Showing papers on "Delaunay triangulation published in 1993"
01 Jul 1993
TL;DR: This paper presents a technique for creating high-quality triangular meshes for regions on curved surfaces, an extension of previous methods developed for regions in the plane.
Abstract: For several commonly-used solution techniques for partial differential equations, the first step is to divide the problem region into simply-shaped elements, creating a mesh. We present a technique for creating high-quality triangular meshes for regions on curved surfaces. This technique is an extension of previous methods we developed for regions in the plane. For both flat and curved surfaces, the resulting meshes are guaranteed to exhibit the following properties: (1) internal and external boundaries are respected, (2) element shapes are guaranteed—all elements are triangles with angles between 30 and 120 degrees (with the exception of badly shaped elements that may be required by the specified boundary), and (3) element density can be controlled, producing small elements in “interesting” areas and large elements elsewhere. An additional contribution of this paper is the development of a practical generalization of Delaunay triangulation to curved surfaces.
467 citations
13 Sep 1993
TL;DR: In this article, it was shown that Hebbian learning with competition leads to lateral connections that correspond to the edges of the induced Delaunay triangulation and leads to a network structure that forms a topology preserving map of a given manifold, independent of the manifold's topology.
Abstract: The problem of forming perfectly topology preserving maps of feature manifolds is studied. First, through introducing “masked Voronoi polyhedra” as a geometrical construct for determining neighborhood on manifolds, a rigorous definition of the term “topology preserving feature map” is given. Starting from this definition, it is shown that a network G of neural units i, i = 1, …, N has to have a lateral connectivity structure A, Aij ∈ {0, 1}, i, j = 1,…, N which corresponds to the “induced Delaunay triangulation” of the synaptic weight vectors wi ∈ ℜDin order to form a perfectly topology preserving map of a given manifold M ⊆ ℜD of features v ∈ M. The lateral connections determine the neighborhood relations between the units in the network, which have to match the neighborhood relations of the features on the manifold. If all the weight vectors wi are distributed over the given feature manifold M, and if this distribution resolves the shape of M, it can be shown that Hebbian learning with competition leads to lateral connections i —j (Aij = 1) that correspond to the edges of the “induced Delaunay triangulation” and, hence, leads to a network structure that forms a perfectly topology preserving map of M, independent of M’s topology. This yields a means for constructing perfectly topology preserving maps of arbitrarily structured feature manifolds.
320 citations
TL;DR: This work is devoted to the description of an efficient unstructured mesh generation method entirely based on the Delaunay triangulation, which has the distinctive characteristic of the proposed method is that point positions and connections are computed simultaneously.
291 citations
IBM1
TL;DR: Simulations show that the final topology is degree-bounded, has a rather regular and uniform structure, and has throughput and reliability that are greater than that of a number of alternative topologies.
Abstract: A distributed topology-control algorithm has been developed for each node in a packet radio network (PRN) to control its transmitting power and logical neighbors for a reliable high-throughput topology. The algorithm first constructs a planar triangulation from locations of all nodes as a starting topology. Then, the minimum angles of all triangles in the planar triangulation are maximized by means of edge switching to improve connectivity and throughput. The resulting triangulation at this stage, the Delaunay triangulation, can be determined locally at each node. The topology is modified by negotiating among neighbors to satisfy a design requirement on the nodal degree parameter. Simulations show that the final topology is degree-bounded, has a rather regular and uniform structure, and has throughput and reliability that are greater than that of a number of alternative topologies. >
275 citations
TL;DR: A linear-time Convex Hull Insertion algorithm is presented to construct TINs for a set of points as well as specific features such as constraint breaklines and exclusion boundaries.
Abstract: The Delaunay triangulation is commonly used to generate triangulated irregular network (TIN) models for a best description of the surface morphology in a variety of applications in geographic information systems (GIS). This paper discusses the definitions and basic properties of the standard and constrained Delaunay triangulations. Several existing Delaunay algorithms are reviewed and classified into three categories according to their procedures: (1) divide-and-conquer methods, (2) incremental insertion methods, and (3) triangulation growth methods. Furthermore, a linear-time Convex Hull Insertion algorithm is presented to construct TINs for a set of points as well as specific features such as constraint breaklines and exclusion boundaries. Empirical results over various sets of up to 50000 points on personal computers show that the proposed algorithm efficiently expedites the construction of TIN models in approximately O(N) for N randomly distributed points.
