,
1.11351636441160673519),
(3.3405490932348202056, -3.4270509831248422723,
-1.1135163644116066184),
(3.3405490932348202056, 3.4270509831248422723, -1.1135163644116066184),
(-2.9152236890588002395, -2.1180339887498948482,
3.3405490932348202056),
(-2.9152236890588002395, 2.1180339887498948482, 3.3405490932348202056),
(2.9152236890588002395, -2.1180339887498948482,
-3.3405490932348205213),
(2.9152236890588002395, 2.1180339887498948482, -3.3405490932348205213),
(-2.2270327288232132368, 0, -1.1135163644116066184),
(-2.2270327288232132368, -4.2360679774997896964,
-1.1135163644116066184),
(-2.2270327288232132368, 4.2360679774997896964,
-1.1135163644116066184),
(2.2270327288232134704, 0, 1.11351636441160673519),
(2.2270327288232134704, -4.2360679774997896964,
1.11351636441160673519),
(2.2270327288232134704, 4.2360679774997896964, 1.11351636441160673519),
(-1.8017073246471935200, -1.3090169943749474241,
1.11351636441160673519),
(-1.8017073246471935200, 1.3090169943749474241,
1.11351636441160673519),
(1.8017073246471935043, -1.3090169943749474241,
-1.1135163644116066184),
(1.8017073246471935043, 1.3090169943749474241, -1.1135163644116066184),
(-1.3763819204711735382, 0, -4.7169310137059934362),
(-1.3763819204711735382, 0, 0.26286555605956679615),
(1.37638192047117353821, 0, 4.7169310137059937438),
(1.37638192047117353821, 0, -0.26286555605956679615),
(-1.1135163644116066184, -3.4270509831248422723,
-3.3405490932348205213),
(-1.1135163644116066184, -0.8090169943749474241,
4.7169310137059937438),
(-1.1135163644116066184, -0.8090169943749474241,
-0.26286555605956679615),
(-1.1135163644116066184, 0.8090169943749474241, 4.7169310137059937438),
(-1.1135163644116066184, 0.8090169943749474241,
-0.26286555605956679615),
(-1.1135163644116066184, 3.4270509831248422723,
-3.3405490932348205213),
(1.11351636441160673519, -3.4270509831248422723,
3.3405490932348202056),
(1.11351636441160673519, -0.8090169943749474241,
-4.7169310137059934362),
(1.11351636441160673519, -0.8090169943749474241,
0.26286555605956679615),
(1.11351636441160673519, 0.8090169943749474241,
-4.7169310137059934362),
(1.11351636441160673519, 0.8090169943749474241,
0.26286555605956679615),
(1.11351636441160673519, 3.4270509831248422723, 3.3405490932348202056),
(-0.85065080835203998877, 0, 1.11351636441160673519),
(0.85065080835203993218, 0, -1.1135163644116066184),
(-0.6881909602355867691, -0.5, -1.1135163644116066184),
(-0.6881909602355867691, 0.5, -1.1135163644116066184),
(-0.6881909602355867691, -4.7360679774997896964,
-1.1135163644116066184),
(-0.6881909602355867691, -2.1180339887498948482,
-1.1135163644116066184),
(-0.6881909602355867691, 2.1180339887498948482,
-1.1135163644116066184),
(-0.6881909602355867691, 4.7360679774997896964,
-1.1135163644116066184),
(0.68819096023558676910, -0.5, 1.11351636441160673519),
(0.68819096023558676910, 0.5, 1.11351636441160673519),
(0.68819096023558676910, -4.7360679774997896964,
1.11351636441160673519),
(0.68819096023558676910, -2.1180339887498948482,
1.11351636441160673519),
(0.68819096023558676910, 2.1180339887498948482,
1.11351636441160673519),
(0.68819096023558676910, 4.7360679774997896964,
1.11351636441160673519),
(-0.42532540417601993887, -1.3090169943749474241,
-4.7169310137059934362),
(-0.42532540417601993887, -1.3090169943749474241,
0.26286555605956679615),
(-0.42532540417601993887, 1.3090169943749474241,
-4.7169310137059934362),
(-0.42532540417601993887, 1.3090169943749474241,
0.26286555605956679615),
(-0.26286555605956679615, -0.8090169943749474241,
1.11351636441160673519),
(-0.26286555605956679615, 0.8090169943749474241,
