Computation of Geodesic Voronoi Diagrams in Riemannian 3-Space using Medial Equations

The Voronoi diagram has been investigated intensively throughout the last decades. This has been done not only in the context of Euclidean geometry but also in curved spaces. Except for [KWR97] these methods typically make use of some fast marching cube algorithms. In this work we will focus on the computation of Voronoi diagrams including Voronoi objects that are contained in a Riemannian manifold M. Further, we assume throughout this paper that M has a differentiable structure consisting of smooth parametrisation functions fi, i 2 I. This is the reason why the approach presented in this work differs from the aforementioned algorithms. More accurate algorithms can be obtained by using to some medial equations that heavily involve normal coordinates. This approach relies on the precise computation of shortest joins of any two given points , q 2 M. For these computations we did not apply shooting methods or related methods. Instead, we used a new perturbation method that operates on a family of deformed manifolds Mt, assuming that M0 has constant sectional curvature. To reduce time and space complexity of the introduced algorithm we suggest to use a randomised incremental construction scheme (RICS). Our approach assumes that those points fulfil a general position requirement for computing the geodesic Voronoi diagram for a set of points. Finally results of some computed Voronoi diagrams will be presented.