def test_point_sort():
assert point_sort([Point(0, 0), Point(1, 0), Point(1, 1)]) == \
[Point2D(1, 1), Point2D(1, 0), Point2D(0, 0)]
fig6 = Polygon((0, 0), (1, 0), (1, 1))
assert polytope_integrate(fig6, x*y) == S(-1)/8
assert polytope_integrate(fig6, x*y, clockwise = True) == S(1)/8
def test_point_sort():
assert point_sort([Point(0, 0), Point(1, 0), Point(1, 1)]) == \
[Point2D(1, 1), Point2D(1, 0), Point2D(0, 0)]
fig6 = Polygon((0, 0), (1, 0), (1, 1))
assert polytope_integrate(fig6, x * y) == Rational(-1, 8)
assert polytope_integrate(fig6, x * y, clockwise=True) == Rational(1, 8)
def test_issue_19234():
polygon = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0))
polys = [1, x, y, x * y, x**2 * y, x * y**2]
assert polytope_integrate(polygon, polys) == \
{1: 1, x: S.Half, y: S.Half, x*y: Rational(1, 4), x**2*y: Rational(1, 6), x*y**2: Rational(1, 6)}
polys = [1, x, y, x * y, 3 + x**2 * y, x + x * y**2]
assert polytope_integrate(polygon, polys) == \
{1: 1, x: S.Half, y: S.Half, x*y: Rational(1, 4), x**2*y + 3: Rational(19, 6), x*y**2 + x: Rational(2, 3)}
def test_max_degree():
polygon = Polygon((0, 0), (0, 1), (1, 1), (1, 0))
polys = [1, x, y, x * y, x**2 * y, x * y**2]
assert polytope_integrate(polygon, polys, max_degree=3) == \
{1: 1, x: S.Half, y: S.Half, x*y: Rational(1, 4), x**2*y: Rational(1, 6), x*y**2: Rational(1, 6)}
assert polytope_integrate(polygon, polys, max_degree=2) == \
{1: 1, x: S.Half, y: S.Half, x*y: Rational(1, 4)}
assert polytope_integrate(polygon, polys, max_degree=1) == \
{1: 1, x: S.Half, y: S.Half}
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 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 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 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 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)}
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)}
], #3
[2, 3, 1], #n02, n31, n2
[2, 0, 3], #n02, n1, n31
[3, 0, 1], #n31, n1, n2
[1, 0, 2]
] #n2, n1, n02
b_H = {}
b_1mH = {}
b_D = {}
for i in range(nDOF1D):
for j in range(nDOF1D - i):
for k in range(nDOF1D - i - j):
basis_ijk = x**i * y**j * z**k
print("basis function ", i, j, k)
T0 = simplify(intpoly.polytope_integrate(sub_tet0, basis_ijk))
T1 = simplify(intpoly.polytope_integrate(sub_tet1, basis_ijk))
T2 = simplify(intpoly.polytope_integrate(sub_tet2, basis_ijk))
frag = simplify(T0 + T1 + T2)
b_H[(i, j, k)] = frag
b_1mH[(i, j,
k)] = intpoly.polytope_integrate(unit_tet, basis_ijk) - frag
theta01_expr = 0.5 - 0.5 * (phi1 + phi0) / (phi1 - phi0)
theta02_expr = 0.5 - 0.5 * (phi2 + phi0) / (phi2 - phi0)
theta31_expr = 0.5 - 0.5 * (phi1 + phi3) / (phi1 - phi3)
theta32_expr = 0.5 - 0.5 * (phi2 + phi3) / (phi2 - phi3)
b_D[(i,j,k)] = frag.diff(theta01,simplify=use_simplify)*(theta01_expr.diff(phi0,simplify=False) + theta01_expr.diff(phi1,simplify=False)) + \
frag.diff(theta02,simplify=use_simplify)*(theta02_expr.diff(phi0,simplify=False) + theta02_expr.diff(phi2,simplify=False)) + \
frag.diff(theta31,simplify=use_simplify)*(theta31_expr.diff(phi3,simplify=False) + theta31_expr.diff(phi1,simplify=False)) + \
frag.diff(theta32,simplify=use_simplify)*(theta32_expr.diff(phi3,simplify=False) + theta32_expr.diff(phi2,simplify=False))
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] == 615780107/594
assert result_dict[expr2] == 13062161/27
assert result_dict[expr3] == 1946257153/924
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] == 615780107 / 594
assert result_dict[expr2] == 13062161 / 27
assert result_dict[expr3] == 1946257153 / 924
x**i * y**j for i in range(nDOF1D) for j in range(nDOF1D - i)
]
nDOF = len(basis)
test_nDOF = sympy.factor(
Sum(Sum(1, (J, 0, N - I - 1)), (I, 0, N - 1)).doit())
unit_triangle = Triangle(Point(0, 0), Point(0, 1), Point(1, 0))
elif nSpace == 1:
basis = [x**i for i in range(nDOF1D)]
nDOF = len(basis)
test_nDOF = sympy.factor(Sum(1, (I, 0, N - 1)).doit())
assert (test_nDOF.evalf(subs={'N': nDOF1D}) == nDOF)
A = np.zeros((nDOF, nDOF), 'd')
for i in range(nDOF):
for j in range(nDOF):
if nSpace == 3:
A[i, j] = intpoly.polytope_integrate(
unit_tet, sympy.expand(basis[j] * basis[i])).evalf()
elif nSpace == 2:
A[i, j] = intpoly.polytope_integrate(
unit_triangle,
sympy.expand(basis[j] * basis[i])).evalf()
elif nSpace == 1:
A[i, j] = sympy.integrate(basis[j] * basis[i],
(x, 0, 1)).evalf()
Ainv = np.linalg.inv(A)
b_H = []
b_dH_x = []
b_dH_y = []
b_dH_z = []
b_1mH = []
if nSpace == 3:
sub_tet = [[(0, 0, 0), (x_0, 0, 0), (0, y_0, 0), (0, 0, z_0)],
def test_max_degree():
polygon = Polygon((0, 0), (0, 1), (1, 1), (1, 0))
polys = [1, x, y, x*y, x**2*y, x*y**2]
assert polytope_integrate(polygon, polys, max_degree=3) == \
{1: 1, x: S(1)/2, y: S(1)/2, x*y: S(1)/4, x**2*y: S(1)/6, x*y**2: S(1)/6}
def test_max_degree():
polygon = Polygon((0, 0), (0, 1), (1, 1), (1, 0))
polys = [1, x, y, x * y, x**2 * y, x * y**2]
assert polytope_integrate(polygon, polys, max_degree=3) == \
{1: 1, x: S(1)/2, y: S(1)/2, x*y: S(1)/4, x**2*y: S(1)/6, x*y**2: S(1)/6}
