Delaunay triangulations of generalized Bolza surfaces

TL;DR

This paper extends Bowyer's algorithm to compute Delaunay triangulations on generalized Bolza surfaces, enabling the calculation of systoles and efficient point set initialization on these hyperbolic surfaces.

Contribution

It introduces an extension of Bowyer's algorithm for generalized Bolza surfaces and provides methods to compute systoles and initialize point sets.

Findings

Successfully extended Bowyer's algorithm to generalized Bolza surfaces.

Computed the systole values for these surfaces.

Developed algorithms for selecting initial point sets.

Abstract

The Bolza surface can be seen as the quotient of the hyperbolic plane, represented by the Poincar\'e disk model, under the action of the group generated by the hyperbolic isometries identifying opposite sides of a regular octagon centered at the origin. We consider generalized Bolza surfaces $\mathbb{M}_g$, where the octagon is replaced by a regular $4g$-gon, leading to a genus $g$ surface. We propose an extension of Bowyer's algorithm to these surfaces. In particular, we compute the value of the systole of $\mathbb{M}_g$. We also propose algorithms computing small sets of points on $\mathbb{M}_g$ that are used to initialize Bowyer's algorithm.