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_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_main_integrate(): triangle = Polygon((0, 3), (5, 3), (1, 1)) facets = triangle.sides hp_params = hyperplane_parameters(triangle) assert main_integrate(x**2 + y**2, facets, hp_params) == Rational(325, 6) assert main_integrate(x**2 + y**2, facets, hp_params, max_degree=1) == \ {0: 0, 1: 5, y: Rational(35, 3), x: 10}
def sample(color, poly): least_dist = 3*256**2 least_img = None min_x, min_y, max_x, max_y = poly.bounds scalex = w/(max_x-min_x) scaley = h/(max_y-min_y) scaledpoly = Polygon(*[((x-min_x)*scalex, (y-min_y)*scaley) for x,y in polyverts(poly)]) for i, img in enumerate(images): #change poly pos, scale, rotate? #s = getcolor(img, poly) s = avgcolors[i] dist = sum([(a-b)**2 for a,b in zip(s,color)]) if dist < least_dist: least_dist = dist least_img = img ret = Image.new("RGBA", (w,h)) tri = polycrop(least_img, scaledpoly) if tri is None: raise Exception("Source directory contains no images") ret.paste(least_img, tri) #return least_img, polycrop(least_img, scaledpoly) ret = ret.resize((max_x-min_x,max_y-min_y)) return ret, (min_x, min_y)
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_geometry(): p1 = Point(1, 2) p2 = Point(2, 3) p3 = Point(0, 0) p4 = Point(0, 1) for c in ( GeometryEntity, GeometryEntity(), Point, p1, Circle, Circle(p1, 2), Ellipse, Ellipse(p1, 3, 4), Line, Line(p1, p2), LinearEntity, LinearEntity(p1, p2), Ray, Ray(p1, p2), Segment, Segment(p1, p2), Polygon, Polygon(p1, p2, p3, p4), RegularPolygon, RegularPolygon(p1, 4, 5), Triangle, Triangle(p1, p2, p3), ): check(c, check_attr=False)
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 parse_polygon(bar: OptionsBar, n: int) -> Polygon: points_arr: List[Tuple[int, int]] = [] for i in range(n): x = int(bar.v_options[i].top.input.text()) y = int(bar.v_options[i].bottom.input.text()) points_arr.append((x, y)) return Polygon(*points_arr)
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_integration_reduction_dynamic(): triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1)) facets = triangle.sides a, b = hyperplane_parameters(triangle)[0] x0 = facets[0].points[0] monomial_values = [[0, 0, 0, 0], [1, 0, 0, 5],\ [y, 0, 1, 15], [x, 1, 0, None]] assert integration_reduction_dynamic(facets, 0, a, b, x, 1, (x, y), 1,\ 0, 1, x0, monomial_values, 3) == Rational(25, 2) assert integration_reduction_dynamic(facets, 0, a, b, 0, 1, (x, y), 1,\ 0, 1, x0, monomial_values, 3) == 0
def is_filtered(self, data): """ {@inheritDocs} """ lon, lat = data.get_location() # Calculate how many times our test ray passes through the polygons for sub_polygon in self.sympy_sub_polygons: intersect_ray = Polygon((lon, lat), (lon + 999999, lat)) intersections = intersect_ray.intersection(sub_polygon) is_inside_polygon = False for intersection in intersections: # Inspect the intersection geometry if type(intersection) is Point: print "Point!" is_inside_polygon = not is_inside_polygon else: print "Line!" if is_inside_polygon: # If the point is inside this sub-polygon return False # Do not filter the point return True # Not in any of the sub-polygons so filter it
def test_geometry(): p1 = Point(1, 2) p2 = Point(2, 3) p3 = Point(0, 0) p4 = Point(0, 1) for c in (GeometryEntity, GeometryEntity(), Point, p1, Circle, Circle(p1, 2), Ellipse, Ellipse(p1, 3, 4), Line, Line(p1, p2), LinearEntity, LinearEntity(p1, p2), Ray, Ray(p1, p2), Segment, Segment(p1, p2), Polygon, Polygon(p1, p2, p3, p4), RegularPolygon, RegularPolygon(p1, 4, 5), Triangle): # XXX: Instance of Triangle hangs because of hasattr in check(). # Triangle(p1,p2,p3) check(c) pass
