Showing papers on "Bowyer–Watson algorithm published in 2002"
TL;DR: An intuitive framework for analyzing Delaunay refinement algorithms is presented that unifies the pioneering mesh generation algorithms of L. Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and helps to solve the difficult problem of meshing nonmanifold domains with small angles.
Abstract: 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 Although small angles inherent in the input geometry cannot be removed, one would like to triangulate a domain without creating any new small angles Unfortunately, this problem is not always soluble A compromise is necessary A Delaunay refinement algorithm is presented that can create a mesh in which most angles are 30^o or greater and no angle is smaller than arcsin[(3/2)sin(@f/2)]~(3/4)@f, where @f=<60^ois the smallest angle separating two segments of the input domain New angles smaller than 30^o appear only near input angles smaller than 60^o In practice, the algorithm's performance is better than these bounds suggest Another new result is that Ruppert's analysis technique can be used to reanalyze one of Chew's algorithms Chew proved that his algorithm produces no angle smaller than 30^o (barring small input angles), but without any guarantees on grading or number of triangles He conjectures that his algorithm offers such guarantees His conjecture is conditionally confirmed here: if the angle bound is relaxed to less than 265^o, Chew's algorithm produces meshes (of domains without small input angles) that are nicely graded and size-optimal
1,313 citations
1 Jan 2002
TL;DR: This paper discusses the three-dimensional analogue, constrained Delaunay tetrahedralizations (also called CDTs), and their advantages in mesh generation; this approach has three advantages over other methods for boundary recovery: it usually requires fewer additional vertices to be inserted, it yields provably good bounds on edge lengths (i.e. edges are not made unnecessarily short), and it interacts well with provable good Delaunays refinement methods for Tetrahedral mesh generation.
Abstract: In two dimensions, a constrained Delaunay triangulation (CDT) respects a set of segments that constrain the edges of the triangulation, while still maintaining most of the favorable properties of ordinary Delaunay triangulations (such as maximizing the minimum angle). CDTs solve the problem of enforcing boundary conformity—ensuring that triangulation edges cover the boundaries (both interior and exterior) of the domain being modeled. This paper discusses the three-dimensional analogue, constrained Delaunay tetrahedralizations (also called CDTs), and their advantages in mesh generation. CDTs maintain most of the favorable properties of ordinary Delaunay tetrahedralizations, but they are more difficult to work with, because some sets of constraining segments and facets simply do not have CDTs. However, boundary conformity can always be enforced by judicious insertion of additional vertices, combined with CDTs. This approach has three advantages over other methods for boundary recovery: it usually requires fewer additional vertices to be inserted, it yields provably good bounds on edge lengths (i.e. edges are not made unnecessarily short), and it interacts well with provably good Delaunay refinement methods for tetrahedral mesh generation.
179 citations
5 Jun 2002
TL;DR: An algorithm which, for any piecewise linear complex (PLC) in 3D, builds a Delaunay triangulation conforming to this PLC, the first practical algorithm devoted to this problem is described.
Abstract: We describe an algorithm which, for any piecewise linear complex (PLC) in 3D, builds a Delaunay triangulation conforming to this PLC.The algorithm has been implemented, and yields in practice a relatively small number of Steiner points due to the fact that it adapts to the local geometry of the PLC. It is, to our knowledge, the first practical algorithm devoted to this problem.
121 citations
TL;DR: A characterization of when a given topological decomposition and angle assignment can be realized as the data of an actual Delaunay decomposition of a hyperbolic surface.
Abstract: Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the \empty disks" circumscribing the regions of the decompo- sition. The main result of this paper is a characterization of when a given topological decomposition and angle assignment can be realized as the data of an actual Delaunay decomposition of a hyperbolic surface.
89 citations
TL;DR: For a set P of points in the plane, the authors introduced a class of triangulations that is an extension of the Delaunay triangulation, where instead of requiring that for each triangle the circle through its vertices contains no points of P inside, they require that at most k points are inside the circle.
Abstract: For a set P of points in the plane, we introduce a class of triangulations that is an extension of the Delaunay triangulation. Instead of requiring that for each triangle the circle through its vertices contains no points of P inside, we require that at most k points are inside the circle. Since there are many different higher-order Delaunay triangulations for a point set, other useful criteria for triangulations can be incorporated without sacrificing the well-shapedness too much. Applications include realistic terrain modeling and mesh generation.