189 citations
01 Jan 1993
TL;DR: A simple new algorithm for triangulating polygons and planar straightline graphs that provides "shape" and "size" guarantees and extends a mesh generation technique of Chew by allowing triangles that vary in size.
Abstract: We present a simple new algorithm for triangulating polygons and planar straightline graphs. It provides "shape" and "size" guarantees: -All triangles have a bounded aspect ratio. - The number of "Steiner points" added is within a constant factor of optimal. Such "quality" triangulations are desirable as meshes for the finite element method, in which the running time generally increases with the number of triangles, and where the convergence and stability may be hurt by very skinny triangles. The technique we use--successive refinement of the Delaunay triangulation--extends a mesh generation technique of Chew by allowing triangles that vary in size. Previous algorithms with shape and size bounds have all been based on quadtrees. The Delaunay refinement algorithm matches their bounds, but uses a fundamentally different approach. It is much simpler, and hence easier to implement, and it generally produces smaller meshes in practice.
177 citations
29 Jul 1993
TL;DR: In this paper, the Delaunay triangulation of object contours is used for 3D reconstruction from cross-sections, and the reconstruction of complex shapes is improved by adding vertices on and inside contours.
Abstract: We propose a solution to the problem of 3D reconstruction from cross-sections, based on the Delaunay triangulation of object contours. Its properties--especially the close relationship to the medial axis--enable us to do a compact tetrahedrization resulting in a nearest-neighbor connection. The reconstruction of complex shapes is improved by adding vertices on and inside contours.
168 citations
01 Jan 1993
TL;DR: The theme of this chapter is a rather simple method that has proved very potent in the analysis of the expected performance of various randomized algorithms and data structures in computational geometry, which is to analyze a randomized algorithm as if it were running backwards in time, from output to input.
Abstract: The theme of this chapter is a rather simple method that has proved very potent in the analysis of the expected performance of various randomized algorithms and data structures in computational geometry. The method can be described as “analyze a randomized algorithm as if it were running backwards in time, from output to input.” We apply this type of analysis to a variety of algorithms, old and new, and obtain solutions with optimal or near optimal expected performance for a plethora of problems in computational geometry, such as computing Delaunay triangulations of convex polygons, computing convex hulls of point sets in the plane or in higher dimensions, sorting, intersecting line segments, linear programming with a fixed number of variables, and others.
141 citations
TL;DR: The Delaunay scheme permits certain edges to be specified in the final triangulation, such as those that correspond to domain boundaries or natural interfaces, and is suitable for mesh generation and contour plotting applications.
119 citations
TL;DR: An algorithm for triangulating 2-D data points that is based on a uniform grid structure and a triangulation strategy that builds triangles in a circular fashion is discussed and two ways to compute the convex hull using the algorithm are presented.
Abstract: An algorithm for triangulating 2-D data points that is based on a uniform grid structure and a triangulation strategy that builds triangles in a circular fashion is discussed. The triangulation strategy lets the algorithm eliminate points from the internal data structure and decreases the time used to find points to form triangles, given an edge. The algorithm has a tested linear time complexity that significantly improves on that of other methods. As a by-product, the algorithm produces the convex hull of the data set at no extra cost. Two ways to compute the convex hull using the algorithm are presented. The first is based on the edge list and the second is based on the grid structure. >
110 citations
TL;DR: In this paper, it was shown that a randomized construction of the Delaunay tree can be done in O(n log n) expected time in the plane and in O (n⌈d2⌉) time in d-dimensional space.
01 Oct 1993
TL;DR: Algorithms and data structures are presented that construct and store the entire family of alpha shapes of a finite point set in three-dimensional space, derived from the Delaunay triangulation of the point set, with a quadratic time and space complexity.
