Delaunay Triangulations of Points on Circles

TL;DR

This paper introduces a canonical, well-defined Delaunay triangulation for points on circles using a max-min angle approach, addressing issues with concyclic points and providing an efficient computation method.

Contribution

It proposes a new canonical triangulation method for concyclic points based on max-min angles, ensuring uniqueness and computational efficiency.

Findings

Defines a canonical triangulation for concyclic points

Ensures uniqueness except for symmetric quadruples

Provides a quasi-linear time algorithm for computation

Abstract

Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but since it is not a generic situation, this difficulty is usually handled by using a (symbolic or explicit) perturbation. As an alternative, we propose to define a canonical triangulation for a set of concyclic points by using a max-min angle characterization of Delaunay triangulations. This point of view leads to a well defined and unique triangulation as long as there are no symmetric quadruples of points. This unique triangulation can be computed in quasi-linear time by a very simple algorithm.