Order-2 Delaunay Triangulations Optimize Angles

Authors

Herbert Edelsbrunner
Alexey Garber, The University of Texas Rio Grande Valley
Morteza Saghafian

Document Type

Article

Publication Date

2024

Abstract

The local angle property of the (order-1) Delaunay triangulations of a generic set in R2 asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-1, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-2 Delaunay triangulations more attractive to applications as well.

Recommended Citation

Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. "Order-2 Delaunay Triangulations Optimize Angles." arXiv preprint arXiv:2310.18238 (2024).