Voronoi Diagrams of Arbitrary Order on the Sphere

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.