Hyperbolic delaunay complexes and voronoi diagrams made practical

Abstract

We study Delaunay complexes and Voronoi diagrams in the Poincaré ball, a conformal model of the hyperbolic space, in any dimension. We elaborate on our earlier work on the space of spheres [18], giving a detailed description of algorithms. We also study algebraic and arithmetic issues, observing that only rational computations are needed. All proofs are based on geometric reasoning, they do not resort to any use of the analytic formula of the hyperbolic distance. This allows for an exact and efficient implementation in 2D. All degenerate cases are handled. The implementation will be submitted to the CGAL editorial board for future integration into the CGAL library.

Hyperbolic delaunay complexes and voronoi diagrams made practical

“…Both are static, i.e., they first compute the whole Euclidean triangulation before performing the extraction. The third one (omitted in this abstract, see [6]) is dynamic: it allows to add a point and to update the hyperbolic Delaunay complex while the Euclidean Delaunay triangulation is updated.…”

“…The construction of the Delaunay complex and the Voronoi diagram in H 2 was implemented using CGAL [14,40,25] (see [6] for a description). The implementation will soon be submitted for future integration in CGAL.…”