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Showing papers on "Delaunay triangulation published in 1990"


Journal ArticleDOI
22 Oct 1990
TL;DR: It is shown how to triangulate a planar point set or a polygonally bounded domain with triangles of bounded aspect ratio, and how to produce a linear-size Delaunay triangulation of a multidimensional point set by adding a linear number of extra points.
Abstract: Several versions of the problem of generating triangular meshes for finite-element methods are studied. It is shown how to triangulate a planar point set or a 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 linear-size Delaunay triangulation of a multidimensional point set by adding a linear number of extra points. All the triangulations have size 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 well-shaped elements and small total size. >

358 citations


Proceedings ArticleDOI
01 Nov 1990
TL;DR: An algorithm for compositing a combination of density clouds and contour surfaces used to represent a scalar function on a 3-D volume subdivided into convex polyhedra, which provides a method for visualizing such data sets.
Abstract: We present an algorithm for compositing a combination of density clouds and contour surfaces used to represent a scalar function on a 3-D volume subdivided into convex polyhedra. The scalar function is interpolated between values defined at the vertices, and the polyhedra are sorted in depth before compositing. For n tetrahedra comprising a Delaunay triangulation, this sorting can always be done in O(n) time. Since a Delaunay triangulation can be efficiently computed for scattered data points, this provides a method for visualizing such data sets. The integrals for opacity and visible intensity along a ray through a convex polyhedron are computed analytically, and this computation is coherent across the polyhedron's projected area.

215 citations


Journal ArticleDOI
S. Rippa1
TL;DR: It is proved that the Delaunay triangulation of the data points minimizes the roughness measure of a PLIS, for any fixed set of function values.

174 citations


Journal ArticleDOI
TL;DR: A method that combines both approaches to fully automatic mesh generation is presented, which provides the linear growth rate and divide-and-conquer approach of the octree method with the simplicity and optimal properties of the Delaunay triangulation.
Abstract: Fully automatic three-dimensional mesh generation is a fundamental requirement for automating the numerical solution of partial differential equations. Two techniques in particular—the octree and Delaunay approaches—have been used towards this end. A method that combines both approaches to fully automatic mesh generation is presented here. The resulting algorithm provides the linear growth rate and divide-and-conquer approach of the octree method with the simplicity and optimal properties of the Delaunay triangulation.

159 citations


Journal ArticleDOI
TL;DR: In this article, a method for generating an unstructured triangular mesh in two dimensions, suitable for computing high Reynolds number flows over arbitrary configurations is presented, based on a Delaunay triangulation, which is performed in a locally stretched space, in order to obtain very high aspect ratio triangles in the boundary layer and the wake regions.

152 citations


Journal ArticleDOI
TL;DR: A coherent way of interpolating three-dimensional data obtained by stereo, for example, with a simplicial polyhedral surface based on the use of the constrained Delaunay triangulation is proposed.

130 citations


Book ChapterDOI
01 Jul 1990
TL;DR: A new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations is given which obviates the need for building a separate point-location structure for nearest-neighbor queries.
Abstract: In this paper we give a new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations. The new algorithm is more “online” than earlier similar methods, takes expected time O(n log n) and space O(n), and is eminently practical to implement. The analysis of the algorithm is also interesting in its own right and can serve as a model for many similar questions in both two and three dimensions. Finally we demonstrate how this approach for constructing Voronoi diagrams obviates the need for building a separate point-location structure for nearest-neighbor queries.

99 citations


Journal ArticleDOI
TL;DR: It is shown that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness, and this characterization leads to a proof that all graphs that can be realized as polytopes inscribed in a sphere are 1-Tough.
Abstract: We show that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness. A graphG is1-tough if for any setP of vertices,c(G?P)≤|G|, wherec(G?P) is the number of components of the graph obtained by removingP and all attached edges fromG, and |G| is the number of vertices inG. This property arises in the study of Hamiltonian graphs: all Hamiltonian graphs are 1-tough, but not conversely. We also show that all Delaunay triangulationsT satisfy the following closely related property: for any setP of vertices the number of interior components ofT?P is at most |P|?2, where an interior component ofT?P is a component that contains no boundary vertex ofT. These appear to be the first nontrivial properties of a purely combinatorial nature to be established for Delaunay triangulations. We give examples to show that these bounds are best possible and are independent of one another. We also characterize the conditions under which a degenerate Delaunay triangulation can fail to be 1-tough. This characterization leads to a proof that all graphs that can be realized as polytopes inscribed in a sphere are 1-tough. One consequence of the toughness results is that all Delaunay triangulations and all inscribable graphs have perfect matchings.