1.11351636441160673519),
(0.26286555605956679615, -0.8090169943749474241,
-1.1135163644116066184),
(0.26286555605956679615, 0.8090169943749474241,
-1.1135163644116066184),
(0.42532540417601996609, -1.3090169943749474241,
4.7169310137059937438),
(0.42532540417601996609, -1.3090169943749474241,
-0.26286555605956679615),
(0.42532540417601996609, 1.3090169943749474241, 4.7169310137059937438),
(0.42532540417601996609, 1.3090169943749474241,
-0.26286555605956679615)
], [9, 66, 47], [44, 62, 77], [20, 91, 49], [33, 47, 83], [3, 77, 84],
[12, 49, 53], [36, 84, 66], [28, 53, 62], [73, 83, 91],
[15, 84, 46], [25, 64, 43], [16, 58, 72], [26, 46, 51],
[11, 43, 74], [4, 72, 91], [60, 74, 84], [35, 91, 64],
[23, 51, 58], [19, 74, 77], [79, 83, 78], [6, 56, 40],
[76, 77, 81], [21, 78, 75], [8, 40, 58], [31, 75, 74],
[42, 58, 83], [41, 81, 56], [13, 75, 43], [27, 51, 47],
[2, 89, 71], [24, 43, 62], [17, 47, 85], [14, 71, 56],
[65, 85, 75], [22, 56, 51], [34, 62, 89], [5, 85, 78],
[32, 81, 46], [10, 53, 48], [45, 78, 64], [7, 46, 66],
[18, 48, 89], [37, 66, 85], [70, 89, 81], [29, 64, 53],
[88, 74, 1], [38, 67, 48], [42, 83, 72], [57, 1, 85],
[34, 48, 62], [59, 72, 87], [19, 62, 74], [63, 87, 67],
[17, 85, 83], [52, 75, 1], [39, 87, 49], [22, 51, 40],
[55, 1, 66], [29, 49, 64], [30, 40, 69], [13, 64, 75],
[82, 69, 87], [7, 66, 51], [90, 85, 1], [59, 69, 72],
[70, 81, 71], [88, 1, 84], [73, 72, 83], [54, 71, 68],
[5, 83, 85], [50, 68, 69], [3, 84, 81], [57, 66, 1],
[30, 68, 40], [28, 62, 48], [52, 1, 74], [23, 40, 51],
[38, 48, 86], [9, 51, 66], [80, 86, 68], [11, 74, 62],
[55, 84, 1], [54, 86, 71], [35, 64, 49], [90, 1, 75],
[41, 71, 81], [39, 49, 67], [15, 81, 84], [61, 67, 86],
[21, 75, 64], [24, 53, 43], [50, 69, 0], [37, 85, 47],
[31, 43, 75], [61, 0, 67], [27, 47, 58], [10, 67, 53],
[8, 58, 69], [90, 75, 85], [45, 91, 78], [80, 68, 0],
[36, 66, 46], [65, 78, 85], [63, 0, 87], [32, 46, 56],
[20, 87, 91], [14, 56, 68], [57, 85, 66], [33, 58, 47],
[61, 86, 0], [60, 84, 77], [37, 47, 66], [82, 0, 69],
[44, 77, 89], [16, 69, 58], [18, 89, 86], [55, 66, 84],
[26, 56, 46], [63, 67, 0], [31, 74, 43], [36, 46, 84],
[50, 0, 68], [25, 43, 53], [6, 68, 56], [12, 53, 67],
[88, 84, 74], [76, 89, 77], [82, 87, 0], [65, 75, 78],
[60, 77, 74], [80, 0, 86], [79, 78, 91], [2, 86, 89],
[4, 91, 87], [52, 74, 75], [21, 64, 78], [18, 86, 48],
[23, 58, 40], [5, 78, 83], [28, 48, 53], [6, 40, 68],
[25, 53, 64], [54, 68, 86], [33, 83, 58], [17, 83, 47],
[12, 67, 49], [41, 56, 71], [9, 47, 51], [35, 49, 91],
[2, 71, 86], [79, 91, 83], [38, 86, 67], [26, 51, 56],
[7, 51, 46], [4, 87, 72], [34, 89, 48], [15, 46, 81],
[42, 72, 58], [10, 48, 67], [27, 58, 51], [39, 67, 87],
[76, 81, 89], [3, 81, 77], [8, 69, 40], [29, 53, 49],
[19, 77, 62], [22, 40, 56], [20, 49, 87], [32, 56, 81],
[59, 87, 69], [24, 62, 53], [11, 62, 43], [14, 68, 71],
[73, 91, 72], [13, 43, 64], [70, 71, 89], [16, 72, 69],
[44, 89, 62], [30, 69, 68], [45, 64, 91]]
# Actual volume is : 51.405764746872634
assert Abs(polytope_integrate(echidnahedron, 1) - 51.4057647468726) < 1e-12
assert Abs(polytope_integrate(echidnahedron, expr) - 253.569603474519) <\
1e-12
# Tests for many polynomials with maximum degree given(2D case).
assert polytope_integrate(cube2, [x**2, y*z], max_degree=2) == \
{y * z: 3125 / S(4), x ** 2: 3125 / S(3)}
assert polytope_integrate(cube2, max_degree=2) == \
{1: 125, x: 625 / S(2), x * z: 3125 / S(4), y: 625 / S(2),
y * z: 3125 / S(4), z ** 2: 3125 / S(3), y ** 2: 3125 / S(3),
z: 625 / S(2), x * y: 3125 / S(4), x ** 2: 3125 / S(3)}
