Delaunay triangulation

About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.

Low distortion delaunay embedding of trees in hyperbolic plane

Abstract: This paper considers the problem of embedding trees into the hyperbolic plane. We show that any tree can be realized as the Delaunay graph of its embedded vertices. Particularly, a weighted tree can be embedded such that the weight on each edge is realized as the hyperbolic distance between its embedded vertices. Thus the embedding preserves the metric information of the tree along with its topology. The distance distortion between non adjacent vertices can be made arbitrarily small --- less than a (1+e) factor for any given e. Existing results on low distortion of embedding discrete metrics into trees carry over to hyperbolic metric through this result. The Delaunay character implies useful properties such as guaranteed greedy routing and realization as minimum spanning trees.

Delaunay-based representation of surfaces defined over arbitrarily shaped domains

Abstract: A method for constructing surface models from a discrete set of arbitrarily distributed data is described. The surface is represented as a network of planar, triangular faces with vertices at the data points, which is built up on a Delaunay triangulation of the data point projections on the x−y plane. In the paper, the problem of triangulating arbitrarily shaped domains is considered. A new definition of Delaunay triangulation of a set of points constrained by a polygonal boundary is introduced, and some of its basic properties are briefly discussed. A new incremental algorithm for constructing a Delaunay triangulation of an arbitrarily shaped domain is described. This method is also demonstrated to be well suited to generate surface approximations at predefined levels of accuracy, which are based on significant subsets of selected data points.

Delaunay graphs are almost as good as complete graphs

Abstract: Let S be any set of N points in the plane and let DT(S) be the graph of the Delaunay triangulation of S. For all points a and b of S, let d(a, b) be the Euclidean distance from a to b and let DT(a, b) be the length of the shortest path in DT(S) from a to b. We show that there is a constant c(≤ 1+√5/2 π ≈ 5.08) independent of S and N such that DT(a, b)/d(a, b) ≪ c.

Conforming Delaunay Triangulation of Stochastically Generated Three Dimensional Discrete Fracture Networks: A Feature Rejection Algorithm for Meshing Strategy

Abstract: We introduce the feature rejection algorithm for meshing (FRAM) to generate a high quality conforming Delaunay triangulation of a three-dimensional discrete fracture network (DFN). The geometric features (fractures, fracture intersections, spaces between fracture intersections, etc.) that must be resolved in a stochastically generated DFN typically span a wide range of spatial scales and make the efficient automated generation of high-quality meshes a challenge. To deal with these challenges, many previous approaches often deformed the DFN to align its features with a mesh through various techniques including redefining lines of intersection as stair step functions and distorting the fracture edges. In contrast, FRAM generates networks on which high-quality meshes occur automatically by constraining the generation of the network. The cornerstone of FRAM is prescribing a minimum length scale and then restricting the generation of the network to only create features of that size and larger. The process is f...

Quality mesh generation in three dimensions

Abstract: We show how to triangulate a three dimensional polyhedral region with holes. Our triangulation is optimal in the following two senses. First, our triangulation achieves the best possible aspect ratio up to a constant. Second, for any other triangulation of the same region into m triangles with bounded aspect ratio, our triangulation has size n = O(m). Such a triangulation is desired as an initial mesh for a finite element mesh refinement algorithm. Previous three dimensional triangulation schemes either worked only on a restricted class of input, or did not guarantee well-shaped tetrahedra, or were not able to bound the output size. We build on some of the ideas presented in previous work by Bern, Eppstein, and Gilbert, who have shown how to triangulate a two dimensional polyhedral region with holes, with similar quality and optimality bounds.