85 citations
TL;DR: A warning panel trailer in which the warning panel, with the plurality of discrete electrical image display means, is raised to various operative positions along a column support and lowered to an inoperative position in an elongated seat on top of a platform of the trailer.
Abstract: Given a set of compact sites on a sphere, we show that their spherical Voronoi diagram can be computed by computing two planar Voronoi diagrams of suitably transformed sites in the plane. We also show that a planar furthest-site Voronoi diagram can always be obtained as a portion of a nearest-site Voronoi diagram of a set of transformed sites. Two immediate applications are an O(n logn) algorithm for the spherical Voronoi diagram of a set of circular arcs on the sphere, and an O(n logn) algorithm for the furthest-site Voronoi diagram for a set of circular arcs in the plane.
77 citations
TL;DR: Results on a two-step improvement of mesh quality in three-dimensional Delaunay triangulations are presented, providing evidence for the practical effectiveness of sliver exudation.
Abstract: We present results on a two-step improvement of mesh quality in three-dimensional Delaunay triangulations. The first step refines the triangulation by inserting sinks and eliminates tetrahedra with large circumradius over shortest edge length ratio. The second step assigns weights to the vertices to eliminate slivers. Our experimental findings pro- vide evidence for the practical effectiveness of sliver exu- dation.
66 citations
TL;DR: A framework for minimum spanning tree construction is established which is based on a general concept of spanning graphs and an O(nlogn) sweep-line algorithm is designed to construct a rectilinearminimum spanning tree without using Delaunay triangulation.
63 citations
17 Jun 2002
TL;DR: It is shown that the complexity of the Delaunay triangulation of points may be quadratic in the worst-case, but it is only linear when the points are distributed on a fixed number of well-sampled facets (e.g. the facets of a polyhedron).
Abstract: Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time therefore depends on the complexity of the Delaunay triangulation of such point sets. Although the complexity of the Delaunay triangulation of points may be quadratic in the worst-case, we show in this paper that it is only linear when the points are distributed on a fixed number of well-sampled facets (e.g. the facets of a polyhedron). Our bound is deterministic and the constants are explicitly given.
61 citations
TL;DR: Three new proximity skeletons related to the Voronoi diagram are introduced and a space subdivision algorithm is presented to construct the new skeletons, having three main advantages: first, it solves at most uni-variate quartic polynomials, which stands in sharp contrast to previous approaches, which require the solution of a non-linear tri-Variate system of equations.
Abstract: We tackle the problem of computing the Voronoi diagram of a 3-D polyhedron whose faces are planar. The main difficulty with the computation is that the diagram’s edges and vertices are of relatively high algebraic degrees. As a result, previous approaches to the problem have been non-robust, difficult to implement, or not provenly correct. We introduce three new proximity skeletons related to the Voronoi diagram: (1) the Voronoi graph(VG), which contains the complete symbolic information of the Voronoi diagram without containing any geometry; (2) the approximate Voronoi graph(AVG), which deals with degenerate diagrams by collapsing sub-graphs of the VG into single nodes; and (3) the proximity structure diagram (PSD), which enhances the VG with a geometric approximation of Voronoi elements to any desired accuracy. The new skeletons are important for both theoretical and practical reasons. Many applications that extract the proximity information of the object from its Voronoi diagram can use the Voronoi graphs or the proximity structure diagram instead. In addition, the skeletons can be used as initial structures for a robust and efficient global or local computation of the Voronoi diagram. We present a space subdivision algorithm to construct the new skeletons, having three main advantages. First, it solves at most uni-variate quartic polynomials. This stands in sharp contrast to previous approaches, which require the solution of a non-linear tri-variate system of equations. Second, the algorithm enables purely local computation of the skeletons in any limited region of interest. Third, the algorithm is simple to implement. 2002 Elsevier Science B.V. All rights reserved.
TL;DR: A method is presented for generalising cartographic lines using an approach based on determination of their structure to allow both sides of multiple lines to be processed, while guaranteeing topological consistency between the resulting generalised lines.