Abstract: Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is often useful or required to compute what one might call the ``shape'''' of the set. For that purpose, this thesis deals with the formal notion of the family of alpha shapes of a finite point set in three-dimensional space. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a real parameter controlling the desired level of detail. Algorithms and data structures are presented that construct and store the entire family of shapes, with a quadratic time and space complexity, in the worst case. Implementations of the algorithms are discussed, with an emphasis on the robust construction of three-dimensional Delaunay triangulations. A general-purpose programming technique, called Simulation of Simplicity, is used to cope with degenerate input data. This method relieves the programmer from the task of providing a consistent treatment for every single special case that can occur. The programs that use the technique tend to be considerably smaller and more robust than others.
TL;DR: In this paper, an efficient simulation method for obtaining a random particle model for concrete is outlined, where the available space (two-dimensional) is divided in separate areas, using a Delaunay triangulation.
Abstract: In this paper, an efficient simulation method for obtaining a random particle model for concrete is outlined. First, the ‘take-and-place method’ and its extension, the ‘directed searching process’, are discussed briefly and the shortcomings are indicated. A new method, the ‘divide-and-fill method’, appears to be more convenient, especially when only a small computer is used. In this simulation method the available space (two-dimensional) is divided in separate areas, using a Delaunay triangulation. These areas are filled with particles taking into account a given grading curve and gravel content. Comparison with physical concrete sections, obtained by means of image analysis, shows that the results of this method closely represent reality. The divide-and-fill method also yields a finite-element mesh in a quasi-automatic way.
TL;DR: It is proved that, for everyG withn vertices andm edges, there is a completion of a Delaunay triangulation ofO(m2n) points that conforms to G.
Abstract: A plane geometric graphC in ?2conforms to another such graphG if each edge ofG is the union of some edges ofC. It is proved that, for everyG withn vertices andm edges, there is a completion of a Delaunay triangulation ofO(m2n) points that conforms toG. The algorithm that constructs the points is also described.
01 Aug 1993
TL;DR: The paper deals with the parallelization of Delaunay triangulation algorithms, giving more emphasis to pratical issues and implementation than to theoretical complexity.
Abstract: The paper deals with the parallelization of Delaunay triangulation algorithms, giving more emphasis to pratical issues and implementation than to theoretical complexity. Two parallel implementations are presented. The first one is built on De Wall, an Ed triangulator based on an original interpretation of the divide & conquer paradigm. The second is based on an incremental construction algorithm. The parallelization strategies are presented and evaluated. The target parallel machine is a distributed computing environment, composed of coarse grain processing nodes. Results of first implementations are reported and compared with the performance of the serial versions running on a Unix workstation.
TL;DR: An abstract view of the edge insertion paradigm is presented, and it is shown that it gives polynomial-time algorithms for several types of optimal triangulations, including minimizing the maximum slope of a piecewise-linear interpolating surface.
Abstract: Edge insertion iteratively improves a triangulation of a finite point set in ?2 by adding a new edge, deleting old edges crossing the new edge, and retriangulating the polygonal regions on either side of the new edge. This paper presents an abstract view of the edge insertion paradigm, and then shows that it gives polynomial-time algorithms for several types of optimal triangulations, including minimizing the maximum slope of a piecewise-linear interpolating surface.
01 Jan 1993
TL;DR: In this article, the Delaunay triangulation of these points is combined with the desirable point placement features, boundary integrity, and robustness traditionally associated with advancing-front-type mesh generation strategies.
Abstract: A new algorithm is described for generating an unstructured mesh about an arbitrary two-dimensional configuration. Mesh points are generated automatically by the algorithm in a manner which ensures a smooth variation of elements, and the resulting triangulation constitutes the Delaunay triangulation of these points. The algorithm combines the mathematical elegance and efficiency of Delaunay triangulation algorithms with the desirable point placement features, boundary integrity, and robustness traditionally associated with advancing-front-type mesh generation strategies. The method offers increased robustness over previous algorithms in that it cannot fail regardless of the initial boundary point distribution and the prescribed cell size distribution throughout the flow-field.
TL;DR: The approach proposed here combines the Delaunay triangulation with ideas from the advancing front method, which uses a background grid to interpolate local mesh size parameters that are taken from the triangulations of the given boundary nodes.
