Abstract: STRIPACK is a Fortran 77 software package that employs an incremental algorithm to construct a Delaunay triangulation and, optionally, a Voronoi diagram of a set of points (nodes) on the surface of the unit sphere. The triangulation covers the convex hull of the nodes, which need not be the entire surface, while the Voronoi diagram covers the entire surface. The package provides a wide range of capabilities including an efficient means of updating the triangulation with nodal additions or deletions. For N nodes, the storage requirement for the triangulation is 13N integer storage locations in addition to 3N nodal corrdinates. Using an off-line algorithm and work space of size 3N, the triangulation can be constructed with time complexity O(NlogN).

Abstract: SUMMARY This paper aims to outline the dierent phases necessary to implement a Delaunay-type automatic mesh generator. First, it summarizes this method and then describes a variant which is numerically robust by mentioning at the same time the problems to solve and the dierent solutions possible. The Delaunay insertion process by itself, the boundary integrity problem, the way to create the eld points as well as the optimization procedures are discussed. The two-dimensional situation is described fully and possible extensions to the three-dimensional case are briey indicated. ? 1997 by John Wiley & Sons, Ltd. There exist a large number of papers dealing with 2-D-mesh generation using the Delaunay method (see Reference 1). Nevertheless, we propose a new scheme with ambition to optimize, if possible, the dierent steps used in the practical application of the algorithm. We have tried to develop for each step an optimal solution. The method has been implemented and results show the performance of the algorithm. So we can construct a well-shaped mesh constituted by a million of triangles in a few minutes on a HP 735 workstation. The 2-D case is fully detailed for clarity but most of the results can be extended without diculty in three dimensions. The problem to be solved is to construct a consistent mesh of a domain essentially from its boundary data segments. The resulting mesh will be the support of a nite element computation which can indicate if it is adapted or must be adapted by any method (it is not the goal of this paper). Nevertheless, it is important to begin with a well-shaped mesh knowing that this request can only be based on geometric considerations, since the sole known information is of geometric nature. Thus, the shape and the size of the elements must be consistent with these data.

Abstract: It becomes an important task to discover significant pattern or characteristics which may implicitly exist in huge spatial dntabases, such as geographical or medical databases. In this paper, we present a spatial data mining method named SMiYN (Spatial data Mining by Triangulated Irregular Network), which is based on Delaunay Triangulation. Sh47ZN demonstrates important advantages over the previous works. First, it discovers even sophisticated pattern like nested doughnuts, and hierarchical structure of cluster distribution. Second, in order to execute SMTIN, we do not need to know a priori the nature of distribution, for example the number of clusters, which is indispensable to other methods. Third, experiments show that SMTIN requires less CPU processing time than other methods such as BIRCH and CLARANS. Finally it is not ordering sensitive and handles efticiently outliers.

Abstract: Shape skeletons are powerful geometric abstractions that provide useful intermediate representations for a number of geometric operations on solid models including feature recognition, shape decomposition, finite element mesh generation, and shape design As a result there has been significant interest in the development of effective methods for skeleton generation of general free-form solids In this paper we describe a method that combines Delaunay triangulation with local numerical optimization schemes for the generation of accurate skeletons of 3D implicit solid models The proposed method accelerates the slow convergence of Voronoi diagrams to the skeleton, which, without optimization, would require impraelically large sample point sets and resulting meshes to attain acceptable accuracy The Delaunay triangulation forms the basis for generating the topological structure of the skeleton The optimization step of the process generates the geometry of the skeleton patches by moving the vertices of Delaunay tetrahedra and relocating their centres to form maximally inscribed spheres The computational advantage of the optimization scheme is that it involves the solution of one small optimization problem per tetrahedron and its complexity is therefore only linear (O(n)) in the number of points used for the skeleton approximation We demonstrate the effectiveness of the method on a number of representative solid models

Abstract: Abstract : This paper describes the derivation of an empirically efficient parallel two-dimensional Delaunay triangulation program from a theoretically efficient CREW PRAM algorithm. Compared to previous work, the resulting implementation is not limited to datasets with a uniform distribution of points, achieves significantly better speedups over good serial code, and is widely portable due to its use of MPI as a communication mechanism. Results are presented for a loosely-coupled cluster of workstations, two distributed-memory multicomputers, and a shared-memory multiprocessor. The Machiavelli toolkit used to transform the nested data parallelism inherent in the divide-and-conquer algorithm into achievable task and data parallelism is also described and compared to previous techniques.

Abstract: Many of the computational geometers' favorite data structures are planar graphs, canonically determined by a set of geometric data, that take Θ(n log n) time to compute. Examples include 2-d Delaunay triangulation, trapezoidations of segments, and constrained Voronoi diagrams, and 3-d convex hulls. Given such a structure, one can determine a permutation of the data in O(n) time such that the data structure can be reconstructed from the permuted data in O(n) time by a simple incremental algorithm.

