Voronoi 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 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.verticesregioni ... end = sv.verticesregion(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()