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.

Optimal delaunay point insertion

Abstract: SUMMARY An efficient algorithm for Delaunay triangulation of a given set of points in d dimensions is presented. Various steps of the point insertion algorithm are reviewed and many acceleration procedures are implemented to speed up the triangulation process. New features include the search for a neighbouring point by a layering scheme, locating the containing simplex by a random walk, formulas of important geometrical quantities of a new simplex based on those of an old one, a novel approach in establishing the adjacency relationship using connection matrices. The resulting scheme seems to be one of the fastest triangulation algorithms known, which enables us to generate tetrahedra in R3 with a linear generation rate of 15 OOO tetrahedra per second for randomly generated points on an HP 735 workstation.

Centroidal Voronoi diagrams for isotropic surface remeshing

Abstract: This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a user-specified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image halftoning, to work directly on mesh triangles and feature edges. We then use the resulting sampling as an initial configuration for building a weighted centroidal Voronoi diagram in a conformal parameter space, where the specified density function is used for weighting. We finally create the mesh by lifting the corresponding constrained Delaunay triangulation from parameter space. A precise control over the sampling is obtained through a flexible design of the density function, the latter being possibly low-pass filtered to obtain a smoother gradation. We demonstrate the versatility of our approach through various remeshing examples.

An O(n log n) Heuristic for Steiner Minimal Tree Problems on the Euclidean Metric

Abstract: An O(n log n) heuristic for the Euclidean Steiner Minimal Tree (ESMT) problem is presented. The algorithm is based on a decomposition approach which first partitions the vertex set into triangles via the Delaunay triangulation, then “recomposes” the suboptimal Steiner Minimal Tree (SMT) according to the Voronoi diagram and Minimum Spanning Tree (MST) of the point set. The ESMT algorithm was implemented in FORTRAN-IV and tested on a number of randomly generated point sets in the plane drawn from a uniform distribution. Comparison of the O(n log n) algorithm with an O(n4) algorithm clearly indicates that the O(n log n) algorithm is as good as the previous O(n4) algorithm in achieving reductions in the ratio SMT/MST of the given vertex set. This is somewhat surprising since the O(n4) algorithm considers more potential Steiner points and alternative tree configurations.

A Method for the Evaluation of Radiation Q Based on Modal Approach

Abstract: A new formula for the evaluation of the modal radiation Q factor is derived. The total Q of selected structures is to be calculated from the set of eigenmodes with associated eigen-energies and eigen-powers. Thanks to the analytical expression of these quantities, the procedure is highly accurate, respecting arbitrary current densities flowing along the radiating device. The electric field integral equation, Delaunay triangulation, method of moments, Rao-Wilton-Glisson basis function and the theory of characteristic modes constitute the underlying theoretical background. In terms of the modal radiation Q, all necessary relations are presented and the essential points of implementation are discussed. Calculation of the modal energies and Q factors enable us to study the effect of the radiating shape separately to the feeding. This approach can be very helpful in antenna design. A few examples are given, including a thin-strip dipole, two coupled dipoles a bowtie antenna and an electrically small meander folded dipole. Results are compared with prior estimates and some observations are discussed. Good agreement is observed for different methods.

AUTOCLUST+ : Automatic clustering of point-data sets in the presence of obstacles

Abstract: Wide spread clustering algorithms use the Euclidean distance to measure spatial proximity. However, obstacles in other GIS data-layers prevent traversing the straight path between two points. AUTOCLUST+ clusters points in the presence of obstacles based on Voronoi modeling and Delaunay Diagrams. The algorithm is free of user-supplied arguments and incorporates global and local variations. Thus, it detects high-quality clusters (clusters of arbitrary shapes, clusters of different densities, sparse clusters adjacent to high-density clusters, multiple bridges between clusters and closely located high-density clusters) without prior knowledge. Consequently, it successfully supports correlation analyses between layers (requiring high-quality clusters) and more general locational optimization problems in the presence of obstacles. All this within O(n log n+ [m + R] log n) expected time, where n is the number of data points, m is the number of line-segments that determine the obstacles and R is the number of Delaunay edges intersecting some obstacles. A series of detailed performance evaluations illustrates the power of AUTOCLUST+ and confirms the virtues of our approach.