package scipy

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

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.

See Also -------- Voronoi : Conventional Voronoi diagrams in N dimensions.

Examples -------- Do some imports and take some points on a cube:

>>> from matplotlib import colors >>> from mpl_toolkits.mplot3d.art3d import Poly3DCollection >>> import matplotlib.pyplot as plt >>> from scipy.spatial import SphericalVoronoi >>> 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() >>> 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: ... random_color = colors.rgb2hex(np.random.rand(3)) ... polygon = Poly3DCollection(sv.vertices[region], alpha=1.0) ... polygon.set_color(random_color) ... ax.add_collection3d(polygon) >>> plt.show()

Sort indices of the vertices to be (counter-)clockwise ordered.

Raises ------ TypeError If the points are not three-dimensional.

Notes ----- For each region in regions, it sorts the indices of the Voronoi vertices such that the resulting points are in a clockwise or counterclockwise order around the generator point.

This is done as follows: Recall that the n-th region in regions surrounds the n-th generator in points and that the k-th Voronoi vertex in vertices is the circumcenter of the k-th triangle in _tri.simplices. For each region n, we choose the first triangle (=Voronoi vertex) in _tri.simplices and a vertex of that triangle not equal to the center n. These determine a unique neighbor of that triangle, which is then chosen as the second triangle. The second triangle will have a unique vertex not equal to the current vertex or the center. This determines a unique neighbor of the second triangle, which is then chosen as the third triangle and so forth. We proceed through all the triangles (=Voronoi vertices) belonging to the generator in points and obtain a sorted version of the vertices of its surrounding region.

Attribute points: get value or raise Not_found if None.

Attribute points: get value as an option.

Attribute radius: get value or raise Not_found if None.

Attribute radius: get value as an option.

Attribute center: get value or raise Not_found if None.

Attribute center: get value as an option.

Attribute vertices: get value or raise Not_found if None.

Attribute vertices: get value as an option.

Attribute regions: get value or raise Not_found if None.

Attribute regions: get value as an option.

Print the object to a human-readable representation.

Print the object to a human-readable representation.

Pretty-print the object to a formatter.