Voronoi diagrams on the sphere

Authors:
Hyeon-Suk Na;Chung-Nim Lee;Otfried Cheong
Affiliations:
Department of Mathematics, Pohang University of Science and Technology, South Korea;Department of Mathematics, Pohang University of Science and Technology, South Korea;Institute of Information and Computing Sciences, Utrecht University, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands
Venue:
Computational Geometry: Theory and Applications
Year:
2002

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Abstract

Given a set of compact sites on a sphere, we show that their spherical Voronoi diagram can be computed by computing two planar Voronoi diagrams of suitably transformed sites in the plane. We also show that a planar furthest-site Voronoi diagram can always be obtained as a portion of a nearest-site Voronoi diagram of a set of transformed sites. Two immediate applications are an O(n logn) algorithm for the spherical Voronoi diagram of a set of circular arcs on the sphere, and an O(n logn) algorithm for the furthest-site Voronoi diagram for a set of circular arcs in the plane.