Abstract: A method is presented for generalising cartographic lines using an approach based on determination of their structure. Constrained Delaunay triangulation is used to construct a skeleton of the space surrounding the lines and hence represent line features in terms of skeleton branches. Several statistical measures are used to characterise the triangulation branches. The measures enable selective generalisation of different types of line feature, leading to the possibility of user-specification of the style of generalisation. In our implementation of the approach, the triangulation is updated dynamically to allow both sides of multiple lines to be processed, while guaranteeing topological consistency between the resulting generalised lines.
6 Jan 2002
TL;DR: It is proved that the Delaunay triangulation of any set of $n$ points in~$\Real^3$ with spread $\Delta$ with complexity $O(\Delta^3)$ has complexity O(\sqrt{n})$.
Abstract: The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in IR3 with spread Δ has complexity O(Δ3). This bound is tight in the worst case for all Δ = O(√n). In particular, the Delaunay triangulation of any dense point set has linear complexity. On the other hand, for any n and Δ = O(n), we construct a regular triangulation of complexity Ω(nΔ) whose n vertices have spread Δ.
1 Jan 2002
TL;DR: A method of point cluster simplification is provided in the paper on the basis of Voronoi polygon establishment in a dynamic way, which finds the distribution range polygon through progressively stripping the outside triangles.
Abstract: Point cluster object contains much structured information in spatial distribution, which is interesting for the research of spatial analysis and map generalization. This paper divides the spatial distribution information of point cluster into three categories : existing, metrical structure and topological structure, and focuses on the discussion of metrical structure. Based on the Delaunay triangulation and Voronoi diagram model, the paper defines four characteristic parameters for metrical structure description: distribution range, distribution density, distribution center and distribution axis. With the consideration of Gestalt principles in visual adjacency cognition, the presented method finds the distribution range polygon through progressively stripping the outside triangles. The distribution density is represented by Voronoi cell size and visualized as gray image. Applying image process method, the distribution center can be extracted from gray image. A method of point cluster simplification is provided in the paper on the basis of Voronoi polygon establishment in a dynamic way.
1 Jan 2002
TL;DR: This thesis presents Delaunay triangulation without addition or displacement of points in 3D space and introduces several existing approaches for increasing the numerical stability of algorithms, including two for an exact evaluation of geometric predicates.
Abstract: The Delaunay triangulation is one of the most popular and most often used methods in problems related to the generation of meshes. A lot of the optimal properties of Delaunay triangulation are known in 2D, where it has been intensively studied during the last twenty years, although the fundamentals were formulated early in the twentieth century (Voronoi, 1908 and Delaunay, 1934). This thesis presents Delaunay triangulation without addition or displacement of points in 3D space. It focuses on its properties and on a summarization of existing sequential algorithms. Also our experience with the implementation of the incremental insertion algorithm is presented and observed features are discussed. The properties of Delaunay triangulation in 3D (or generally in higher dimensions) are not as good as in 2D and different kinds of methods are used mainly to remove the tetrahedra of undesirable shape. Although this area of research was not within our main scope, we present an existing simple method for tetrahedra shape improvement. We have implemented this method and our results are presented and discussed. In the implementation of algorithms, which have to deal with inprecise floating-point arithmetic on real computers, the question of numerical stability becomes very important for the proper function of the implementation. We introduce several existing approaches for increasing the numerical stability of algorithms, two of them for an exact evaluation of geometric predicates are presented in more details. We made a comparison of them and we mention the results of incorporating one of them in our implementation.
TL;DR: An algorithm is presented for constructing three‐dimensional Delaunay tessellations in periodic domains, and although the general framework is similar to point insertion in a convex hull, a number of new issues are introduced by periodicity.
Abstract: An algorithm is presented for constructing three-dimensional Delaunay tessellations in periodic domains. Applications include mesh generation for periodic transport problems and geometric decomposition for modelling particulate structures. The algorithm is a point insertion technique, and although the general framework is similar to point insertion in a convex hull, a number of new issues are introduced by periodicity. These issues are discussed in detail in the context of the computational algorithm. Examples are given for the tessellation of random points and random sphere packings. Performance data for the algorithm are also presented. These data show an empirical scaling of the computation time with size of O(N1.11) and tessellation rates of 7000–14000 tetrahedrons per second for the problems studied (up to 105 points). A breakdown of the performance is given, which shows the computational load is shared most heavily by two specific parts of the point-insertion procedure. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: This paper suggests two improvements to the Delaunay triangulation construction algorithm, the first one speeds up the computation without increasing memory requirements and the second decreases memory requirements, trading space for small slow down.
