""" Spherical Voronoi Code .. versionadded:: 0.18.0 """ # # Copyright (C) Tyler Reddy, Ross Hemsley, Edd Edmondson, # Nikolai Nowaczyk, Joe Pitt-Francis, 2015. # # Distributed under the same BSD license as SciPy. # import warnings import numpy as np import scipy from . import _voronoi from scipy.spatial import cKDTree __all__ = ['SphericalVoronoi'] def calculate_solid_angles(R): """Calculates 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.""" # Original method uses a triple product `R1 . (R2 x R3)` for the numerator. # This is equal to the determinant of the matrix [R1 R2 R3], which can be # computed with better stability. numerator = np.linalg.det(R) denominator = 1 + (np.einsum('ij,ij->i', R[:, 0], R[:, 1]) + np.einsum('ij,ij->i', R[:, 1], R[:, 2]) + np.einsum('ij,ij->i', R[:, 2], R[:, 0])) return np.abs(2 * np.arctan2(numerator, denominator)) class SphericalVoronoi: """ 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.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() """ def __init__(self, points, radius=1, center=None, threshold=1e-06): if radius is None: radius = 1. warnings.warn('`radius` is `None`. ' 'This will raise an error in a future version. ' 'Please provide a floating point number ' '(i.e. `radius=1`).', DeprecationWarning) self.radius = float(radius) self.points = np.array(points).astype(np.double) self._dim = len(points[0]) if center is None: self.center = np.zeros(self._dim) else: self.center = np.array(center, dtype=float) # test degenerate input self._rank = np.linalg.matrix_rank(self.points - self.points[0], tol=threshold * self.radius) if self._rank < self._dim: raise ValueError("Rank of input points must be at least {0}".format(self._dim)) if cKDTree(self.points).query_pairs(threshold * self.radius): raise ValueError("Duplicate generators present.") radii = np.linalg.norm(self.points - self.center, axis=1) max_discrepancy = np.abs(radii - self.radius).max() if max_discrepancy >= threshold * self.radius: raise ValueError("Radius inconsistent with generators.") self._calc_vertices_regions() def _calc_vertices_regions(self): """ 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 """ # get Convex Hull conv = scipy.spatial.ConvexHull(self.points) # get circumcenters of Convex Hull triangles from facet equations # for 3D input circumcenters will have shape: (2N-4, 3) self.vertices = self.radius * conv.equations[:, :-1] + self.center self._simplices = conv.simplices # calculate regions from triangulation # for 3D input simplex_indices will have shape: (2N-4,) simplex_indices = np.arange(len(self._simplices)) # for 3D input tri_indices will have shape: (6N-12,) tri_indices = np.column_stack([simplex_indices] * self._dim).ravel() # for 3D input point_indices will have shape: (6N-12,) point_indices = self._simplices.ravel() # for 3D input indices will have shape: (6N-12,) indices = np.argsort(point_indices, kind='mergesort') # for 3D input flattened_groups will have shape: (6N-12,) flattened_groups = tri_indices[indices].astype(np.intp) # intervals will have shape: (N+1,) intervals = np.cumsum(np.bincount(point_indices + 1)) # split flattened groups to get nested list of unsorted regions groups = [list(flattened_groups[intervals[i]:intervals[i + 1]]) for i in range(len(intervals) - 1)] self.regions = groups def sort_vertices_of_regions(self): """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 self._simplices. For each region n, we choose the first triangle (=Voronoi vertex) in self._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. """ if self._dim != 3: raise TypeError("Only supported for three-dimensional point sets") _voronoi.sort_vertices_of_regions(self._simplices, self.regions) def _calculate_areas_3d(self): self.sort_vertices_of_regions() sizes = [len(region) for region in self.regions] csizes = np.cumsum(sizes) num_regions = csizes[-1] # We create a set of triangles consisting of one point and two Voronoi # vertices. The vertices of each triangle are adjacent in the sorted # regions list. point_indices = [i for i, size in enumerate(sizes) for j in range(size)] nbrs1 = np.array([r for region in self.regions for r in region]) # The calculation of nbrs2 is a vectorized version of: # np.array([r for region in self.regions for r in np.roll(region, 1)]) nbrs2 = np.roll(nbrs1, 1) indices = np.roll(csizes, 1) indices[0] = 0 nbrs2[indices] = nbrs1[csizes - 1] # Normalize points and vertices. pnormalized = (self.points - self.center) / self.radius vnormalized = (self.vertices - self.center) / self.radius # Create the complete set of triangles and calculate their solid angles triangles = np.hstack([pnormalized[point_indices], vnormalized[nbrs1], vnormalized[nbrs2] ]).reshape((num_regions, 3, 3)) triangle_solid_angles = calculate_solid_angles(triangles) # Sum the solid angles of the triangles in each region solid_angles = np.cumsum(triangle_solid_angles)[csizes - 1] solid_angles[1:] -= solid_angles[:-1] # Get polygon areas using A = omega * r**2 return solid_angles * self.radius**2 def _calculate_areas_2d(self): # Find start and end points of arcs arcs = self.points[self._simplices] - self.center # Calculate the angle subtended by arcs cosine = np.einsum('ij,ij->i', arcs[:, 0], arcs[:, 1]) sine = np.abs(np.linalg.det(arcs)) theta = np.arctan2(sine, cosine) # Get areas using A = r * theta areas = self.radius * theta # Correct arcs which go the wrong way (single-hemisphere inputs) signs = np.sign(np.einsum('ij,ij->i', arcs[:, 0], self.vertices - self.center)) indices = np.where(signs < 0) areas[indices] = 2 * np.pi * self.radius - areas[indices] return areas def calculate_areas(self): """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`. .. versionadded:: 1.5.0 Returns ------- areas : double array of shape (npoints,) The areas of the Voronoi regions. """ if self._dim == 2: return self._calculate_areas_2d() elif self._dim == 3: return self._calculate_areas_3d() else: raise TypeError("Only supported for 2D and 3D point sets")