Towards exact numerical Voronoi diagrams

Voronoi diagrams are extremely versatile as a data structure for many geometric applications. Computing this diagram “exactly” for a polyhedral set in 3D has been a quest of computational geometers for over two decades; this quest is still unrealized. We will locate the difficulty in this quest, thanks to a recent result of Everett et al (2009). More generally, it points to the need for alternative computational models, and other notions of exactness. In this paper, we consider an alternative approach based on the well-known Subdivision Paradigm. A brief review of such algorithms for Voronoi diagrams is given. Our unique emphasis is the use of purely numerical primitives. We avoid exact (algebraic) primitives because (1) they are hard to implement correctly, and (2) they fail to take full advantage of the resolution-limited properties of subdivision. We encapsulate our numerical approach using the concept of soft primitives that conservatively converge to the exact ones in the limit. We illustrate our approach by designing the first purely numerical algorithm for the Voronoi complex of a nondegenerate polygonal set. We also discuss the critical role of filters in such algorithms. A preliminary version of our algorithm has been implemented.