Section: New Results

Non-Euclidean Computational Geometry

Participants : Iordan Iordanov, Monique Teillaud, Gert Vegter.

The work on Delaunay triangulations of flat $d$-dimensional orbifolds, started several years ago in the Geometrica project team in Sophia Antipolis, was finalized this year [13].

We give a definition of the Delaunay triangulation of a point set in a closed Euclidean $d$-manifold, i.e. a compact quotient space of the Euclidean space for a discrete group of isometries (a so-called Bieberbach group or crystallographic group). We describe a geometric criterion to check whether a partition of the manifold actually forms a triangulation (which subsumes that it is a simplicial complex). We provide an incremental algorithm to compute the Delaunay triangulation of the manifold defined by a given set of input points, if it exists. Otherwise, the algorithm returns the Delaunay triangulation of a finite-sheeted covering space of the manifold. The algorithm has optimal randomized worst-case time and space complexity. It extends to closed Euclidean orbifolds. To the best of our knowledge, this is the first general result on this topic.

Motivated by applications in various fields, some packages to compute periodic Delaunay triangulations in the 2D and 3D Euclidean spaces have been introduced in the CGAL library and have attracted a number of users. To the best of our knowledge, no software is available to compute periodic triangulations in a hyperbolic space, though they are also used in diverse fields, such as physics, solid modeling, cosmological models, neuromathematics.

This would be a natural extension: 2D Euclidean periodic triangulations can be seen as triangulations of the two-dimensional (flat) torus of genus one; similarly, periodic triangulations in the hyperbolic plane can be seen as triangulations of hyperbolic surfaces. A closed orientable hyperbolic surface is the quotient of the hyperbolic plane under the action of a Fuchsian group only containing hyperbolic translations. Intuition is challenged there, in particular because such groups are non-Abelian in general.

We have obtained some theoretical results on Delaunay triangulations of general closed orientable hyperbolic surfaces, and we have investigated algorithms in the specific case of the Bolza surface, a hyperbolic surface with the simplest possible topology, as it is homeomorphic to a genus-two torus [20]. We are now studying more practical aspects and we propose a first implementation of an incremental construction of Delaunay triangulations of the Bolza surface [30].