73 citations


Journal ArticleDOI
TL;DR: This paper shows that the in_front/behind relation defined for the faces of C with respect to any fixed viewpointx is acyclic, which has applications to hidden line/surface removal and other problems in computational geometry.
Abstract: LetC be a cell complex ind-dimensional Euclidean space whose faces are obtained by orthogonal projection of the faces of a convex polytope ind+ 1 dimensions. For example, the Delaunay triangulation of a finite point set is such a cell complex. This paper shows that the in_front/behind relation defined for the faces ofC with respect to any fixed viewpointx is acyclic. This result has applications to hidden line/surface removal and other problems in computational geometry.

69 citations


Journal ArticleDOI
TL;DR: The mutual arrangement of the Delaunay simplices (configurations of four nearest atoms) has been studied for molecular dynamic (MD) models of liquid and quenched rubidium obtained by M. Tanaka as mentioned in this paper.
Abstract: The mutual arrangement of the Delaunay simplices (configurations of four nearest atoms) has been studied for molecular dynamic (MD) models of liquid and quenched rubidium obtained by M. Tanaka [J. Phys. Soc. Jpn. 5 5, 3108 (1986)]. The Delaunay simplices with large circumradii and low local density of atoms, the simplices with small circumradii representing dense atomic configurations, and the simplices close in shape to perfect tetrahedron representing ‘‘rigid’’ arrangements of atomic quadruplets were delineated. The Delaunay simplices form clusters; consideration of the latter constitutes a site percolation problem on the Voronoi network [N. N. Medvedev, V. P. Voloshin, and Yu. I. Naberukhin, J. Phys. A: Math. Gen. 2 1, L247 (1986)]. Analysis of the MD results in these terms shows that low density atomic configurations in the liquid phase form a percolative cluster. Such a cluster does not occur in a solid phase. On the contrary, there is a percolative cluster in the solid sample, but formed by high density configurations which are nearly tetrahedral.

65 citations


Journal ArticleDOI
TL;DR: An algorithm is described which uses matching Delaunay triangles to achieve a form of geometric correction that can cope with high frequency distortion and has the capacity to correct images which exhibit very complex patterns of distortion.
Abstract: A major factor limiting the usefulness of airborne scanner imagery such as ATM has been the inability of conventional geometric correction procedures to remove high frequency distortion induced by platform and relief effects. An algorithm is described which uses matching Delaunay triangles to achieve a form of geometric correction that can cope with high frequency distortion. Early experimental results show that the procedure has the capacity to correct images which exhibit very complex patterns of distortion.

Journal ArticleDOI
TL;DR: A method for interpolating scattered data using C1 piecewise cubic surfaces based on data-dependent triangulations based on the Delaunay triangulation is discussed.

Journal ArticleDOI
TL;DR: It is shown that any linear triangulation of a simple polygon can be realized as a (combinatorially equivalent) Delaunay triangulations.

Book ChapterDOI
01 Jan 1990
TL;DR: The purpose of this survey article is to review the state-of-the-art methods in curve and surface fitting and Computer-Aided Geometric Design, with the hope that some of the recently developed methods may be useful in designing new and improved reconstruction techniques.
Abstract: This paper is concerned with the construction of a mathematical model of a three dimensional object, starting from cross-sectional data. The problem is to build a model which is suitable for displaying and manipulating an image of the object using a graphics workstation. This problem is of great importance in a number of fields, particularly in medical imaging. While it has been intensely studied by various researchers, it seems that approximation theorists have had little involvement. The purpose of this survey article is to review the state-of-the-art, with the hope that some of the recently developed methods in curve and surface fitting and Computer-Aided Geometric Design may be useful in designing new and improved reconstruction techniques.

Journal ArticleDOI
TL;DR: A method is proposed whereby boundary data are supplemented with points to ensure that imposed boundary edges are preserved during the Delaunay triangulation.
Abstract: The Delaunay triangulation has recently received attention as a viable method for construction computational meshes. However, an arbitrary boundary definition which must be preserved in the triangulation process will not, in general, satisfy the geometrical definition on which the Delaunay construction is founded. The effect of this is that the integrity of the given boundary edges will be violated and the computational mesh will not conform to the applied geometrical shape. A method is proposed whereby boundary data are supplemented with points to ensure that imposed boundary edges are preserved during the Delaunay triangulation. The method is illustrated on a geometry of an estuary which exhibits highly complex geometrical features.

Journal ArticleDOI
TL;DR: The linear elastic Delaunay network model developed in a previous paper is used to obtain further results on mechanical properties of graph-representable materials and an increase of effective moduli and a decrease of their scatter are observed.
Abstract: The linear elastic Delaunay network model developed in a previous paper is used to obtain further results on mechanical properties of graph-representable materials. First, we investigate the error involved in the uniform strain approximation — a computationally inexpensive approach widely employed in the determination of effective moduli of granular and fibrous media. Although this approximation gives an upper bound on the macroscopic moduli, it results in very good estimates of their second order statistics. In order to derive a lower bound another window definition has to be introduced. Also, an energy-based derivation of both bounds is given. The final result relates to a modification of a Delaunay network so that its vertices correspond to the centroids of cells of the corresponding Voronoi tessellation; an increase of effective moduli and a decrease of their scatter are observed.

Journal ArticleDOI
TL;DR: A Delaunay triangulation of a set of nodes is a collection of triangles whose vertices are at the nodes and whose union fills the convex hull of the set of node, making it useful for solving closest point problems.
Abstract: A Delaunay triangulation of a set of nodes is a collection of triangles whose vertices are at the nodes and whose union fills the convex hull of the set of nodes. It also has several geometrical properties, making it useful for solving closest point problems. The generalization presented here allows the triangulation to cover nonconvex regions including those with holes. Although a variety of such generalizations are possible, the one presented here is shown to retain important closest point characteristics. Thus it is useful for determining shortest paths within planar regions with polygonal boundaries.


Journal ArticleDOI
TL;DR: Some of the trade-offs involved in different implementation strategies, focusing on the tasks of learning discrete classifications and smooth nonlinear mappings are examined, and an important source of computational inefficiency is identified.

Proceedings ArticleDOI
01 Jan 1990
TL;DR: A number of generalizations of the Voronoi diagram have been presented, including an algorithm for general distance measures that Minkowski called convex distance functions, where the “unit circle” can be defined to be any convex shape.
Abstract: The Voronoi Diagram for a set of data points (called sites) is a subdivision of the plane into regions, one region for each site. The region belonging to a given site consists of that portion of the plane closer to it than to any of the other sites. Shamos and Hoey [SH75] gave an optimal O(n log n) algorithm for computing the Voronoi diagram and showed that it leads to optimal algorithms for a number of other problems in computer science. The diagram has been re-discovered many times for applications in geography, meteorology, biology, anthropology, archeology, astronomy, geology, physics, metallurgy, and statistics. Boots [B87] gives a summary of many of these applications, with dozens of references. The geometric dual of the Voronoi diagram is called the Delaunay diagram. It has the property that every edge is contained inside a circle with no sites (other than its endpoints) in its interior or on its boundary. If no four sites are cocircular, the Delaunay diagram is a triangulation of the sites. If not, it can be extended to a triangulation by arbitrarily triangulating regions with more than three sides. Any such triangulation is called a Delaunay triangulation. It has the property that every edge is contained in a circle with no sites in its interior. Such a circle is calledpoint-free. A number of generalizations of the Voronoi diagram have been presented. The ones relevant to this paper define the diagram for other distance measures. Hwang gives an algorithm for the Ll metric [H79]. Lee and Wong give an algorithm for the Ll and L, metrics bW80]. Lee gives an algorithm for general Lp metrics jL80]. Widmayer, Wu, and Wong give an algorithm for metrics where paths between points are limited to a fixed number of orientations [WWW87]. Chew and Drysdale give an algorithm for general distance measures that Minkowski called convex distance functions, where the “unit circle” can be defined to be any convex shape [CD85]. All of the other metrics Unfortunately, the algorithms discussed above are not easy to implement. Some do not handle special cases like cocircular points or points lying on a line parallel to a flat side of the convex shape. The data structures and algorithms to represent and manipulate bisector curves are complex. There can be problems with numerical stability. Robert Collins implemented Chew and Drysdale’s algorithm in 115 pages of very well documented Mesa code, with about three quarters of the code handling computing bisectors, * Dept. Math & CS, Dartmouth College, Hanover, NH 03755 + Supported in part by National Science Foundation Grant DMC-8704147.

