School of Mathematical and Statistical Sciences Faculty Publications and Presentations
Document Type
Article
Publication Date
2-2025
Abstract
The local angle property of the (order-1) Delaunay triangulations of a generic set in R2" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; position: relative;">R2R2 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" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; position: relative;">R2R2, 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" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; position: relative;">R2R2, 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." Advances in Mathematics 461 (2025): 110055. https://doi.org/10.1016/j.aim.2024.110055
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Publication Title
Advances in Mathematics
DOI
https://doi.org/10.1016/j.aim.2024.110055
Comments
Original published version available at https://doi.org/10.1016/j.aim.2024.110055