Random Delaunay triangulations and metric uniformization

Random Delaunay triangulations, the Thurston-Andreev theorem, and metric uniformization

In this thesis a connection between the worlds of discrete and continuous conformal geometry is explored. Specifically, a disk pattern production theroem is proved using an energy which measures how ``uniform'' the angle data of a triangulation is, see also math.DG/0002150. Then this energy is averaged over all the Delaunay triangulation of a Riemannian surface to form an energy measuring how ``uniform'' a metric is, see also math.DG/0010316.

Embedded Delaunay triangulations for point clouds of surfaces in $\mathbb{R}^3$

In the following article we discuss Delaunay triangulations for a point cloud on an embedded surface in $\mathbb{R}^3$. We give sufficient conditions on the point cloud to show that the diagonal switch algorithm finds an embedded Delaunay triangulation.

An obstruction to Delaunay triangulations in Riemannian manifolds

Delaunay has shown that the Delaunay complex of a finite set of points $P$ of Euclidean space $\mathbb{R}^m$ triangulates the convex hull of $P$, provided that $P$ satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds. However, Delaunay's genericity assumption no longer guarantees that the Delaunay complex will yield a triangulation; stronger assumptions on $P$ are required. A natural one is to assume that $P$ is sufficiently dense. Although results in this direction have been claimed, we show that sample density alone i…

Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes

We present a constructive proof of Alexandrov's theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a relation with the weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of a positively curved generalized convex polytope. The latter is shown to be e…

On Delaunay Triangulations of Gromov Sets

Let $Y$ be a subset of a metric space $X.$ We say that $Y$ is $η$-Gromov provided $Y$ is $η$-separated and not properly contained in any other $η$-separated subset of $X.$ In this paper, we review a result of Chew which says that any $η$-Gromov subset of $\mathbb{R}^{2} $ admits a triangulation $\mathcal{T}$ whose smallest angle is at least $π/6 $ and whose edges have length between $η$ and $2η.$ We then show that given any $k = 1,2,3\ldots$, there is a subdivision $\mathcal{T} _{k}$ of $\mathcal{T}$ whose edges have length in $\left[ \fracη{10 k},\frac{2η}{10 k} \right] $ and whose minimum an…

曲面上带装饰的分段欧氏度量的离散单值化定理

A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces

本文提出曲面上高斯曲率的新离散化方法，并利用Bobenko-Lutz离散共形理论与带约束变分原理，在非正欧拉数曲面上证明了该离散曲率的离散单值化定理。