Delaunay triangulations of hyperbolic surfaces

Triangulations are among the most important and well-studied objects in computational geometry. A triangulation is a subdivision of a surface into triangles. This allows the use of computer algorithms to analyze the geometry of the surface or perform simulations. A Delaunay triangulation is a particular kind of triangulation that is often used because of its favorable properties.

In this thesis we studied Delaunay triangulations of hyperbolic surfaces. Hyperbolic surfaces are surfaces with a constant negative curvature and can be used to model shapes or structures that, intuitively speaking, cannot be "flattened" in the Euclidean plane.

In the thesis we describe the properties of a specific class of hyperbolic surfaces that allow a well-known algorithm for computing Delaunay triangulations to be generalized to these surfaces. In particular, we compute the systole of these surfaces, which is an important parameter in the algorithm. Moreover, we provide upper and lower bounds for the minimal number of vertices of Delaunay triangulations of hyperbolic surfaces and show that these bounds are asymptotically optimal.