Abstract: This paper introduces a new algorithm for constructing a 2D Delaunay triangulation. It is based on a sweep-line paradigm, which is combined with a local optimization criterion-a characteristic of incremental insertion algorithms. The sweep-line status is represented by a so-called advancing front, which is implemented as a hash-table. Heuristics have been introduced to prevent the construction of tiny triangles, which would probably be legalized. This algorithm has been compared with other popular Delaunay algorithms and it is the fastest algorithm among them. In addition, this algorithm does not use a lot of memory for supporting data structure, it is easy to understand and simple to implement.

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.

Abstract: We propose a new refinement algorithm to generate size-optimal quality-guaranteed Delaunay triangulations in the plane. The algorithm takes O(n log n + m) time, where n is the input size and m is the output size. This is the first time-optimal Delaunay refinement algorithm.

Abstract: We extend the notion of higher-order Delaunay triangulations to constrained higher-order Delaunay triangulations and provide various results. We can determine the order k of a given triangulation in O(min(nk log n log k, n3/2logO(1) n)) time. We show that the completion of a set of useful order-k Delaunay edges may have order 2k - 2, which is worst-case optimal. We give an algorithm for the lowest-order completion for a set of useful order-k Delaunay edges when k ≤ 3. For higher orders the problem is open.

Abstract: For hydrologic applications, terrain models should have few local minima, and drainage lines should coincide with edges. We show that triangulating a set of points with elevations such that the number of local minima of the resulting terrain is minimized is NP-hard for degenerate point sets. The same result applies when there are no degeneracies for higher-order Delaunay triangulations. Two heuristics are presented to reduce the number of local minima for higher-order Delaunay triangulations, which start out with the Delaunay triangulation. We give efficient algorithms for their implementation, and test on real-world data how well they perform. We also study another desirable drainage characteristic, namely few valley components.

Abstract: This paper discusses a surface reconstruction method using the Delaunay triangulation algorithm. Surface reconstruction is used in various engineering applications to generate CAD model in reverse engineering, STL files for rapid prototyping and NC codes for CAM system from physical objects. The suggested method has two other components in addition to the triangulation: the weighted alpha shapes algorithm and the peel-off algorithm. The weighted alpha shapes algorithm is applied to restrict the growth of tetrahedra, where the weight is calculated based on the density of points. The peel-off algorithm is employed to enhance the reconstruction in detail. The results show that the increase in execution time due to the two additional processes is very small compared to the ordinary triangulation, which demonstrates that the proposed surface reconstruction method has great advantage in execution time for a large set of points.

Abstract: We discuss the deletion of a single vertex in a Delaunay tetrahedralization (DT). While some theoretical solutions exist for this problem, the many degeneracies in three dimensions make them impossible to be implemented without the use of extra mechanisms. In this paper, we present an algorithm that uses a sequence of bistellar flips to delete a vertex in a DT, and we present two different mechanisms to ensure its robustness.

Abstract: This paper presents a new incremental insertion algorithm for constructing a Delaunay triangulation. Firstly, the nearest point is found in order to speed up the location of a triangle containing a currently inserted point. A hash table and 1–3 deterministic skip lists, combined with a walking strategy, are used for this task. The obtained algorithm is compared with the most popular Delaunay triangulation algorithms. The algorithm has the following attractive features: it is fast and practically independent of the distribution of input points, it is not memory demanding, and it is numerically stable and easy to implement.

Abstract: We present a new linear time algorithm to compute a good order for the point set of a Delaunay triangulation in the plane. Such a good order makes reconstruction in linear time with a simple algorithm possible. Similarly to the algorithm of Snoeyink and van Kreveld [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459-471], our algorithm constructs such orders in O(log n) phases by repeatedly removing a constant fraction of vertices from the current triangulation. Compared to [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459-471] we improve the guarantee on the number of removed vertices in each such phase. If we restrict the degree of the points (at the time they are removed) to 6, our algorithm removes at least 1/3 of the points while the algorithm from [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459-471] gives a guarantee of 1/10. We achieve this improvement by removing the points sequentially using a breadth first search (BFS) based procedure that--in contrast to [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459-471]--does not (necessarily) remove an independent set.Besides speeding up the algorithm, removing more points in a single phase has the advantage that two consecutive points in the computed order are usually closer to each other. For this reason, we believe that our approach is better suited for vertex coordinate compression.We implemented prototypes of both algorithms and compared their running time on point sets uniformly distributed in the unit cube. Our algorithm is slightly faster. To compare the vertex coordinate compression capabilities of both algorithms we round the resulting sequences of vertex coordinates to 16-bit integers and compress them with a simple variable length code. Our algorithm achieves about 14% better vertex data compression them the algorithm from [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459-471].

Abstract: Fingerprint matching is a key issue in research of an automatic fingerprint identification system. On the basis of triangulation in computational geometry, we develop a kind of method for fingerprint matching based on Delaunay Triangulation net in this paper. Through carrying on Delaunay Triangulation to the topological structure of fingerprint minutiae, minutiae with closer distance link to each other on the space according to the Delaunay criterion and form the Delaunay Triangulation net. Then look for some reference minutiae pairs correctly from the net. According to the reference minutiae pairs, match fingerprint on point pattern. The experimental results on FVC2000 indicate the validity of algorithm.

Abstract: Let P be a Poisson point process in Rd with intensity 1. We show that the simple random walk on the cells of the Voronoi diagram of P is almost surely recurrent in dimensions d = 1 and d = 2 and is almost surely transient in dimension d ≥ 3.

Abstract: Skin surfaces are used for the modeling and visualization of molecules. They form a class of tangent continuous surfaces defined in terms of a set of balls (the atoms of the molecule) and a shrink factor. More recently, skin surfaces have been used to approximate arbitrary surfaces. We present an algorithm that approximates a skin surface with a topologically correct mesh. The complexity of the mesh is linear in the size of the Delaunay triangulation of the balls, which is worst case optimal. We also adapt two existing refinement algorithms to improve the quality of the mesh and show that the same algorithm can be used for meshing a union of balls.

Abstract: The Voronoi diagram (VD) and the Delaunay triangulation (DT) can be used for modelling different kinds of data for different purposes. These two structures are attractive alternatives to rasters to discretise a continuous phenomenon such as the percentage of gold in the soil, the temperature of the water, or the elevation of a terrain. They can also be used to represent the boundaries of real-world features, for example geological modelling of strata or cadastral models of apartment buildings. The VD and the DT are an appealing solution because of their duality (they represent the same thing, just from a different point of view) and because both structures have interesting properties (see Aurenhammer [Aur91] for a review of the properties and potential applications).

Abstract: We present the first time-optimal algorithm for computing the Voronoi Diagram under the metric induced by a transportation network with discrete entry and exit points. For input with n sites, k stations, and e transportation lines, the algorithm computes the

Abstract: This study presents an efficient algorithm of Delaunay triangulation by grid subdivision. The proposed algorithm show a superior performance in terms of execution time to the incremental algorithm and uniform grid method mainly due to the efficient way of searching a mate. In the proposed algorithm, uniform grids are divided into sub-grids depending on the density of points and areas with high chance of finding a mate is explored first. Most of the previous researches have focused on theoretical aspects of the triangulation, but this study presents empirical results of computer implementation in 2-dimension and 3-dimension, respectively.

Abstract: This paper assumes a set of identical wireless hosts, each one aware of its location. The network is described by a unit distance graph whose vertices are points on the plane two of which are connected if their distance is at most one. The goal of this paper is to design local distributed solutions that require a constant number of communication rounds, independently of the network size or diameter. This is achieved through a combination of distributed computing and computational complexity tools. Starting with a unit distance graph, the paper shows: 1. How to extract a triangulated planar spanner; 2. Several algorithms are proposed to construct spanning trees of the triangulation. Also, it is described how to construct three spanning trees of the Delaunay triangulation having pairwise empty intersection, with high probability. These algorithms are interesting in their own right, since trees are a popular structure used by many network algorithms; 3. A load balanced distributed storage strategy on top of ...

Abstract: It is presented a LOD-based algorithm of TIN model, which is quite suitable for terrain rendering in large 3D landscape. The emphasis focuses on issues of transition between TIN models with different resolutions, and generation of these models, such as how to design data structures and how to optimize algorithms when constructing a Delaunay triangulation.

Abstract: We show that triangulating a set of points with elevations such that the number of local minima of the resulting terrain is minimized is NP-hard for degenerate point sets. The same result applies when there are no degeneracies for higher-order Delaunay triangulations. Two heuristics are presented to minimize the number of local minima for higher-order Delaunay triangulations, and they are compared experimentally.

Abstract: Voronoi diagrams have been known to have numerous applications in various fields in science and engineering. While the Voronoi diagram for points has been extensively studied in two and higher dimensions, the Voronoi diagram for spheres in three or higher dimensions has not been studied sufficiently. In this paper, we propose an algorithm to construct Euclidean Voronoi diagrams for spheres in 3D. Starting from the ordinary Voronoi diagram for the centers of spheres, the proposed region expansion algorithm constructs the desired diagram by expanding Voronoi regions for one sphere after another via a series of topology operations. Adopted data structure for the proposed algorithm is a variation of radial data structure. While the worst-case time complexity is O(n3 log n) for the whole diagram, its expected time complexity can be much lower.

Abstract: Although there are many applications of Euclidean Voronoi diagram for spheres in a 3D space in various disciplines from sciences and engineering, it has not been studied as much as it deserves. In this paper, we present an edge-tracing algorithm to compute the Euclidean Voronoi diagram of 3-dimensional spheres in O(mn) in the worst-case, where m is the number of edges of Voronoi diagram and n is the number of spheres. After building blocks for the algorithm, we show an example of Voronoi diagram for atoms using actual protein data and discuss its applications for protein analysis.

Abstract: The using of TIN (triangulated irregular network) is becoming more and more extensive.This paper discusses the method of building TIN.The author improves the arithmetic in his program, and the efficiency of building TIN is advanced sharply.

Abstract: Most of the curves and surfaces encountered in geometric modelling are defined as the set of common zeroes of a set of polynomials (algebraic varieties) or subsets of algebraic varieties defined by one or more algebraic inequalities (semi-algebraic sets). Many problems from different fields involve proximity queries like finding the nearest neighbour, finding all the neighbours, or quantifying the neighbourliness of two objects. The Voronoi diagram of a set of sites is a decomposition of the space into proximal regions: each site's Voronoi region is the set of points closer to that site than to any other site. The Delaunay graph of a set of sites is the dual graph of the Voronoi diagram of that set of sites, which stores the spatial adjacency relationships among sites induced by the Voronoi diagram. The Voronoi diagram has been used for solving the earlier mentioned proximity queries. The ordinary Voronoi diagram of point sites has been extended or generalised in several directions (underlying space, metrics, sites), and the resulting generalised Voronoi diagrams have found many practical applications. The Voronoi diagrams have not yet been generalised to algebraic curves or semi-algebraic sets. In this paper, we present a conflict locator for the certified incremental maintenance of the Delaunay graph of semi-algebraic sets.

Abstract: The Voronoi diagram is one of the most fundamental data structures in computational geometry, which is concerned with the design and analysis of algorithms for geometrical problems. In this paper, a parallel algorithm for constructing the Voronoi diagram on CREW (Concurrent Read and Exclusive Write) model is proposed. This is an improved algorithm based on Preilowski and Mumbeck’s work. In their algorithm, they apply the Neighbor-Point-Theorem and present a parallel approach to check neighbor points. In this article, we propose an improved approach, Wave-Front algorithm, which is a quite different way to check neighbor points. The algorithm is then implemented in both sequential and multithreaded models.Since the Wave-Front algorithm has inherently concurrent tasks that can be executed simultaneously, multithreaded version was executed to observe the performance. Computational results indicate the effectiveness of the threaded model.

Abstract: A new algorithm is proposed for generating interior nodal points within an arbitrary two-dimensional domain. The algorithm is based on a background triangle mesh and the contours on the background mesh. The newly generated points are exactly within the domain with smooth point densities and a good quality of the final mesh. An improved Delaunay triangulation algorithm using advancing front technique is also proposed. The present algorithm is more general and robust. A FORTRAN 90 program based on the present algorithms is developed and has been successfully applied in many projects. Examples are given to show the effectiveness and robustness of the algorithms. Copyright © 2005 John Wiley & Sons, Ltd.

Abstract: We propose an algorithm to construct a three-dimensional (3D) discrete Voronoi diagram using the incremental method on a digitized space and apply it to measure the topology preservation rate of self-organizing maps. The 3D discrete Voronoi diagram needs many pixels and requires much memory and much time to calculate. In this sense, the fast 3D version of the incremental method on a digitized space is useful. Recently, studies of self-organizing feature maps have made it clear that a close relationship exists between feature extraction and the obtaining of masked Delaunay triangulation of input data. Thus, the Voronoi division is becoming an indispensable technique for the study of self organizing feature maps. © 2005 Wiley Periodicals, Inc. Syst Comp Jpn, 36(5): 55–67, 2005; Published online in Wiley InterScience (). DOI 10.1002sscj.20205