Abstract: Delaunay triangulation has been much used in such applications as volume rendering, shape representation, terrain modeling and so on. The main disadvantage of Delaunay triangulation is large computation time required to obtain the triangulation on an input points set. This time can be reduced by using more than one processor, and several parallel algorithms for Delaunay triangulation have been proposed. In this paper, we propose an improved parallel algorithm for Delaunay triangulation, which partitions the bounding convex region of the input points set into a number of regions by using Delaunay edges and generates Delaunay triangles in each region by applying an incremental construction approach. Partitioning by Delaunay edges makes it possible to eliminate merging step required for integrating subresults. It is shown from the experiments that the proposed algorithm has good load balance and is more efficient than Cignoni et al.'s algorithm (1993) and our previous algorithm (1996).

Abstract: A view-dependent multiresolution triangulation algorithm is presented for a real-time flythrough. The triangulation of the terrain is generated incrementally on-the-fly during the rendering time. We show that since the view changes smoothly only a few incremental modifications are required to update the triangulation to a new view. The resulting triangles form a multiresolution Delaunay triangulation which satisfies a predetermined view-dependent error tolerance. The presented method provides a guaranteed quality mesh since it has control over the global geometric approximation error of the multiresolution view-dependent triangulation.

Abstract: Voronoi diagrams, Delaunay triangulations, and vertical decompositions in the plane are structures that are canonically defined for a set of n points or line segments. Construction requires O(n log n) time in the worst case. We show that an order exists in each case such that construction takes only linear time. The order can be determined in linear time from the Voronoi diagram, Delaunay triangulation, or vertical decomposition. This result has applications in compression and decompression of data, and as a consequence transmission overhead can be reduced. The method is simple and our implementation can be seen at http: /liruu. cs .ubc. ca/spiderlsnoey ink/terrain/Demo .html

Abstract: This paper describes the logic of a dynamic algorithm for a general 2D Delaunay triangulation of arbitrarily prescribed interior and boundary nodes. The complexity of the geometry is completely arbitrary. The scheme is free of specific restrictions on the input of the geometrical data. The scheme generates triangles whose associated circumcircles contain no nodal points except their vertices. There is no predefined limit for the number of points and the boundaries. The direction of generation of the triangles cannot be determined a priori as opposed to the moving front techniques. An automatic node placement scheme reflecting the initial boundary point spacings is used. The successive refinement scheme results in such a point distribution that the triangulation algorithm need not perform any geometric intersection check for overlapped triangles and penetrated boundaries. Further computational saving is provided by using a special binary tree (ADT) in which the points are ordered such that contiguous points in the list are neighbours in physical space. The method consists of a set of simple rules to understand. The dynamic nature of the Object Oriented Programming (OOP) of the algorithms provides efficient memory management on the insertion, deletion and searching processes. The computational effort bears a linear relation-ship between the CPU time and the total number of nodes. Some of the existing methods in the literature regarding triangular mesh generation are discussed in context. © 1997 by John Wiley & Sons, Ltd.

Abstract: We describe two problem-specific approaches to remove geometric degeneracies that we call {\it perturbing the problem} and {\it perturbing the world}. Using as our primary examples 2-d and 3-d Delaunay triangulation with Euclidean and polygonal metrics, we show that these approaches lead to relatively simple and efficient perturbations of the points that do not depend on a fixed ordering or index. Thus, they produce canonical output, which is important for producing test suites and verifiers for randomized or dynamic geometric algorithms.

Abstract: A computational method, DAS, is proposed for symmetry analysis of a planar figure closed by a simply connected curve. DAS determines both the symmetric axis and the symmetric point pairs on the curve, consistently, based on the duality of two geometric plane divisions, the Delaunay triangulation and the Voronoi diagram.

Abstract: An efficient way to render terrain data is to display an approximating surface whose resolution has been enhanced in areas of interest. We present a new reduced expansion algorithm which extracts such a selectively-refined mesh faster than our previous minimal surface algorithm. A mesh can be generated from the output of any surface approximation process which can be viewed as a set of refinement operations acting on a base triangulation. We prove that our new algorithm generates a complete, although not necessarily minimal, surface which will satisfy any given resolution criteria and which may, in addition, satisfy the Delaunay triangulation criterion.

Abstract: We present a method for computing Voronoi Diagrams for a relevant class of metric surfaces, namely all Euclidean and spherical two orbifolds. Since these surfaces are quotients of the Euclidean plane (sphere) by a discrete group of motions, the computation of Voronoi Diagram is reduced to the computation of this diagram for periodic sets of points on the Euclidean plane (sphere). This is accomplished by further reduction to the standard case of a finite set of points.