Abstract: SUMMARY The past decade has known an increasing interest in the solution of the Euler equations on unstructured grids due to the simplicity with which an unstructured grid can be tailored around very complex geometries and be adapted to the solution. It is desirable that the mesh can be generated with minimum input from the user, ideally, just specifying the boundary geometry and, perhaps, a function to prescribe some desired mesh size. The internal nodes should then be found automatically by the grid generation code. The approach we propose here combines the Delaunay triangulation with ideas from the advancing front method of Peraire et al. and Lijhner et al. Both methods are briefly reviewed in Section 1. Our method uses a background grid to interpolate local mesh size parameters that is taken from the triangulation of the given boundary nodes. Geometric criteria are used to find a set of nodes in a frontal manner. This set is subsequently introduced into the existing mesh, thus providing an updated Delaunay triangulation. The procedure is repeated until no more improvement of the grid can be achieved by inserting new nodes.
TL;DR: This work presents an efficient, numerically stable update algorithm for the topological structure of the Voronoi diagram in a dynamic scene, using only O(log n) time for each change, and develops fast algorithms for inserting and deleting points at the edge of the dynamic scene.
TL;DR: In this paper, the authors characterize the extent to which Kapouleas'examples are typical of all embedded constant mean curvature surfaces and give an approximate solution to the problem by centering radius R = 2 spheres at all vertices and connecting these smoothly by Delaunay segments having R < 2 (but near 2) with axes along the graph edges.
Abstract: In a recent ground-breaking paper Kapouleas constructed this century's first examples of complete, finite topology, properly embedded surfaces with (non-zero) constant mean curvature [Ka l ] . Geometrically these surfaces are quite easy to visualize, as we explain below.The purpose of the present paper is to characterize the extent to which Kapouleas ' examples are typical of all embedded constant mean curvature surfaces. First, and throughout this paper, we scale space so that the (constant) mean curvature (the trace of the second fundamental form) of our surface L" is identically one, and we call such a surface an MC1 surface. The only complete embedded MC1 surfaces of finite topology known before Kapouleas ' work arise from the 1parameter family of axially-symmetric (and periodic) examples found by Delaunay in 1841: Using the maximum axial radius R of such a surface as a parameter, this family begins with the cylinder (R = 1), and evolves through embedded, periodic surfaces until the family degenerates to a linear chain of touching spheres (R = 2); the Delaunay family continues for all R > 2 as periodic immersions. Kapouleas ' construction of embedded surfaces begins with a piecewise linear graph in R 3, having the property that edges are either finite segments (with length a multiple of 4) or half-infinite rays, and with the further property that at each vertex some linear combination of their direction vectors, with positive coefficients, adds to zero. (See Fig. 1.) He constructs an approximate solution to the problem by centering radius R = 2 spheres at all vertices, and connecting these smoothly by Delaunay segments having R < 2 (but near 2) with axes along the graph-edges. (Since the periods of the Delaunay surfaces are slightly longer than 4, he must assume also a "flexibility" condition on the graph, to guarantee that such a gluing
TL;DR: Two coupled particle-finite volume methods which use the properties of Delaunay-Voronoi?
TL;DR: A Convex Hull Insertion algorithm for constructing the Delaunay triangulation and the Voronoi diagram of randomly distributed points in the Euclidean plane is described.
TL;DR: In this article, it was shown that such an algorithm does not exist and recursion-theoretic limitations of any algorithm designed to perform approximate calculations of sums over all possible triangulations of a compact four-dimensional manifold.
Abstract: Computations in dynamical triangulation models of four-dimensional Quantum Gravity involve weighted averaging over sets of all distinct triangulations of compact four-dimensional manifolds. In order to be able to perform such computations one needs an algorithm which for any givenN and a given compact four-dimensional manifoldM constructs all possible triangulations ofM with ≤N simplices. Our first result is that such algorithm does not exist. Then we discuss recursion-theoretic limitations of any algorithm designed to perform approximate calculations of sums over all possible triangulations of a compact four-dimensional manifold.
01 Jan 1993
TL;DR: The paper proposes an automatic way to generate the coarse meshes given the finest one using the Delaunay-Voronoi algorithm and it is shown that it owns good properties with respect to multigrid algorithms.
