Robust Construction of Voronoi Diagrams of Spherical Balls in Three-Dimensional Space

Abstract

Voronoi diagrams are useful for spatial reasoning among particles and there are many prior studies on their construction. However, most prior works were for the ordinary Voronoi diagrams of points in R2 and R3. Here we propose a robust algorithm for constructing the Voronoi diagram of spherical balls in R3, where the output is guaranteed to be at least topologically consistent. This topologyoriented incremental algorithm constructs the Voronoi diagram in O(n3) time in the worst case, whereas its empirical time behavior shows a strong linear fashion for all data we tested. The proposed algorithm is the three-dimensional generalization of its counterpart in the plane. It is implemented, thoroughly tested, and compared with two well-known programs. Current implementation processes approximately 350 balls per second using one core of ordinary desktop computer. This paper also contains an extensive review on the Voronoi diagrams of 2D circular disks and 3D spherical balls. We anticipate the algorithm will be widely used to solve application problems from many disciplines in science and engineering. The library is freely available from the github repository. (c) 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND