Delaunay Triangulations of Degenerate Point Sets

TL;DR

This paper investigates how small random perturbations affect the Delaunay triangulation of degenerate point sets, focusing on grid and polygon configurations, with empirical and theoretical insights into the resulting triangulations.

Contribution

It provides a detailed analysis of the effects of normally distributed perturbations on degenerate point sets' Delaunay triangulations, especially for grid and polygon configurations.

Findings

Perturbations can lead to unique DTs in degenerate cases.

Empirical findings reveal surprising properties of perturbed DTs.

Theoretical explanations support observed phenomena.

Abstract

The Delaunay triangulation (DT) is one of the most common and useful triangulations of point sets $P$ in the plane. DT is not unique when $P$ is degenerate, specifically when it contains quadruples of co-circular points. One way to achieve uniqueness is by applying a small (or infinitesimal) perturbation to $P$. We consider a specific perturbation of such degenerate sets, in which the coordinates of each point are independently perturbed by normally distributed small quantities, and investigate the effect of such perturbations on the DT of the set. We focus on two special configurations, where (1) the points of $P$ form a uniform grid, and (2) the points of $P$ are vertices of a regular polygon. We present interesting (and sometimes surprising) empirical findings and properties of the perturbed DTs for these cases, and give theoretical explanations to some of them.