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Voronoi diagrams  a survey of a fundamental geometric data structure
 ACM COMPUTING SURVEYS
, 1991
"... This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. ..."
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Cited by 560 (5 self)
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This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
Provably Good Mesh Generation
 J. Comput. Syst. Sci
, 1990
"... We study several versions of the problem of generating triangular meshes for finite element methods. We show how to triangulate a planar point set or polygonally bounded domain with triangles of bounded aspect ratio; how to triangulate a planar point set with triangles having no obtuse angles; how t ..."
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Cited by 193 (11 self)
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We study several versions of the problem of generating triangular meshes for finite element methods. We show how to triangulate a planar point set or polygonally bounded domain with triangles of bounded aspect ratio; how to triangulate a planar point set with triangles having no obtuse angles; how to triangulate a point set in arbitrary dimension with simplices of bounded aspect ratio; and how to produce a linearsize Delaunay triangulation of a multidimensional point set by adding a linear number of extra points. All our triangulations have size (number of triangles) within a constant factor of optimal, and run in optimal time O(n log n+k) with input of size n and output of size k. No previous work on mesh generation simultaneously guarantees wellshaped elements and small total size. 1. Introduction Geometric partitioning problems ask for the decomposition of a geometric input into simpler objects. These problems are fundamental in many areas, such as solid modeling, computeraided ...
Mesh Generation And Optimal Triangulation
, 1992
"... We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some cri ..."
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Cited by 180 (8 self)
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We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.
Geometric structures for threedimensional shape representation
 ACM Trans. Graph
, 1984
"... Different geometric structures are investigated in the context of discrete surface representation. It is shown that minimal representations (i.e., polyhedra) can be provided by a surfacebased method using nearest neighbors structures or by a volumebased method using the Delaunay triangulation. Bot ..."
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Cited by 166 (3 self)
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Different geometric structures are investigated in the context of discrete surface representation. It is shown that minimal representations (i.e., polyhedra) can be provided by a surfacebased method using nearest neighbors structures or by a volumebased method using the Delaunay triangulation. Both approaches are compared with respect to various criteria, such as space requirements, computation time, constraints on the distribution of the points, facilities for further calculations, and agreement with the actual shape of the object.
Applicationlayer multicast with Delaunay triangulations
 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS
, 2002
"... Applicationlayer multicast supports group applications without the need for a networklayer multicast protocol. Here, applications arrange themselves in a logical overlay network and transfer data within the overlay. In this paper, we present an applicationlayer multicast solution that uses a Del ..."
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Cited by 145 (3 self)
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Applicationlayer multicast supports group applications without the need for a networklayer multicast protocol. Here, applications arrange themselves in a logical overlay network and transfer data within the overlay. In this paper, we present an applicationlayer multicast solution that uses a Delaunay triangulation as an overlay network topology. An advantage of using a Delaunay triangulation is that it allows each application to locally derive nexthop routing information without requiring a routing protocol in the overlay. A disadvantage of using a Delaunay triangulation is that the mapping of the overlay to the network topology at the network and data link layer may be suboptimal. We present a protocol, called Delaunay triangulation (DT protocol), which constructs Delaunay triangulation overlay networks. We present measurement experiments of the DT protocol for overlay networks with up to 10 000 members, that are running on a local PC cluster with 100 Linux PCs. The results show that the protocol stabilizes quickly, e.g., an overlay network with 10 000 nodes can be built in just over 30 s. The traffic measurements indicate that the average overhead of a node is only a few kilobits per second if the overlay network is in a steady state. Results of throughput experiments of multicast transmissions (using TCP unicast connections between neighbors in the overlay network) show an achievable throughput of approximately 15 Mb/s in an overlay with 100 nodes and 2 Mb/s in an overlay with 1000 nodes.
On Levels of Detail in Terrains
, 1995
"... In many applications it is important that one can view a scene at different levels of detail. A prime example is flight simulation: a high level of detail is needed when flying low, whereas a low level of detail suffices when flying high. More precisely, one would like to visualize the part of the s ..."
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Cited by 78 (2 self)
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In many applications it is important that one can view a scene at different levels of detail. A prime example is flight simulation: a high level of detail is needed when flying low, whereas a low level of detail suffices when flying high. More precisely, one would like to visualize the part of the scene that is close at a high level of detail, and the part that is far away at a low level of detail. We propose a hierarchy of detail levels for a polyhedral terrain (or, triangulated irregular network) that allows this: given a view point, it is possible to select the appropriate level of detail for each part of the terrain in such a way that the parts still fit together continuously. The main advantage of our structure is that it uses the Delaunay triangulation at each level, so that triangles with very small angles are avoided. This is the first method that uses the Delaunay triangulation and still allows to combine different levels into a single representation.
