(ph 5PSrSSKrSSKrSSKJr SSKJr S/rSr "SS5r g) z3 Spherical Voronoi Code .. versionadded:: 0.18.0 N)_voronoi)cKDTreeSphericalVoronoic [RRU5nS[R"SUSS2S4USS2S45[R"SUSS2S4USS2S45-[R"SUSS2S4USS2S45--n[R"S[R "X5-5$)zCalculates the solid angles of plane triangles. Implements the method of Van Oosterom and Strackee [VanOosterom]_ with some modifications. Assumes that input points have unit norm.rij,ij->iNr)nplinalgdeteinsumabsarctan2)R numerator denominators S/var/www/html/venv/lib/python3.13/site-packages/scipy/spatial/_spherical_voronoi.pycalculate_solid_anglesrs a IryyQq!tWa1g>yyQq!tWa1g>?yyQq!tWa1g>?@K 66!bjj88 99c@\rSrSrSrS SjrSrSrSrSr S r S r g) r$aVoronoi diagrams on the surface of a sphere. .. versionadded:: 0.18.0 Parameters ---------- points : ndarray of floats, shape (npoints, ndim) Coordinates of points from which to construct a spherical Voronoi diagram. radius : float, optional Radius of the sphere (Default: 1) center : ndarray of floats, shape (ndim,) Center of sphere (Default: origin) threshold : float Threshold for detecting duplicate points and mismatches between points and sphere parameters. (Default: 1e-06) Attributes ---------- points : double array of shape (npoints, ndim) the points in `ndim` dimensions to generate the Voronoi diagram from radius : double radius of the sphere center : double array of shape (ndim,) center of the sphere vertices : double array of shape (nvertices, ndim) Voronoi vertices corresponding to points regions : list of list of integers of shape (npoints, _ ) the n-th entry is a list consisting of the indices of the vertices belonging to the n-th point in points Methods ------- calculate_areas Calculates the areas of the Voronoi regions. For 2D point sets, the regions are circular arcs. The sum of the areas is ``2 * pi * radius``. For 3D point sets, the regions are spherical polygons. The sum of the areas is ``4 * pi * radius**2``. Raises ------ ValueError If there are duplicates in `points`. If the provided `radius` is not consistent with `points`. Notes ----- The spherical Voronoi diagram algorithm proceeds as follows. The Convex Hull of the input points (generators) is calculated, and is equivalent to their Delaunay triangulation on the surface of the sphere [Caroli]_. The Convex Hull neighbour information is then used to order the Voronoi region vertices around each generator. The latter approach is substantially less sensitive to floating point issues than angle-based methods of Voronoi region vertex sorting. Empirical assessment of spherical Voronoi algorithm performance suggests quadratic time complexity (loglinear is optimal, but algorithms are more challenging to implement). References ---------- .. [Caroli] Caroli et al. Robust and Efficient Delaunay triangulations of points on or close to a sphere. Research Report RR-7004, 2009. .. [VanOosterom] Van Oosterom and Strackee. The solid angle of a plane triangle. IEEE Transactions on Biomedical Engineering, 2, 1983, pp 125--126. See Also -------- Voronoi : Conventional Voronoi diagrams in N dimensions. Examples -------- Do some imports and take some points on a cube: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.spatial import SphericalVoronoi, geometric_slerp >>> from mpl_toolkits.mplot3d import proj3d >>> # set input data >>> points = np.array([[0, 0, 1], [0, 0, -1], [1, 0, 0], ... [0, 1, 0], [0, -1, 0], [-1, 0, 0], ]) Calculate the spherical Voronoi diagram: >>> radius = 1 >>> center = np.array([0, 0, 0]) >>> sv = SphericalVoronoi(points, radius, center) Generate plot: >>> # sort vertices (optional, helpful for plotting) >>> sv.sort_vertices_of_regions() >>> t_vals = np.linspace(0, 1, 2000) >>> fig = plt.figure() >>> ax = fig.add_subplot(111, projection='3d') >>> # plot the unit sphere for reference (optional) >>> u = np.linspace(0, 2 * np.pi, 100) >>> v = np.linspace(0, np.pi, 100) >>> x = np.outer(np.cos(u), np.sin(v)) >>> y = np.outer(np.sin(u), np.sin(v)) >>> z = np.outer(np.ones(np.size(u)), np.cos(v)) >>> ax.plot_surface(x, y, z, color='y', alpha=0.1) >>> # plot generator points >>> ax.scatter(points[:, 0], points[:, 1], points[:, 2], c='b') >>> # plot Voronoi vertices >>> ax.scatter(sv.vertices[:, 0], sv.vertices[:, 1], sv.vertices[:, 2], ... c='g') >>> # indicate Voronoi regions (as Euclidean polygons) >>> for region in sv.regions: ... n = len(region) ... for i in range(n): ... start = sv.vertices[region][i] ... end = sv.vertices[region][(i + 1) % n] ... result = geometric_slerp(start, end, t_vals) ... ax.plot(result[..., 0], ... result[..., 1], ... result[..., 2], ... c='k') >>> ax.azim = 10 >>> ax.elev = 40 >>> _ = ax.set_xticks([]) >>> _ = ax.set_yticks([]) >>> _ = ax.set_zticks([]) >>> fig.set_size_inches(4, 4) >>> plt.show() NcUc [S5e[U5Ul[R"U5R [R 5UlURRSUl Uc&[R"UR5Ul O[R"U[S9Ul [RRURURS- X@R-S9UlURUR:a[SUR35e[UR5R!X@R-5(a [S5e[RR#URUR- SS9n[R$"XPR- 5R'5nXdUR-:a [S 5eUR)5 g) NzM`radius` is `None`. Please provide a floating point number (i.e. `radius=1`).r)dtyper)tolz&Rank of input points must be at least zDuplicate generators present.axisz$Radius inconsistent with generators.) ValueErrorfloatradiusr arrayastypefloat64pointsshape_dimzeroscenterr matrix_rank_rankr query_pairsnormrmax_calc_vertices_regions)selfr#rr' thresholdradiimax_discrepancys r__init__SphericalVoronoi.__init__s} >23 3Fm hhv&--bjj9 KK%%a( >((499-DK((67DKYY**4;;Q+G/8;;/F+H :: !Edii[QR R 4;;  + +I ,C D D<= = t{{T[[8qA&&!4599; $++5 5CD D ##%rc [RRUR5nURUR SS2SS24-UR -UlURUl [R"[UR55n[R"U/UR-5R5nURR5n[R "USS9nX5R#[R$5n[R&"[R("US-55n[+[U5S- 5Vs/sHn[-XgUXxS-5PM n nXlgs snf)a  Calculates the Voronoi vertices and regions of the generators stored in self.points. The vertices will be stored in self.vertices and the regions in self.regions. This algorithm was discussed at PyData London 2015 by Tyler Reddy, Ross Hemsley and Nikolai Nowaczyk N mergesort)kindr)scipyspatial ConvexHullr#r equationsr'vertices simplices _simplicesr arangelen column_stackr%ravelargsortr!intpcumsumbincountrangelistregions) r.convsimplex_indices tri_indices point_indicesindicesflattened_groups intervalsigroupss rr-'SphericalVoronoi._calc_vertices_regionss5}}'' 4 dnnQV&<rIr.s rrW)SphericalVoronoi.sort_vertices_of_regionss38 99>MN N))$//4<<HrcUR5 URVs/sHn[U5PM nn[R"U5nUSn[ U5VVVs/sHupV[ U5HnUPM M nnnn[R"URVV s/sH oHoPM M sn n5n [R"U S5n [R"US5n SU S'XS- X'URUR- UR- n URUR- UR- n[R"XXX/5RUSS45n[U5n[R"U5US- nUSS===USS-sss&UURS--$s snfs snnnfs sn nf)Nr5rrrUr )rWrIr@r rE enumeraterGr rollr#r'rr<hstackreshaper)r.regionsizescsizes num_regionsrQsizejrMrnbrs1nbrs2rN pnormalized vnormalized trianglestriangle_solid_angles solid_angless r_calculate_areas_3d$SphericalVoronoi._calculate_areas_3ds %%'+/<<8<V<85!Rj +4E*:/*:wq"'+Q"-*: /$,,G,1!!,GHq!''&!$ z*{{T[[0DKK? }}t{{2dkkA II{9*1*1!"#*';1*="> !7y Ayy!67 C QRL"--dkk1n,,G9/HsG !G$G c.URURUR- n[R"USS2S4USS2S4- S-SS9n[R "SUSUR S--- - 5nUR U-n[R"[R"SUSS2S4URUR- 55n[R"US:5nS[R-UR -XF- XF'U$)Nrrr rr) r#r>r'r sumarccosrsignr r<wherepi)r.arcsdthetaareassignsrNs r_calculate_areas_2d$SphericalVoronoi._calculate_areas_2d.s{{4??+dkk9 FFDAJad+1 : !qA)9$:;<= e# *d1a4j.2mmdkk.IKL((519%RUUT[[05>A rcURS:XaUR5$URS:XaUR5$[S5e)anCalculates the areas of the Voronoi regions. For 2D point sets, the regions are circular arcs. The sum of the areas is ``2 * pi * radius``. For 3D point sets, the regions are spherical polygons. The sum of the areas is ``4 * pi * radius**2``. .. versionadded:: 1.5.0 Returns ------- areas : double array of shape (npoints,) The areas of the Voronoi regions. r rUz'Only supported for 2D and 3D point sets)r%rzrmrVrXs rcalculate_areas SphericalVoronoi.calculate_areas@sD 99>++- - YY!^++- -EF Fr)r%r)r>r'r#rrIr<)rNgư>) __name__ __module__ __qualname____firstlineno____doc__r2r-rWrmrzr}__static_attributes__rrrr$s,AD&>BI@%-N$Gr) rnumpyr r8r scipy.spatialr__all__rrrrrrs3 !   :qGqGr