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 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)}