Topic
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.
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24 Jul 1998
TL;DR: The algorthim is the first for this problem with provable guarantees for the reconstruction of surfaces from unorganized sample points in IR3, based on the three-dimensional Voronoi diagram.
Abstract: Author(s): Amenta, Nina; Bern, Marshall; Kamvysselis, Manolis | Abstract: We describe our experience with a new algorithm for the reconstruction of surfaces from unorganized sample points in IR3. The algorthim is the first for this problem with provable guarantees. Given a ''good sample'' from a smooth surface, the output is guaranteed to be topologically correct and convergent to the original surface as the sampling sensity increases. The definition of a good sample is itself interesting: the required sampling density varies locally, rigorously capturing the intuitive notion that featureless areas can be reconstructed from fewer samples. The output mesh interpolates, rather than approximates, the input points. Our algorithm is based on the three-dimensional Voronoi diagram. Given a good program for thsi fundamental subroutine, the algorithm is quite easy to implement.
1,037 citations
TL;DR: This competitive Hebbian rule provides a novel approach to the problem of constructing topology preserving feature maps and representing intricately structured manifolds and makes this novel approach particularly useful in all applications where neighborhood relations have to be exploited or the shape and topology of submanifolds have to been take into account.
912 citations
01 Dec 1983
TL;DR: Two algorithms are given, one that constructs the Voronoi diagram of the given sites, and another that inserts a new site in O(n) time, based on the use of the Vor onoi dual, the Delaunay triangulation, and are simple enough to be of practical value.
Abstract: We discuss the following problem: given n points in the plane (the “sites”), and an arbitrary query point q, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites, and then locating the query point in one of its regions. We give two algorithms, one that constructs the Voronoi diagram in O(n lg n) time, and another that inserts a new site in O(n) time. Both are based on the use of the Voronoi dual, the Delaunay triangulation, and are simple enough to be of practical value. The simplicity of both algorithms can be attributed to the separation of the geometrical and topological aspects of the problem, and to the use of two simple but powerful primitives, a geometric predicate and an operator for manipulating the topology of the diagram. The topology is represented by a new data structure for generalized diagrams, that is embeddings of graphs in two-dimensional manifolds. This structure represents simultaneously an embedding, its dual, and its mirror-image. Furthermore, just two operators are sufficient for building and modifying arbitrary diagrams.
788 citations
01 May 1995
TL;DR: Compared with previous quadtree-based algorithms for quality mesh generation, the Delaunay refinement approach is much simpler and generally produces meshes with fewer triangles.
Abstract: We present a simple new algorithm for triangulating polygons and planar straightline graphs, It provides "shape" and "size" guarantees: ?All triangles have a bounded aspect ratio.?The number of triangles is within a constant factor of optimal. Such "quality" triangulations are desirable as meshes for the finite element method, in which the running time generally increases with the number of triangles, and where the convergence and stability may be hurt by very skinny triangles. The technique we use-successive refinement of a Delaunay triangulation-extends a mesh generation technique of Chew by allowing triangles of varying sizes. Compared with previous quadtree-based algorithms for quality mesh generation, the Delaunay refinement approach is much simpler and generally produces meshes with fewer triangles. We also discuss an implementation of the algorithm and evaluate its performance on a variety of inputs.
752 citations
01 Jan 1992
TL;DR: The Voronoi diagram of a set of sites partitions space into regions one per site the region for a site s consists of all points closer to s than to any other site as discussed by the authors.
Abstract: The Voronoi diagram of a set of sites partitions space into regions one per site the region for a site s consists of all points closer to s than to any other site The dual of the Voronoi diagram the Delaunay triangulation is the unique triangulation so that the circumsphere of every triangle contains no sites in its interior Voronoi diagrams and Delaunay triangulations have been rediscovered or applied in many areas of math ematics and the natural sciences they are central topics in computational geometry with hundreds of papers discussing algorithms and extensions Section discusses the de nition and basic properties in the usual case of point sites in R with the Euclidean metric while section gives basic algorithms Some of the many extensions obtained by varying metric sites environment and constraints are discussed in section Section nishes with some interesting and nonobvious structural properties of Voronoi diagrams and Delaunay triangulations
651 citations