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""" 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. #
""" Determines distance of generators from theoretical sphere surface.
""" actual_squared_radii = (((points[...,0] - center[0]) ** 2) + ((points[...,1] - center[1]) ** 2) + ((points[...,2] - center[2]) ** 2)) max_discrepancy = (np.sqrt(actual_squared_radii) - radius).max() return abs(max_discrepancy)
""" Calculates the cirumcenters of the circumspheres of tetrahedrons.
An implementation based on http://mathworld.wolfram.com/Circumsphere.html
Parameters ---------- tetrahedrons : an array of shape (N, 4, 3) consisting of N tetrahedrons defined by 4 points in 3D
Returns ---------- circumcenters : an array of shape (N, 3) consisting of the N circumcenters of the tetrahedrons in 3D
"""
num = tetrahedrons.shape[0] a = np.concatenate((tetrahedrons, np.ones((num, 4, 1))), axis=2)
sums = np.sum(tetrahedrons ** 2, axis=2) d = np.concatenate((sums[:, :, np.newaxis], a), axis=2)
dx = np.delete(d, 1, axis=2) dy = np.delete(d, 2, axis=2) dz = np.delete(d, 3, axis=2)
dx = np.linalg.det(dx) dy = -np.linalg.det(dy) dz = np.linalg.det(dz) a = np.linalg.det(a)
nominator = np.vstack((dx, dy, dz)) denominator = 2*a return (nominator / denominator).T
""" Projects the elements of points onto the sphere defined by center and radius.
Parameters ---------- points : array of floats of shape (npoints, ndim) consisting of the points in a space of dimension ndim center : array of floats of shape (ndim,) the center of the sphere to project on radius : float the radius of the sphere to project on
returns: array of floats of shape (npoints, ndim) the points projected onto the sphere """
lengths = scipy.spatial.distance.cdist(points, np.array([center])) return (points - center) / lengths * radius + center
""" Voronoi diagrams on the surface of a sphere.
.. versionadded:: 0.18.0
Parameters ---------- points : ndarray of floats, shape (npoints, 3) Coordinates of points to construct a spherical Voronoi diagram from radius : float, optional Radius of the sphere (Default: 1) center : ndarray of floats, shape (3,) 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, 3) the points in 3D to generate the Voronoi diagram from radius : double radius of the sphere Default: None (forces estimation, which is less precise) center : double array of shape (3,) center of the sphere Default: None (assumes sphere is centered at origin) vertices : double array of shape (nvertices, 3) 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]_. A 3D Delaunay tetrahedralization is obtained by including the origin of the coordinate system as the fourth vertex of each simplex of the Convex Hull. The circumcenters of all tetrahedra in the system are calculated and projected to the surface of the sphere, producing the Voronoi vertices. The Delaunay tetrahedralization 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.
The surface area of spherical polygons is calculated by decomposing them into triangles and using L'Huilier's Theorem to calculate the spherical excess of each triangle [Weisstein]_. The sum of the spherical excesses is multiplied by the square of the sphere radius to obtain the surface area of the spherical polygon. For nearly-degenerate spherical polygons an area of approximately 0 is returned by default, rather than attempting the unstable calculation.
Empirical assessment of spherical Voronoi algorithm performance suggests quadratic time complexity (loglinear is optimal, but algorithms are more challenging to implement). The reconstitution of the surface area of the sphere, measured as the sum of the surface areas of all Voronoi regions, is closest to 100 % for larger (>> 10) numbers of generators.
References ----------
.. [Caroli] Caroli et al. Robust and Efficient Delaunay triangulations of points on or close to a sphere. Research Report RR-7004, 2009. .. [Weisstein] "L'Huilier's Theorem." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/LHuiliersTheorem.html
See Also -------- Voronoi : Conventional Voronoi diagrams in N dimensions.
Examples --------
>>> 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], ]) >>> center = np.array([0, 0, 0]) >>> radius = 1 >>> # calculate spherical Voronoi diagram >>> sv = SphericalVoronoi(points, radius, center) >>> # sort vertices (optional, helpful for plotting) >>> sv.sort_vertices_of_regions() >>> # generate plot >>> 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()
"""
""" Initializes the object and starts the computation of the Voronoi diagram.
points : The generator points of the Voronoi diagram assumed to be all on the sphere with radius supplied by the radius parameter and center supplied by the center parameter. radius : The radius of the sphere. Will default to 1 if not supplied. center : The center of the sphere. Will default to the origin if not supplied. """
self.points = points if np.any(center): self.center = center else: self.center = np.zeros(3) if radius: self.radius = radius else: self.radius = 1
if pdist(self.points).min() <= threshold * self.radius: raise ValueError("Duplicate generators present.")
max_discrepancy = sphere_check(self.points, self.radius, self.center) if max_discrepancy >= threshold * self.radius: raise ValueError("Radius inconsistent with generators.") self.vertices = None self.regions = None self._tri = None self._calc_vertices_regions()
""" 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 """
# perform 3D Delaunay triangulation on data set # (here ConvexHull can also be used, and is faster) self._tri = scipy.spatial.ConvexHull(self.points)
# add the center to each of the simplices in tri to get the same # tetrahedrons we'd have gotten from Delaunay tetrahedralization # tetrahedrons will have shape: (2N-4, 4, 3) tetrahedrons = self._tri.points[self._tri.simplices] tetrahedrons = np.insert( tetrahedrons, 3, np.array([self.center]), axis=1 )
# produce circumcenters of tetrahedrons from 3D Delaunay # circumcenters will have shape: (2N-4, 3) circumcenters = calc_circumcenters(tetrahedrons)
# project tetrahedron circumcenters to the surface of the sphere # self.vertices will have shape: (2N-4, 3) self.vertices = project_to_sphere( circumcenters, self.center, self.radius )
# calculate regions from triangulation # simplex_indices will have shape: (2N-4,) simplex_indices = np.arange(self._tri.simplices.shape[0]) # tri_indices will have shape: (6N-12,) tri_indices = np.column_stack([simplex_indices, simplex_indices, simplex_indices]).ravel() # point_indices will have shape: (6N-12,) point_indices = self._tri.simplices.ravel()
# array_associations will have shape: (6N-12, 2) array_associations = np.dstack((point_indices, tri_indices))[0] array_associations = array_associations[np.lexsort(( array_associations[...,1], array_associations[...,0]))] array_associations = array_associations.astype(np.intp)
# group by generator indices to produce # unsorted regions in nested list groups = [] for k, g in itertools.groupby(array_associations, lambda t: t[0]): groups.append(list(list(zip(*list(g)))[1]))
self.regions = groups
""" 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 projected circumcenter of the tetrahedron obtained by the k-th triangle in _tri.simplices (and the origin). 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. """
_voronoi.sort_vertices_of_regions(self._tri.simplices, self.regions) |