Edge topology construction of Voronoi diagrams of spheres in non-general position

Abstract

Although 3D Voronoi diagrams and medial axis transforms have numerous applications in biology, robotics, and manufacturing, most researchers use Voronoi diagrams of points instead of the true 3D input geometry, due to issues of robustness and scalability. In this paper, we present a robust sample-based GPU algorithm for calculating the full topology of Voronoi diagrams of non-general position spheres. Prior work demonstrated that the presence, geometry, and combinatorial basis of spheres that contribute to Voronoi vertices can be efficiently computed by shooting rays from each input sphere, mapping ray intersections with the nearest bisector surface to parametric bounding cubes, and analyzing the results. In this paper, we propose an algorithm on this parametric bounding cube to compute Voronoi edges in addition to the vertices. We successfully extract the full topology of the Voronoi diagram, including special cases such as isolated Voronoi edges that do not contain Voronoi vertices, more than three Voronoi edges emanating from a Voronoi vertex, and Voronoi edges that are shared by more than three Voronoi cells. Our GPU implementation efficiently and robustly handles all input, whether in general or non-general position, and finds all Voronoi vertices and edges, modulo the sampling density, including isolated disconnected edges.