Hyperbolic Voronoi Diagram: A Fast Method

Voronoi diagrams have useful applications in various fields and are one of the most fundamental concepts in computational geometry. Although Voronoi diagrams in the plane have been studied extensively, using different notions of sites and metrics, little is known for other geometric spaces. In this paper, we present a simple method to construct the Voronoi diagram of a set of points in the Poincare hyperbolic disk, which is a 2-dimensional manifold with negative curvature. Our trick is to define and use some well-formed geometric maps which take care of connection between the Euclidean plane and Poincare hyperbolic disk. Finally, we give a brief report of our implementation.