Related papers: Formal study of plane Delaunay triangulation
We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called \emph{pseudo-simultaneously flippable edges}. Such edges are related to the notion of convex…
An elegant procedure which characterizes a decomposition of some class of binomial configurations into two other, resembling a definition of Pascal's Triangle, was given in \cite{gevay}. In essence, this construction was already presented…
A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface $S$ is geometric, that is, it is combinatorially equivalent to a geodesic…
Urban reconstruction from a video captured by a surveying vehicle constitutes a core module of automated mapping. When computational power represents a limited resource and, a detailed map is not the primary goal, the reconstruction can be…
Two-dimensional Delaunay triangulation is a fundamental aspect of computational geometry. This paper presents a novel algorithm that is specifically designed to ensure the correctness of 2D Delaunay triangulation, namely the Polygonal…
This paper introduces a Delaunay triangulation algorithm based on the external incremental method. Unlike traditional random incremental methods, this approach uses convex hull and points as basic operational units instead of triangles.…
We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result…
Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every $n$-vertex triangulation with at least six vertices has a simultaneous flip into a 4-connected triangulation, and that it can be computed in O(n)…
We show that a uniformly acute triangulation of the plane is rigid under Luo's discrete conformal change, extending previous results on hexagonal triangulations. Our result is a discrete analogue of the conformal rigidity of the plane. We…
We present a self-contained short proof of the seminal result of Dillencourt (SoCG 1987 and DCG 1990) that Delaunay triangulations, of planar point sets in general position, are 1-tough. An important implication of this result is that…
A triangulation of a point configuration is regular if it can be given by a height function, that is every point gets lifted to a certain height and projecting the lower convex hull gives the triangulation. Checking regularity of a…
Consider a graph drawn on a surface (for example, the plane minus a finite set of obstacle points), possibly with crossings. We provide an algorithm to decide whether such a drawing can be untangled, namely, if one can slide the vertices…
The Hilbert metric is a distance function defined for points lying within the interior of a convex body. It arises in the analysis and processing of convex bodies, machine learning, and quantum information theory. In this paper, we show how…
In this paper, we prove that the set of triangulations of a polygon can be equipped with an order to become a lattice. First, we define this order. In [HN99], authors defined the flip operator and then prove some properties of the graph of…
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…
We aim to give a strict proof of the existence and uniqueness of the weighted Voronoi decomposition and the dual weighted Delaunay triangulation on Euclidean and hyperbolic polyhedral surface as well as hyperbolic surface with geodesic…
We compute the convex hull in $\mathbb{C}^2$ of an arbitrary finite subgroup of ${\mathbb{C}^*}^2$. The combinatorics are dictated by continued fractions in a natural way. This reproves a theorem of Smilansky, with a slightly stronger…
We introduce a parametrized notion of genericity for Delaunay triangulations which, in particular, implies that the Delaunay simplices of $\delta$-generic point sets are thick. Equipped with this notion, we study the stability of Delaunay…
We prove that a planar graph is generically rigid in the plane if and only if it can be embedded as a pseudo-triangulation. This generalizes the main result of math.CO/0307347 which treats the minimally generically rigid case. The proof…
Geometric predicates are at the core of many algorithms, such as the construction of Delaunay triangulations, mesh processing and spatial relation tests. These algorithms have applications in scientific computing, geographic information…