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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.


Papers
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Journal ArticleDOI
TL;DR: A family of simple incremental algorithms for constructing short paths on the Delaunay graph is defined and potential applications to routeing in mobile communication networks are discussed.
Abstract: Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path. We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

67 citations

Journal ArticleDOI
TL;DR: An abstract view of the edge insertion paradigm is presented, and it is shown that it gives polynomial-time algorithms for several types of optimal triangulations, including minimizing the maximum slope of a piecewise-linear interpolating surface.
Abstract: Edge insertion iteratively improves a triangulation of a finite point set in ?2 by adding a new edge, deleting old edges crossing the new edge, and retriangulating the polygonal regions on either side of the new edge. This paper presents an abstract view of the edge insertion paradigm, and then shows that it gives polynomial-time algorithms for several types of optimal triangulations, including minimizing the maximum slope of a piecewise-linear interpolating surface.

67 citations

Book ChapterDOI
01 Jan 2005
TL;DR: This chapter focuses on the automatic mesh generation methods based on the advancing front method and the Delaunay triangulation method, which are the basis of many existing mesh generation programs and the basis for current research.
Abstract: The process of creating a finite element mesh is often termed as mesh generation. Mesh generation has always been a time-consuming and error-prone process. This is true in the practical science and engineering computations, where meshes have to be generated for three-dimensional geometries of various levels of complexity. The attempt to create a fully automatic mesh generator, which is a particular mesh generation algorithm that is capable of generating valid finite element meshes over arbitrary domains and needs only the information of the specified geometric boundary of the domain and the required distribution of the element size, started in the early 1970s. Since then, many methodologies have been proposed and different algorithms have been devised in the development of automatic mesh generators. This chapter focuses on the automatic mesh generation methods based on the advancing front method and the Delaunay triangulation method. These are the basis of many existing mesh generation programs and the basis of current research. This chapter discusses the algorithmic procedures of the advancing front method in two dimensions and the Delaunay triangulation method in three dimensions. It also discusses curve and surface mesh generations.

67 citations

BookDOI
01 Nov 2008
TL;DR: This work presents a methodology for Automated Cartographic Data Input, Drawing and Editing Using Kinetic Delaunay/Voronoi Diagrams, and results show efficient Swarm Neighborhood Management and Robust Point-Location in Generalized Voronoi diagrams.
Abstract: Computational Geometry Methods and Intelligent Computing.- Generalized Voronoi Diagrams: State-of-the-Art in Intelligent Treatment of Applied Problems.- Shapes of Delaunay Simplexes and Structural Analysis of Hard Sphere Packings.- The ?-Shape and ?-Complex for Analysis of Molecular Structures.- Computational Geometry Analysis of Quantum State Space and Its Applications.- Efficient Swarm Neighborhood Management Using the Layered Delaunay Triangulation.- Intelligent Solutions for Curve Reconstruction Problem.- A Methodology for Automated Cartographic Data Input, Drawing and Editing Using Kinetic Delaunay/Voronoi Diagrams.- Density-Based Clustering Based on Topological Properties of the Data Set.- Modeling Optimal Beam Treatment with Weighted Regions for Bio-medical Applications.- Advanced Treatment of Topics of Special Interest.- Constructing Centroidal Voronoi Tessellations on Surface Meshes.- Simulated Annealing and Genetic Algorithms in Quest of Optimal Triangulations.- Higher Order Voronoi Diagrams and Distance Functions in Art and Visualization.- Robust Point-Location in Generalized Voronoi Diagrams.- Conclusions and Future Trends in Intelligent Treatment of Applied Problems.

67 citations

Proceedings ArticleDOI
22 Jan 1995
TL;DR: It is shown that any set of n points in IR has a Steiner Delaunay triangulation with O(ndd/2e) simplices, none of which has an obtuse dihedral angle.
Abstract: We show that any set of n points in IR has a Steiner Delaunay triangulation with O(ndd/2e) simplices, none of which has an obtuse dihedral angle. This result improves a naive bound of O(n). No bound depending only on n is possible if we require the maximum dihedral angle to measure at most 90◦−2 or the minimum dihedral to measure

67 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202393
2022203
2021130
2020185
2019204
2018223