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# Thesis: Delaunay triangulations on hyperbolic surfaces

Ebbens, Y.M. (2017) Thesis: Delaunay triangulations on hyperbolic surfaces. Master's Thesis / Essay, Mathematics.

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## Abstract

There exist several algorithms to compute the Delaunay triangulation of a point set in a Euclidean space. In the literature the incremental algorithm has been extended to Euclidean orbifolds and hyperbolic surfaces, but it is only guaranteed to work in a finitely sheeted covering space. Bounds for the minimum number of sheets that are needed are only known for the Bolza surface. In this thesis we will first prove that the lower bound for the number of sheets is Omega(g) in general and Omega(g^3) if the surface can be represented by a fundamental region with 4g concircular vertices, where g is the genus of the surface. Then we will prove that there does not exist an upper bound for the number of sheets necessary for surfaces of genus 2. The systole of a surface plays an important role in determining the complexity of triangulating the corresponding surface. We will state a conjecture for the systole of hyperbolic surfaces of genus g represented by regular 4g-gons. Finally, to avoid many sheeted covering spaces a different method uses a well chosen set of points, which guarantees that the output is simplicial. We will show a lower bound for the number of points of such a point set, which is of order Omega(sqrt(g)) in general and of order Omega(g) if the systole of a family of surfaces is bounded.

Item Type: Thesis (Master's Thesis / Essay) Mathematics Master's Thesis / Essay English 15 Feb 2018 08:31 15 Feb 2018 08:31 http://fse.studenttheses.ub.rug.nl/id/eprint/15727

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