Optimality of Delaunay Triangulations

Author

Estefania A. Sierra, The University of Texas Rio Grande Valley

Date of Award

5-2021

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

First Advisor

Dr. Oleg Musin

Second Advisor

Dr. Alexey Glazryin

Third Advisor

Dr. Jacob White

Abstract

In this paper, we begin by defining and examining the properties of a Voronoi diagram and extend it to its dual, the Delaunay triangulations. We explore the algorithms that construct such structures. Furthermore, we define several optimal functionals and criterions on the set of all triangulations of points in Rd that achieve their minimum on the Delaunay triangulation. We found a new result and proved that Delaunay triangulation has lexicographically the least circumradii sequence. We discuss the CircumRadii-Area (CRA) conjecture that the circumradii raised to the power of alpha times the area of the triangulation holds true for all α ≥ 1. We took it upon ourselves to prove that CRA conjecture is true for α =1, FRV quadrilaterals, and TRV quadrilaterals. Lastly, we demonstrate counterexamples for alpha<1.

Comments

Copyright 2021 Estefania A. Sierra. All Rights Reserved.

https://go.openathens.net/redirector/utrgv.edu?url=https://www.proquest.com/dissertations-theses/optimality-delaunay-triangulations/docview/2572588657/se-2?accountid=7119

Recommended Citation

Sierra, Estefania A., "Optimality of Delaunay Triangulations" (2021). Theses and Dissertations. 970.
https://scholarworks.utrgv.edu/etd/970