def test_issue_5933(): from sympy.geometry.polygon import (Polygon, RegularPolygon) from sympy.simplify.radsimp import denom x = Polygon(*RegularPolygon((0, 0), 1, 5).vertices).centroid.x assert abs(denom(x).n()) > 1e-12 assert abs(denom(radsimp(x))) > 1e-12 # in case simplify didn't handle it
def is_inside(this: Polygon, other: Polygon) -> bool: for point in this.vertices: if not other.encloses_point(point): return False return True
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, 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_integration_reduction(): triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1)) facets = triangle.sides a, b = hyperplane_parameters(triangle)[0] assert integration_reduction(facets, 0, a, b, 1, (x, y), 0) == 5 assert integration_reduction(facets, 0, a, b, 0, (x, y), 0) == 0
def randompoly(w,h,vertcount=4): return Polygon(*((randint(0,w-1), randint(0,h-1)) for i in range(vertcount)))
def construct(inpath, outpath): target = Image.open(inpath).convert("RGB") w,h = target.size print(w,h) images = [] avgcolors = [] for fp in getfiles(): #print(fp) img = Image.open(fp).convert("RGB") img = img.resize((w,h)) images.append(img) avgcolors.append(getcolor(img, Polygon((0,0),(0,w),(0,h),(w,h)))) print(f"{len(images)} images.") #TODO sort/index avgcolors for faster search, change images list order equally out = Image.new("RGBA", (w,h)) def sample(color, poly): least_dist = 3*256**2 least_img = None min_x, min_y, max_x, max_y = poly.bounds scalex = w/(max_x-min_x) scaley = h/(max_y-min_y) scaledpoly = Polygon(*[((x-min_x)*scalex, (y-min_y)*scaley) for x,y in polyverts(poly)]) for i, img in enumerate(images): #change poly pos, scale, rotate? #s = getcolor(img, poly) s = avgcolors[i] dist = sum([(a-b)**2 for a,b in zip(s,color)]) if dist < least_dist: least_dist = dist least_img = img ret = Image.new("RGBA", (w,h)) tri = polycrop(least_img, scaledpoly) if tri is None: raise Exception("Source directory contains no images") ret.paste(least_img, tri) #return least_img, polycrop(least_img, scaledpoly) ret = ret.resize((max_x-min_x,max_y-min_y)) return ret, (min_x, min_y) points = [(0,0), (w,0), (0,h), (w,h)] if CLOUD is None: for i in range(NUMPOINTS): points.append((randint(0,w-1), randint(0,h-1))) else: cloud = Image.open(CLOUD) for x in range(cloud.size[0]): for y in range(cloud.size[1]): if cloud.getpixel((x,y))[:3] != (255,255,255): points.append((x,y)) print(f"{len(points)} points.") if len(points) > NUMPOINTS: n = len(points)-NUMPOINTS print(f"Sampling {n} down to {NUMPOINTS}.") points = poprand(points, n) ADDPOINTS = 200 print(f"Adding {ADDPOINTS} random points.") for i in range(ADDPOINTS): points.append((randint(0,w-1), randint(0,h-1))) triangles = pytess.triangulate(points) #triangles = pytess.voronoi(points) try: for t, triangle in progressbar(list(enumerate(triangles)), "Triangles: ", 40): #print(f"{t}/{len(triangles)}") poly = Polygon(*triangle) targetcolor = getcolor(target, poly) #img, mask = sample(targetcolor, poly) #out.paste(img, (0,0), mask) img, coords = sample(targetcolor, poly) out.paste(img, coords, img) #img.save(f"conv/{t}-img.png") #mask.save(f"conv/{t}-mask.png") except KeyboardInterrupt: pass out.convert("RGB").save(outpath)
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}