TL;DR: In this article, a series of dynamic update methods for Voronoi diagram types are described, which can be used to add or delete pointsets without complete reconstruction of the underlying data structure.
Abstract: This paper describes a series of dynamic update methods th at can be applied to a family of Voronoi diagram types, so that changes can be updated incrementa lly, without the usual recourse to complete reconstruction of their underlying data structure. Mor e efficient incremental update methods are described for the ordinary Voronoi diagram, the farthest-point Voronoi diagram, the order-k Voronoi diagram, the ordered order- k Voronoi diagram and the k th nearest-point Voronoi diagram. A discussion is also given of one case where increme ntal update is not practical, that of the multiplicatively weighted Voronoi diagram. Update methods rely on a previously reported generic triangle-based data structure (Gahegan & Lee, 2000) f rom which local topology can be reconstructed following changes to the underlying pointset. An a pplication, which implements these ideas, is available for download via the Internet as proof of concept. Results show that the algorithmic complexity of dynamic update methods vary considerably according to the Voronoi type, but offer in all cases (except the multiplicativel y weighted Voronoi diagram) a substantial increase in performance, enabling Voronoi methods to addre ss larger pointsets and more complex modelling problems without suffering from efficiency problems.
TL;DR: The paper describes a new parallel algorithm of Delaunay triangulation based on randomized incremental insertion that was developed for architectures with a lower degree of parallelism, such as several-processor workstations, and tested on up to 8 processors.
Abstract: The paper describes a new parallel algorithm of Delaunay triangulation based on randomized incremental insertion. The algorithm is practical, simple and can be modified also for constrained triangulation or tetrahedralization. It was developed for architectures with a lower degree of parallelism, such as several-processor workstations, and tested on up to 8 processors.
TL;DR: In this paper, a moving finite element method based on Delaunay automatic triangulation was developed for the analysis of dynamic crack bifurcation, and the generation phase simulation was carried out, based on the experimentally recorded fracture histories by an ultra-high speed camera.
Abstract: The governing condition of dynamic crack bifurcation phenomena had not been fully elucidated until our recent experimental studies. We found from the experimental results that the energy flux per unit time into a propagating crack tip or into a fracture process zone governs the crack bifurcation. Regarding the numerical simulation of dynamic crack bifurcation, to the authors’ knowledge, no accurate simulations have been carried out, due to several unresolved difficulties. In order to overcome the difficulties, for the analysis of dynamic crack bifurcation, we developed a moving finite element method based on Delaunay automatic triangulation. Using the moving finite element method, the generation phase simulation was carried out, based on the experimentally recorded fracture histories by an ultra-high speed camera. To evaluate fracture parameters for shortly branched cracks, a switching method of the path independent dynamic J integral was also developed. The simulated results agree excellently with those of the experiment. Furthermore, the numerical simulation revealed detailed variations of various fracture parameters before and after the crack bifurcation. keyword: Dynamic fracture, dynamic crack bifurcation, dynamic crack branching, Delaunay triangulation, moving element method, automatic mesh generation, dynamic J integral, path independent integral.
5 Jun 2002
TL;DR: This paper considers the situation when the points are drawn from a 2-dimensional Poisson distribution with rate n over a fixed union of triangles in $\myRe^3.$ and shows that with high probability the complexity of their Voronoi diagram is $\Otn.
Abstract: (MATH) It is well known that the complexity, i.e., the number of vertices, edges and faces, of the 3-dimensional Voronoi diagram of n points can be as bad as Θ(n 2). Interest has recently arisen as to what happens, both in deterministic and probabilistic situations, when the 3-dimensional points are restricted to lie on the surface of a 2-dimensional object. In this paper we consider the situation when the points are drawn from a 2-dimensional Poisson distribution with rate n over a fixed union of triangles in $\myRe^3.$ We show that with high probability the complexity of their Voronoi diagram is $\Otn.(MATH) This implies, for example, that the complexity of the Voronoi diagram of points chosen from the surface of a general fixed polyhedron in $\myRe3 will also be $\Otn with high probability.
