Flipping Geometric Triangulations on Hyperbolic Surfaces

TL;DR

This paper proves the connectivity of the flip graph of geometric triangulations on hyperbolic surfaces and provides bounds on the flips needed to reach Delaunay triangulations, advancing understanding of surface triangulation transformations.

Contribution

It establishes the connectivity of the flip graph for geometric triangulations on hyperbolic surfaces and bounds the flips required to achieve Delaunay triangulations.

Findings

Flip graph of geometric triangulations is connected.

Upper bounds on edge flips to reach Delaunay triangulations.

Connectivity holds for flat tori and closed hyperbolic surfaces.

Abstract

We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint locally geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a closed hyperbolic surface is connected. We give upper bounds on the number of edge flips that are necessary to transform any geometric triangulation on such a surface into a Delaunay triangulation.