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Journal ArticleDOI

Two algorithms for constructing a Delaunay triangulation

01 Jun 1980-International Journal of Parallel Programming (Kluwer Academic Publishers-Plenum Publishers)-Vol. 9, Iss: 3, pp 219-242
TL;DR: This paper provides a unified discussion of the Delaunay triangulation and two algorithms are presented for constructing the triangulations over a planar set ofN points.
Abstract: This paper provides a unified discussion of the Delaunay triangulation. Its geometric properties are reviewed and several applications are discussed. Two algorithms are presented for constructing the triangulation over a planar set ofN points. The first algorithm uses a divide-and-conquer approach. It runs inO(N logN) time, which is asymptotically optimal. The second algorithm is iterative and requiresO(N 2) time in the worst case. However, its average case performance is comparable to that of the first algorithm.

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Citations
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Journal ArticleDOI
TL;DR: The Voronoi diagram as discussed by the authors divides the plane according to the nearest-neighbor points in the plane, and then divides the vertices of the plane into vertices, where vertices correspond to vertices in a plane.
Abstract: Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor

4,236 citations

Journal ArticleDOI
TL;DR: L'accentuation est mise sur la representation de donnees dans les applications de traitement d'images, d'infographie, les systemes d'informations geographiques and the robotique.
Abstract: Apercu sur les quadarbres et les structures de donnees hierarchiques. Elles sont basees sur le principe de decomposition recursive. L'accentuation est mise sur la representation de donnees dans les applications de traitement d'images, d'infographie, les systemes d'informations geographiques et la robotique. On examine en detail un certain nombre d'operations dans lesquelles de telles structures de donnees trouvent leur utilisation

2,273 citations

Book ChapterDOI
27 May 1996
TL;DR: Triangle as discussed by the authors is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunayer refinement algorithm for quality mesh generation, and it is shown that the problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is impossible for some PSLGs.
Abstract: Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of adaptive exact arithmetic to ensure robustness with minimal sacrifice of speed. The problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is shown to be impossible for some PSLGs, contradicting the claim that a variant of the Delaunay refinement algorithm solves this problem.

2,268 citations

Book
02 Jan 1991

1,377 citations

Journal ArticleDOI
Steven Fortune1
TL;DR: A geometric transformation is introduced that allows Voronoi diagrams to be computed using a sweepline technique and is used to obtain simple algorithms for computing the Vor onoi diagram of point sites, of line segment sites, and of weighted point sites.
Abstract: We introduce a geometric transformation that allows Voronoi diagrams to be computed using a sweepline technique. The transformation is used to obtain simple algorithms for computing the Voronoi diagram of point sites, of line segment sites, and of weighted point sites. All algorithms haveO(n logn) worst-case running time and useO(n) space.

1,209 citations

References
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Book
02 Jan 1991

1,377 citations

Proceedings ArticleDOI
13 Oct 1975
TL;DR: The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space, and is used to obtain O(N log N) algorithms for most of the problems considered.
Abstract: A number of seemingly unrelated problems involving the proximity of N points in the plane are studied, such as finding a Euclidean minimum spanning tree, the smallest circle enclosing the set, k nearest and farthest neighbors, the two closest points, and a proper straight-line triangulation. For most of the problems considered a lower bound of O(N log N) is shown. For all of them the best currently-known upper bound is O(N2) or worse. The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space. The Voronoi diagram is used to obtain O(N log N) algorithms for all of the problems.

1,140 citations