Journal ArticleDOI
S. Rippa1, B. Schiff1
TL;DR: It is shown in this paper that the well-known Delaunay triangulation of a set of points in R2 is a minimum energy triangulations for the energy functional associated with the nonhomogeneous Laplace equation.
Abstract: We consider the family of finite element spaces of fixed dimension, defined over all triangulations of a given polygonal domain in R2, based on a common set of nodes (vertices). A minimum energy triangulation, relative to a given elliptic problem, is a triangulation for which the finite element solution has the minimal energy. The minimum energy triangulation can be considered optimal, as it minimizes the solution error in the natural norm associated with the problem. It is shown in this paper that the well-known Delaunay triangulation of a set of points in R2 is a minimum energy triangulation for the energy functional associated with the nonhomogeneous Laplace equation. Minimum energy triangulations may be very expensive to compute. In this paper, therefore, we present algorithms for constructing locally minimal energy triangulations and outline efficient schemes for computing sub-optimal triangulations. In both cases the basic idea is to improve an initial triangulation by using local operations on the edges of the triangulation. It is shown for several model problems that such sub-minimal energy triangulations can significantly improve the quality of the approximate solution.

Journal ArticleDOI
TL;DR: A novel algorithm for computing optimal constrained triangulation is presented which is equally applicable to 2-D and 3-D optimal constrained Triangulation and Delaunay triangulations and has been applied to finite-element mesh generation.
Abstract: A novel algorithm for computing optimal constrained triangulation is presented which is equally applicable to 2-D and 3-D optimal constrained triangulation and Delaunay triangulation. This algorithm has no degenerate and near-degenerate problems. The same amount of time is needed to add a new point to an existing mesh of any element number provided that the element it belongs to has been predetermined, as in the self-adaptive finite-element analysis process. This algorithm has been applied to finite-element mesh generation. Test results are given. >

Book ChapterDOI
01 Jul 1990
TL;DR: This work presents the first quadratic-time algorithm for the greedy triangulation of a finite planar point set, and the first linear-time algorithms for the greedier triangulating of a convex polygon.
Abstract: We present the first quadratic-time algorithm for the greedy triangulation of a finite planar point set, and the first linear-time algorithm for the greedy triangulation of a convex polygon.

Proceedings ArticleDOI
01 May 1990
TL;DR: The results generalize to arbitrary fixed dimensions, to geometric bodies other than spheres, and to geometric structures other than Delaunay triangulations.
Abstract: We show that for any set Π of n points in three-dimensional space there is a set Q of 𝒪(n1/2 log3 n) points so that the Delaunay triangulation of Π ∪ Q has at most 𝒪(n3/2 log3 n) edges — even though the Delaunay triangulation of Π may have Ω(n2) edges. The main tool of our construction is the following geometric covering result: For any set Π of n points in three-dimensional space and any set S of m spheres, where each sphere passes through a distinct point pair in Π, there exists a point x, not necessarily in Π, that is enclosed by Ω(m2/n2 log3 n2/m) of the spheres in S.Our results generalize to arbitrary fixed dimensions, to geometric bodies other than spheres, and to geometric structures other than Delaunay triangulations.

Journal ArticleDOI
TL;DR: It is shown that the “oriented walk” search, when the total number of points is less than 417 or when the points are presorted by distance or coordinates, is successful.
Abstract: A new, dynamic, hierarchical subdivision and recursive algorithm for computing Delaunay triangulations is presented. The algorithm has four main steps: (1) location of the already formed triangle that contains the point (2) identification of other adjoining triangles whose circumcircle contains the point (3) formation of the new triangles, and (4) database update. Different search procedures are analyzed. It is shown that the “oriented walk” search, when the total number of points is less than 417 or when the points are presorted by distance or coordinates. The algorithm has point-deletion capabilities which are discussed in detail.