Abstract: For finite-element non-structured type meshes, the non-nested multigrid algorithms require to build a sequence of independent meshes. The paper proposes an automatic way to generate the coarse meshes given the finest one. The method first eliminates a set of points from the current mesh level and then uses the Delaunay-Voronoi algorithm to triangulate the remaining set of points. The algorithm is presented and it is shown that it owns good properties with respect to multigrid algorithms. Several examples of its application to bi-dimensional meshes are presented.
TL;DR: It is shown that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time $O(n^2 )$.
Abstract: It is shown that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time $O(n^2 )$. The algorithm is reasonably easy to implement and is based on the theorem that there is a triangulation with minmax edge length that contains the relative neighborhood graph of the points as a subgraph. With minor modifications the algorithm works for arbitrary normed metrics.
TL;DR: It is proved that local transformations can be used to construct a Delaunay triangulation of a set of nk-dimensional points for any $k \geq 2$ and algorithms using this approach are presented.
Abstract: In [SIAM J. Sci. Statist. Comput.,10 (1989), pp. 718–741] and [Comput. Aided Geom. Des., 8 (1991), pp. 123–142] the author presented algorithms that use local transformations to construct a Delaunay triangulation of a set of n three-dimensional points. This paper proves that local transformations can be used to construct a Delaunay triangulation of a set of nk-dimensional points for any $k \geq 2$, and presents algorithms using this approach. The empirical time complexities of these algorithms are discussed for sets of random points from the uniform distribution as well as worst-case time complexities. These time complexities are about the same or better than those of other algorithms for constructing k-dimensional Delaunay triangulations (when $k \geq 3$).
01 Dec 1993
TL;DR: A parallel algorithm for constructing the Delaunay triangulation of a set of vertices in three-dimensional space is presented, which achieves a fast running time and good scalability over a wide range of problem sizes and machine sizes.
Abstract: A parallel algorithm for constructing the Delaunay triangulation of a set of vertices in three-dimensional space is presented. The algorithm achieves a high degree of parallelism by starting the construction from every vertex and expanding over all open faces thereafter. In the expansion of open faces, the search is made faster by using a bucketing technique. The algorithm is designed under a data-parallel paradigm. It uses segmented list structures and virtual processing for load-balancing. As a result, the algorithm achieves a fast running time and good scalability over a wide range of problem sizes and machine sizes. A topological check is incorporated to eliminate inconsistencies due to degeneracies and numerical errors. The algorithm is implemented on Connection Machines CM-2 and CM-5, and experimental results are presented.
TL;DR: An algorithm to construct a tetrahedrization that incorporates specified convex, planar, polygonal regions, including triangles, as unions of facets of tetrahedral regions is described.
TL;DR: An algorithm is proposed for the efficient generation of admissible meshes suitable for the integration of the device equations with the usual numerical schemes based on the iterative refinement of coarse elements avoiding the "obtuse angle problem" by construction.
Abstract: This paper addresses the problem of the allocation of spatial grids for complex nonplanar three-dimensional (3-D) semiconductor device structures. We have characterized the class of meshes suitable for the integration of the device equations with the usual numerical schemes as being a subclass of the class of Delaunay meshes. We propose an algorithm for the efficient generation of such admissible meshes based on the iterative refinement of coarse elements. The generated meshes permit an exact geometrical modeling of rather general domain boundaries of modern silicon devices avoiding the "obtuse angle problem" by construction. >
01 Mar 1993
TL;DR: A strategy for adaptive mesh generation is proposed that consists of the use of a suitably defined density function, which can either be defined by the user or be calculated from a previous approximate solution, to guide the generation of a new mesh.
Abstract: A strategy for adaptive mesh generation is proposed. The method consists of the use of a suitably defined density function, which can either be defined by the user or be calculated from a previous approximate solution, to guide the generation of a new mesh. This new mesh is built starting from a minimal number of triangular elements, which are then, in several sweeps, repeatedly refined according to the density function. The Delaunay algorithm is used in each stage to keep the shape of the triangles as equilateral as possible. >