Delaunay triangulations of generalized Bolza surfaces

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.