The Farthest Point Strategy for Progressive Image Sampling
 IEEE Trans. on Image Processing
, 1997
"... Abstract — A new method of farthest point strategy (FPS) for progressive image acquisition—an acquisition process that enables an approximation of the whole image at each sampling stage—is presented. Its main advantage is in retaining its uniformity with the increased density, providing efficient me ..."
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Cited by 61 (1 self)
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Abstract — A new method of farthest point strategy (FPS) for progressive image acquisition—an acquisition process that enables an approximation of the whole image at each sampling stage—is presented. Its main advantage is in retaining its uniformity with the increased density, providing efficient means for sparse image sampling and display. In contrast to previously presented stochastic approaches, the FPS guarantees the uniformity in a deterministic minmax sense. Within this uniformity criterion, the sampling points are irregularly spaced, exhibiting antialiasing properties comparable to those characteristic of the best available method (Poisson disk). A straightforward modification of the FPS yields an imagedependent adaptive sampling scheme. An efficient O(N log N) algorithm for both versions is introduced, and several applications of the FPS are discussed. Index Terms—Antialiasing, progressive sampling, progressive transmission. I.
The Expected Extremes In A Delaunay Triangulation
, 1991
"... We give an expectedcase analysis of Delaunay triangulations. To avoid edge e#ects we consider a unitintensity Poisson process in Euclidean dspace, and then limit attention to the portion of the triangulation within a cube of side n 1/d .Ford equal to two, we calculate the expected maximum edge ..."
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Cited by 32 (3 self)
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We give an expectedcase analysis of Delaunay triangulations. To avoid edge e#ects we consider a unitintensity Poisson process in Euclidean dspace, and then limit attention to the portion of the triangulation within a cube of side n 1/d .Ford equal to two, we calculate the expected maximum edge length, the expected minimum and maximum angles, and the average aspect ratio of a triangle. We also show that in any fixed dimension the expected maximum vertex degree is #(log n/ log log n). Altogether our results provide some measure of the suitability of the Delaunay triangulation for certain applications, such as interpolation and mesh generation. Keywords: Computational geometry, Delaunay triangulation, probabilistic analysis 1. Introduction Suppose that # is a set of points (called sites) in Euclidean dspace, such that no d + 2 sites lie on a sphere (a general position assumption). In the Delaunay triangulation of #, a set of d + 1 sites defines a dsimplex of the triangulation exac...
An Upper Bound for Conforming Delaunay Triangulations
 Discrete Comput. Geom
, 1993
"... A plane geometric graph C in ! 2 conforms to another such graph G if each edge of G is the union of some edges of C. It is proved that for every G with n vertices and m edges, there is a completion of a Delaunay triangulation of O(m 2 n) points that conforms to G. The algorithm that construct ..."
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Cited by 32 (6 self)
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A plane geometric graph C in ! 2 conforms to another such graph G if each edge of G is the union of some edges of C. It is proved that for every G with n vertices and m edges, there is a completion of a Delaunay triangulation of O(m 2 n) points that conforms to G. The algorithm that constructs the points is also described. Keywords. Discrete and computational geometry, plane geometric graphs, Delaunay triangulations, point placement. Appear in: Discrete & Computational Geometry, 10 (2), 197213 (1993) 1 Research of the first author is supported by the National Science Foundation under grant CCR8921421 and under the Alan T. Waterman award, grant CCR9118874. Any opinions, finding and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the view of the National Science Foundation. Work of the second author was conducted while he was on study leave at the University of Illinois. 2 Department of Computer Scienc...
PolynomialSize Nonobtuse Triangulation Of Polygons
, 1992
"... We describe methods for triangulating polygonal regions of the plane so that no triangle has a large angle. Our main result is that a polygon with n sides can be triangulated with O(n 2 ) nonobtuse triangles. We also show that any triangulation (without Steiner points) of a simple polygon has a ..."
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Cited by 31 (8 self)
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We describe methods for triangulating polygonal regions of the plane so that no triangle has a large angle. Our main result is that a polygon with n sides can be triangulated with O(n 2 ) nonobtuse triangles. We also show that any triangulation (without Steiner points) of a simple polygon has a refinement with O(n 4 ) nonobtuse triangles. Finally we show that a triangulation whose dual is a path has a refinement with only O(n 2 ) nonobtuse triangles. Keywords: Computational geometry, mesh generation, triangulation, angle condition. 1. Introduction One of the classical motivations for problems in computational geometry has been automatic mesh generation for finite element methods. In particular, mesh generation has motivated a number of triangulation algorithms, such as finding a triangulation that minimizes the maximum angle. 1 A triangulation algorithm takes a geometric input, typically a point set or polygonal region, and produces an output that is a triangulation of ...