Near-Delaunay Metrics

TL;DR

This paper introduces metrics to evaluate how close a triangulation is to Delaunay, useful when constraints prevent using the actual Delaunay triangulation, and compares their effectiveness through experiments.

Contribution

It proposes new near-Delaunay metrics based on Delaunay properties that are invariant under similarity transformations and compares their judgments and optimization results.

Findings

Different metrics can yield varying judgments of closeness to Delaunay.

Optimizing for these metrics produces similar but not identical triangulations.

Metrics are effective in assessing triangulation quality under constraints.

Abstract

We study metrics that assess how close a triangulation is to being a Delaunay triangulation, for use in contexts where a good triangulation is desired but constraints (e.g., maximum degree) prevent the use of the Delaunay triangulation itself. Our near-Delaunay metrics derive from common Delaunay properties and satisfy a basic set of design criteria, such as being invariant under similarity transformations. We compare the metrics, showing that each can make different judgments as to which triangulation is closer to Delaunay. We also present a preliminary experiment, showing how optimizing for these metrics under different constraints gives similar, but not necessarily identical results, on random and constructed small point sets.