Hyperbolic Voronoi Diagram

Abstract

Voronoi diagrams are among the most extensively studied objects in computational geometry with useful applications in different areas of science. To understand impacts of non-Euclidean geometry on computational geometry, this paper investigates the Voronoi diagram in hyperbolic space specially the one in the Poincare hyperbolic disk, which is a 2-dimensional manifold with negative curvature. We first prove some lemma in Poincare hyperbolic disk and then give an incremental algorithm to construct Voronoi diagram.