Journal ArticleDOI
TL;DR: In this article, a suite de procedures de changement de diagonale de Delaunay are presented. André et al. deduit plusieurs caracterisations des triangulations de Delauau.
Abstract: Apres avoir rappele quelques resultats sur les maillages de Voronoi et Delaunay, on prouve, sous certaines hypotheses, que toute triangulation d'un polygone peut etre modifiee par une suite de procedures de changement de diagonale afin d'obtenir une triangulation de Delaunay. On en deduit plusieurs caracterisations des triangulations de Delaunay

Journal ArticleDOI
TL;DR: An algorithm with worst case time complexity O(log/sup 2/N) in two dimensions and O(m/sup 1/2/log N) in three dimensions with N input points and m as the number of tetrahedra in triangulation is given.
Abstract: An algorithm with worst case time complexity O(log/sup 2/N) in two dimensions and O(m/sup 1/2/log N) in three dimensions with N input points and m as the number of tetrahedra in triangulation is given. Its AT/sup 2/ VLSI complexity on Thompson's logarithmic delay model, (1983) is O(N/sup 2/log/sup 6/N) in two dimensions and O(m/sup 2/Nlog/sup 4/ N) in three dimensions. >

Journal ArticleDOI
TL;DR: In this article, simplicial configurations of a tetrahedral and quartoctahedral form are considered for molecular-dynamic models of liquid and quenched rubidium and the results of a detailed analysis of finite clusters, depending on the fraction of coloured sites for different colours, are proposed.
Abstract: Simplicial atomic configurations (four nearest atoms, defining the Delaunay simplices) are considered for molecular-dynamic models of liquid and quenched rubidium. The aggregates, involving simplicial configurations of a tetrahedral and quartoctahedral form are studied. The problem is reduced by the general Voronoi-Delaunay approach to the study of clusters of the coloured sites on the Voronoi network in terms of percolation theory. The results of a detailed analysis of finite clusters, depending on the fraction of coloured sites for different colours, are proposed. The simplicial atomic configurations, close in shape to regular tetrahedra, determine the T-coloured sites on the Voronoi network and those close in shape to regular quartoctahedra determine the O-coloured ones. Small, medium and large clusters and the backbones of finite ones are studied for this model. The percolation thresholds are determined for the T and O colouring. A topological similarity between the aggregates of tetrahedral simplices in liquid and quenched states is revealed. The significant role of the quartoctahedral simplicial configurations in the quenched state structure is emphasized.

Proceedings ArticleDOI
13 May 1990
TL;DR: Two O(N log N) heuristic approaches for planning viewpoints in 2D known environments are described, both based on the generalized Delaunay triangulation of a polygon with holes.
Abstract: Two O(N log N) heuristic approaches for planning viewpoints in 2D known environments are described. Both are based on the generalized Delaunay triangulation of a polygon with holes. The main difference between them is the heuristic strategies used to merge the triangles to form the partitioning or covering star polygons. When the environment needs to be explored, an O(N/sup 2/ log N) heuristic scheme which successively selects the viewpoints and views for mobile robots is used. Upper bounds on the number of planned viewpoints and views for each scheme are estimated. >

Journal ArticleDOI
TL;DR: Lawson’s triangulation method, which maximizes the smallest angle over all triangulations, controls error bounds for linear interpolates as well as for certain derivative estimates in polynomial interpolation schemes.
Abstract: One way of evaluating a triangulation method is to study how well it controls error bounds and stability in computations. Lawson’s method, which maximizes the smallest angle over all triangulations, controls error bounds for linear interpolates as well as for certain derivative estimates. In particular, no other triangulation improves these bounds by more than a factor of 2. Although Lawson’s triangulation method does not minimize the largest edge over all triangulations, it nearly does; this edge is no longer than ${2 / {\sqrt 3 }}$ times the longest edge in any other triangulation. Thus, Lawson’s triangulation method controls error bounds in polynomial interpolation schemes.