The Stability of Delaunay Triangulations

TL;DR

This paper introduces a new concept of genericity for Delaunay triangulations, analyzing their stability under perturbations of points and metrics, and quantifying conditions for the triangulation to remain unchanged.

Contribution

It proposes a parametrized genericity notion ensuring Delaunay simplices are thick and studies the stability of Delaunay triangulations under perturbations.

Findings

Delaunay simplices are thick for delta-generic point sets.

Quantified bounds for perturbations preserving the triangulation.

Established conditions under which Delaunay triangulations remain stable.

Abstract

We introduce a parametrized notion of genericity for Delaunay triangulations which, in particular, implies that the Delaunay simplices of $\delta$-generic point sets are thick. Equipped with this notion, we study the stability of Delaunay triangulations under perturbations of the metric and of the vertex positions. We quantify the magnitude of the perturbations under which the Delaunay triangulation remains unchanged.