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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 567 (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.
Voronoi Diagrams and Delaunay Triangulations
 Computing in Euclidean Geometry
, 1992
"... The Voronoi diagram is a fundamental structure in computationalgeometry and arises naturally in many different fields. This chapter surveys properties of the Voronoi diagram and its geometric dual, the Delaunay triangulation. The emphasis is on practical algorithms for the construction of Voronoi ..."
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Cited by 195 (3 self)
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The Voronoi diagram is a fundamental structure in computationalgeometry and arises naturally in many different fields. This chapter surveys properties of the Voronoi diagram and its geometric dual, the Delaunay triangulation. The emphasis is on practical algorithms for the construction of Voronoi diagrams. 1 Introduction Let S be a set of n points in ddimensional euclidean space E d . The points of S are called sites. The Voronoi diagram of S splits E d into regions with one region for each site, so that the points in the region for site s2S are closer to s than to any other site in S. The Delaunay triangulation of S is the unique triangulation of S so that there are no elements of S inside the circumsphere of any triangle. Here `triangulation' is extended from the planar usage to arbitrary dimension: a triangulation decomposes the convex hull of S into simplices using elements of S as vertices. The existence and uniqueness of the Delaunay triangulation are perhaps not obvio...
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 179 (7 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.
Tetrahedral Mesh Generation by Delaunay Refinement
 Proc. 14th Annu. ACM Sympos. Comput. Geom
, 1998
"... Given a complex of vertices, constraining segments, and planar straightline constraining facets in E 3 , with no input angle less than 90 ffi , an algorithm presented herein can generate a conforming mesh of Delaunay tetrahedra whose circumradiustoshortest edge ratios are no greater than two ..."
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Cited by 115 (7 self)
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Given a complex of vertices, constraining segments, and planar straightline constraining facets in E 3 , with no input angle less than 90 ffi , an algorithm presented herein can generate a conforming mesh of Delaunay tetrahedra whose circumradiustoshortest edge ratios are no greater than two. The sizes of the tetrahedra can provably grade from small to large over a relatively short distance. An implementation demonstrates that the algorithm generates excellent meshes, generally surpassing the theoretical bounds, and is effective in eliminating tetrahedra with small or large dihedral angles, although they are not all covered by the theoretical guarantee. 1 Introduction Meshes of triangles or tetrahedra have many applications, including interpolation, rendering, and numerical methods such as the finite element method. Most such applications demand more than just a triangulation of the object or domain being rendered or simulated. To ensure accurate results, the triangles or tetr...
Delaunay Refinement Algorithms for Triangular Mesh Generation
 Computational Geometry: Theory and Applications
, 2001
"... Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of tria ..."
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Cited by 101 (0 self)
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Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and the grading of triangles from small to large sizes. This article presents an intuitive framework for analyzing Delaunay refinement algorithms that unifies the pioneering mesh generation algorithms of L. Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and most importantly, helps to solve the difficult problem of meshing nonmanifold domains with small angles.
Aspects of Unstructured Grids and FiniteVolume Solvers for the Euler and NavierStokes Equations (Part 4)
, 1995
"... this report, the model was tested on various subsonic and transonic flow problems: flat plates, airfoils, wakes, etc. The model consists of a single advectiondiffusion equation with source term for a field variable which is the product of turbulence Reynolds number and kinematic viscosity, e RT . ..."
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Cited by 57 (0 self)
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this report, the model was tested on various subsonic and transonic flow problems: flat plates, airfoils, wakes, etc. The model consists of a single advectiondiffusion equation with source term for a field variable which is the product of turbulence Reynolds number and kinematic viscosity, e RT . This variable is proportional to the eddy viscosity except very near a solid wall. The model equation is of the form: D( e RT ) Dt =(c ffl 2 f 2 (y + ) \Gamma c ffl 1 ) q e RT P +( + t oe R )r 2 ( e RT ) \Gamma 1 oe ffl (r t ) \Delta r( e RT ): (6:3:3) In this equation P is the production of turbulent kinetic energy and is related to the mean flow velocity rateofstrain and the kinematic eddy viscosity t . Equation (6.3.3) depends on distance to solid walls in two ways. First, the damping function f 2 appearing in equation (6.3.3) depends directly on distance to the wall (in wall units). Secondly, t depends on e R t and damping functions which require distance to the wall
Bubble Mesh: Automated Triangular Meshing of NonManifold Geometry by Sphere Packing
 ACM SYMPOSIUM ON SOLID MODELING AND APPLICATIONS
, 1995
"... This paper presents a new computational method for fully automated triangular mesh generation, consistently applicable to wireframe, surface, solid, and nonmanifold geometries. The method, called bubble meshing, is based on the observation that a pattern of tightly packed spheres mimics a Voronoi d ..."
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Cited by 56 (11 self)
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This paper presents a new computational method for fully automated triangular mesh generation, consistently applicable to wireframe, surface, solid, and nonmanifold geometries. The method, called bubble meshing, is based on the observation that a pattern of tightly packed spheres mimics a Voronoi diagram, from which a set of wellshaped Delaunay triangles and tetrahedra can be created by connecting the centers of the spheres. Given a domain geometry and a nodespacing function, spheres are packed on geometric entities, namely, vertices, edges, faces, and volumes, in ascending order of dimension. Once the domain is filled with spheres, mesh nodes are placed at the centers of these spheres and are then connected by constrained Delaunay triangulation and tetrahedrization. To obtain a closely packed configuration of spheres, the authors devised a technique for physically based mesh relaxation with adaptive population control. The process of mesh relaxation significantly reduces the number of illshaped triangles and tetrahedra.
Curves and Surfaces for CAGD
, 1993
"... This article provides a historical account of the major developments in the area of curves and surfaces as they entered the area of CAGD – Computer Aided Geometric Design – until the middle 1980s. We adopt the definition that CAGD deals with the construction and representation of freeform curves, s ..."
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Cited by 52 (0 self)
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This article provides a historical account of the major developments in the area of curves and surfaces as they entered the area of CAGD – Computer Aided Geometric Design – until the middle 1980s. We adopt the definition that CAGD deals with the construction and representation of freeform curves, surfaces, or volumes. 1.
Variable Resolution Terrain Surfaces
, 1996
"... A model for the multiresolution decomposition of planar domains into triangles is introduced, which is more general than other multiresolution models proposed in the literature, and can be efficiently applied to the representation of a polyhedral terrain at variable resolution. The model is based on ..."
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Cited by 50 (6 self)
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A model for the multiresolution decomposition of planar domains into triangles is introduced, which is more general than other multiresolution models proposed in the literature, and can be efficiently applied to the representation of a polyhedral terrain at variable resolution. The model is based on a collection of fragments of plane triangulations arranged into a partially ordered set. Different decompositions of a domain can be obtained by combining different fragments from the model. A data structure to encode the model is presented, and an efficient algorithm is proposed that can extract in linear time a polyhedral terrain representation, whose accuracy over the domain is variable according to a given threshold function. Furthermore, the size of the extracted representation is minimum among all possible polyhedral representations that can be built from the model, and that satisfy the threshold function. A major application of these results is in real time rendering of terrains in f...
Mesh Generation
 Handbook of Computational Geometry. Elsevier Science
, 2000
"... this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary. ..."
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Cited by 48 (6 self)
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this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary.