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arXiv · 2207.13788

Voronoi Diagrams of Arbitrary Order on the Sphere

Mercè Claverol·Andrea de las Heras Parrilla·Clemens Huemer
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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.

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Mercè Claverol, Andrea de las Heras Parrilla, Clemens Huemer. 2022-07-27. Voronoi Diagrams of Arbitrary Order on the Sphere. https://arxiv.org/abs/2207.13788

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