Voronoi Diagrams of Arbitrary Order on the Sphere

TL;DR

This paper generalizes the construction of spherical Voronoi diagrams from order 1 to any order k on the sphere, providing new formulas for their combinatorial properties and revealing additional structural features.

Contribution

It extends existing methods to arbitrary order k, derives formulas for vertices, edges, faces, and uncovers new properties of spherical Voronoi diagrams.

Findings

Formulas for vertices, edges, and faces of SV_k(U)

Construction method for arbitrary order k

SV_k(U) has a small orientable cycle double cover

Abstract

For a given set of points $U$ on a sphere $S$, the order $k$ spherical Voronoi diagram $SV_k(U)$ decomposes the surface of $S$ into regions whose points have the same $k$ nearest points of $U$. Hyeon-Suk Na, Chung-Nim Lee, and Otfried Cheong (Comput. Geom., 2002) applied inversions to construct $SV_1(U)$. We generalize their construction for spherical Voronoi diagrams from order $1$ to any order $k$. We use that construction to prove formulas for the numbers of vertices, edges, and faces in $SV_k(U)$. These formulas were not known before. We obtain several more properties for $SV_k(U)$, and we also show that $SV_k(U)$ has a small orientable cycle double cover.