6 Jan 2002
TL;DR: Three production-quality Delaunay triangulation programs are compared on some 'real-world' sets of points lying on or near 2D surfaces.
Abstract: The Delaunay triangulation of a set of points in 3D can have size Θ(n2) in the worst case, but this is rarely if ever observed in practice. We compare three production-quality Delaunay triangulation programs on some 'real-world' sets of points lying on or near 2D surfaces.
Patent•
4 Sep 2002TL;DR: In this paper, a method of representing spatial relations among objects in the environment uses a Delaunay triangulation as the data structure to store the spatial relations when the objects are represented in the form of simplified objects such as cuboids.
Abstract: A method of representing spatial relations among objects in the environment uses a Delaunay triangulation as the data structure to store the spatial relations when the objects are represented in the form of simplified objects such as cuboids. The method receives image data corresponding to the environment and recognizes the objects in the image data, and updates the Delaunay triangulation so that the Delaunay triangulation is consistent with the recognized objects. Furthermore, a proximity query can be carried out using the Delaunay triangulation.
21 Nov 2002
TL;DR: The case where the Voronoi neighbors of p are in convex position is studied, and it is proved that there is at most one local maximum.
Abstract: Given a set S of s points in the plane, where do we place a new point, p, in order to maximize the area of its region in the Voronoi diagram of S and p? We study the case where the Voronoi neighbors of p are in convex position, and prove that there is at most one local maximum.
1 Jan 2002
TL;DR: This paper presents a field based method to deal with the displacement of polygon cluster in both aspects above, on the basis of the skeleton of Delaunay triangulation.
Abstract: As an important operator in polygon cluster generalization, the displacement has two applications. One is to resolve the proximity conflicts to guarantee the legibility constraint. Another is to act as the operator prior to other generalization operators, such as aggregation. This paper presents a field based method to deal with the displacement of polygon cluster in both aspects above. On the basis of the skeleton of Delaunay triangulation, a displacement field is built in which the propagation force is taken into account. Taking the building cluster as the example, the study offers the computation of displacement direction and offset distance for the building displacement, which is driven by the street widening. The vector operation is performed based on the grade and other field concepts.
TL;DR: This framework supports the design of robust algorithms for computing the Delaunay triangulation and the Voronoi diagram with imprecise input by showing that the map which sends three partial points to the partial disc passing through them is computable.
IBM1
TL;DR: The min-max Voronoi diagram of a set S of polygonal objects is studied and it is shown that it is equivalent to the Vor onoi diagram under the Hausdorff distance function.
Abstract: We study the min-max Voronoi diagram of a set S of polygonal objects, a generalization of Voronoi diagrams based on the maximum distance between a point and a polygon. We show that the min-max Voronoi diagram is equivalent to the Voronoi diagram under the Hausdorff distance function. We investigate the combinatorial properties of this diagram and give improved combinatorial bounds and algorithms. As a byproduct we introduce the min-max hull which relates to the min-max Voronoi diagram in the way a convex hull relates to the ordinary Voronoi diagram.
Journal Article•
TL;DR: Owing to the difficulty of constructing regular Voronoi diagram based on vector method, a raster based approach is developed, which employs the Arc/Info, to compute a few regular Vor onoi diagrams.
Abstract: Voronoi diagram is a method of partitioning a space, which has powerful potential in many fields. This paper presents two regular Voronoi diagrams: line weighted Voronoi diagram and area weighted Voronoi diagram. Owing to the difficulty of constructing regular Voronoi diagram based on vector method, it develops a raster based approach, which employs the Arc/Info, to compute a few regular Voronoi diagrams. Our programs can construct the following Voronoi diagrams: the ordinary Voronoi diagram, line Voronoi diagram, area Voronoi diagram, and the multiplicatively or additively or compoundly weighted Voronoi diagram generated by points or any figures(lines or polygons), in the plane. As there are a huge amount of grids, computing time in constructing line or polygon weighted Voronoi diagram is a bit more, and this problem is just the future research effort. Lastly, it makes an attempt to apply ordinary Voronoi diagram and weighted Voronoi diagram for delimitating city's affected coverage in Henan province whose experimental result shows that weighted Voronoi diagram is an efficient technique for delimitating economic object's affected coverage.
24 Feb 2002