Abstract: We propose a new iterative algorithm for the computation of the vertices of a Voronoi diagram for a set of geometric objects of the euclidean plane. Each one of these vertices is the centre of the circle “touching” a triple of objects (passing through points or tangent to any other geometric object). The algorithm starts with an initial triple of points pertaining to each one of the three objects. It computes its circumcentre and the closest point (called foot) of each object from the circumcentre. These three feet form the starting triple for the next iteration. We geometrically demonstrate a necessary and sufficient condition for the general case. This iterative algorithm is used as a new method for constructing a dynamic Voronoi diagram for a set of points and straight line segments (see Gold and al.

Abstract: The Voronoi diagram (VD) is a popular tool for partitioning the support of an image. An algorithm is presented for constructing VD when the seed set, which determines the Voronoi regions, can be modified by adding and removing seeds. The number of pixels and seeds revisited for updating the diagram and the neighbor relationships among seeds is minimized. A result on cocircular seeds is presented. The adjacency, or dual, graph of the VD is readily obtained. The use of the algorithm for constructing skeletons by influence zones (SKIZ) is demonstrated.

Abstract: We consider the following problem: given a planar straight-line graph, find a covering triangulation whose maximum angle is as small as possible. A covering triangulation is a triangulation whose vertex set contains the input vertex set and whose edge set contains the input edge set. The covering triangulation problem differs from the usual Steiner triangulation problem in that we may not add a vertex on any input edge. Covering triangulations provide a convenient method for triangulating multiple regions sharing a common boundary, as each region can be triangulated independently. We give an explicit lower bound γopt on the maximum angle in any covering triangulation of a particular input graph in terms of its local geometry. Our algorithm produces a covering triangulation whose maximum angle γ is provably close to γopt. Specifically, we show that , i.e., our γ is not much closer to π than is γopt. To our knowledge, this result represents the first nontrivial bound on a covering triangulation's maximum angle. Our algorithm adds O(n) Steiner points and runs in time O(n log n), where n is the number of vertices of the input. We have implemented an O(n2) time version of our algorithm.

Abstract: A generalized unstructured mesh generation procedure using Delaunay triangulation has been developed for adaptive finite element applications. The main features of the method include: (i) a fast and efficient initial triangulation; (ii) interior node insertion with good control over the grid size and aspect ratio distribution; (iii) a flux-based error across element boundaries. The scheme has been applied successfully to heat transfer problems involving steep temperature gradients.

Abstract: This paper compares three different node insertion strategies which can be used to incrementally construct Delaunay triangulation models Delaunay networks may be used to efficiently store low dimensional nonlinear models, and can therefore be used in a wide range of real-time applications However, there are no direct node selection methods, and it can be shown that the network's generalisation abilities are strongly affected by the triangular partitioning of the input space The three iterative, constructive node insertion algorithms (maximum error, local weighted error and one-step-ahead optimal search) are compared using two data sets, and conclusions are drawn about the quality of the extracted triangulation and the algorithms' computational costs

Abstract: In this paper we describe and discuss a kernel for higher-dimensional computational geometry and we present its application in the calculation of convex hulls and Delaunay triangulations. The kernel is available in form of a software library module programmed in C++ extending LEDA. We introduce the basic data types like points, vectors, directions, hyperplanes, segments, rays, lines, spheres, affine transformations, and operations connecting these types. The description consists of a motivation for the basic class layout as well as topics like layered software design, runtime correctness via checking routines and documentation issues. Finally we shortly describe the usage of the kernel in the application domain. © 1998 Elsevier Science B.V.

Abstract: A fast and easy to implement divide-and-conquer algorithm is presented for the construction of the Constrained Delaunay triangulation of a polygonal domain. The algorithm simplifies the complicated merging step inherent to divide-and-conquer algorithms for the computation of triangulations. Furthermore, no triangles are computed outside the valid region of the domain. A grid structure accelerates the computation of the visibility among vertices with respect to the boundary polygons as well as the computation of Constrained Delaunay triangles.

Abstract: An image coding technique that combines irregular subsampling and triangulation is proposed. First, a Delaunay triangulation is constructed over a set of visually significant sampling points selected incrementally as the triangulation progresses. Then a data-dependent triangulation is generated, starting from the Delaunay triangulation, by a series of edge swaps based on the quality of the approximation. Bilinear interpolation is used to approximate the triangular regions. The simulation results show the superiority of the proposed algorithm over JPEG at low bit rates (0.206 bpp and 0.113 bpp) as well as its robustness.