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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.


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Book ChapterDOI
16 Dec 1999
TL;DR: This work considers online routing strategies for routing between the vertices of embedded planar straight line graphs and proposes two deterministic memoryless routing strategies and a randomized memoryless strategy that works for all triangulations.
Abstract: We consider online routing strategies for routing between the vertices of embedded planar straight line graphs. Our results include (1) two deterministic memoryless routing strategies, one that works for all Delaunay triangulations and the other that works for all regular triangulations, (2) a randomized memoryless strategy that works for all triangulations, (3) an O(1) memory strategy that works for all convex subdivisions, (4) an O(1) memory strategy that approximates the shortest path in Delaunay triangulations, and (5) theoretical and experimental results on the competitiveness of these strategies.

268 citations

Journal ArticleDOI
TL;DR: A new algorithm for the semi-automatic triangulation of arbitrary, multiply connected planar domains based upon a modification of a finite element mesh genration algorithm recently developed is described.
Abstract: The object of this paper is to describe a new algorithm for the semi-automatic triangulation of arbitrary, multiply connected planar domains. The strategy is based upon a modification of a finite element mesh genration algorithm recently developed. 1 The scheme is designed for maximum flexibility and is capable of generating meshes of triangular elements for the decomposition of virtually any multiply connected planar domain. Moreover, the desired density of elements in various regions of the problem domain is specified by the user, thus allowing him to obtain a mesh decomposition appropriate to the physical loading and/or boundary conditions of the particular problem at hand. Several examples are presented to illustrate the applicability of the algorithm. An extension of the algorithm to the triangulation of shell structures is indicated.

261 citations

01 Jan 1998
TL;DR: The optimized algorithm is faster, with an expected cost of O((m+n) logm).
Abstract: Several algorithms for approximating terrains and other height fields using polygonal meshes are described, compared, and optimized. These algorithms take a height field as input, typically a rectangular grid of elevation data H(x, y), and approximate it with a mesh of triangles, also known as a triangulated irregular network, or TIN. The algorithms attempt to minimize both the error and the number of triangles in the approximation. Applications include fast rendering of terrain data for flight simulation and fitting of surfaces to range data in computer vision. The methods can also be used to simplify multi-channel height fields such as textured terrains or planar color images. The most successful method we examine is the greedy insertion algorithm. It begins with a simple triangulation of the domain and, on each pass, finds the input point with highest error in the current approximation and inserts it as a vertex in the triangulation. The mesh is updated either with Delaunay triangulation or with data-dependent triangulation. Most previously published variants of this algorithm had expected time cost of O(mn) or O(n logm+m), where n is the number of points in the input height field and m is the number of vertices in the triangulation. Our optimized algorithm is faster, with an expected cost of O((m+n) logm). On current workstations, this allows one million point terrains to be simplified quite accurately in less than a minute. We are releasing a C++ implementation of our algorithm.

260 citations

Proceedings ArticleDOI
P Chew1
01 Aug 1986
TL;DR: Given a source, a destination, and a set of polygonal obstacles of size n, an size data structure can be used to find a reasonable approximation to the shortest path between the source and the destination in &Ogr;(n log n) time.
Abstract: Given a set S of points in the plane, there is a triangulation of S such that a path found within this triangulation has length bounded by a constant times the straight-line distance between the endpoints of the path. Specifically, for any two points a and b of S there is a path along edges of the triangulation with length less than √10 times |ab|, where |ab| is the straight-line Euclidean distance between a and b. Thus, a shortest path in this planar graph is less than about 3 times longer than the corresponding straight-line distance. The triangulation that has this property is the L1 metric Delaunay triangulation for the set 5. This result can be applied to motion planning in the plane. Given a source, a destination, and a set of polygonal obstacles of size n, an O(n) size data structure can be used to find a reasonable approximation to the shortest path between the source and the destination in O(n log n) time.

259 citations

Patent
14 May 2002
TL;DR: In this article, a system and method for the rapid creation of an optimized mesh model of a real world object, terrain or other three-dimensional surface is presented, which can be used for rapid removal of points or rapid regeneration of the mesh.
Abstract: A system and method for the rapid creation of an optimized mesh model of a real world object, terrain or other three-dimensional surface. The mesh construction technique provides dynamic “up resolution/down resolution” mesh construction capabilities. The system inserts points into the mesh incrementally, ordering the points before each insertion so that dynamic resolution construction can be maintained. The point ordering process determines the distance each point has from the surface of a given mesh configuration such that the next point added will always be the point bringing the most significant detail to the mesh. This procedure of “optimal simplification” optimizes the mesh by guaranteeing the fewest number of points for the most detail at any given resolution. The present invention also provides a system and method to ensure an optimal quality of mesh at any level of insertion or deletion, following in an exemplary configuration a regularized systemized checking function to maintain a mesh of optimal quality such as by Delaunay triangulation principles. The system stores a history of the insertion and deletion steps in a compact list, which can be used for rapid removal of points or rapid regeneration of the mesh.

259 citations


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