An efficient sweep-line Delaunay triangulation algorithm

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.     

A note on lee and schachter's algorithm for delaunay triangulation

Lee and Schachter have presented an algorithm for the Delaunay triangulation of a set of points whose convex hull is a rectangular region. An addendum to that algorithm is presented which gives the Delaunay triangulation of a set of points with an arbitrary convex hull. Timing results are also given.     

Constrained delaunay triangulations

Given a set ofn vertices in the plane together with a set of noncrossing, straight-line edges, theconstrained Delaunay triangulation (CDT) is the triangulation of the vertices with the following properties: (1) the prespecified edges are included in the triangulation, and (2) it is as close as possible to the Delaunay triangulation. We show that the CDT can be built in optimalO(n logn) time using a divide-and-conquer technique. This matches the time required to build an arbitrary (unconstrained) Delaunay triangulation and the time required to build an arbitrary constrained (non-Delaunay) triagulation. CDTs, because of their relationship with Delaunay triangulations, have a number of properties that make them useful for the finite-element method. Applications also include motion planning in the presence of polygonal obstacles and constrained Euclidean minimum spanning trees, spanning trees subject to the restriction that some edges are prespecified.An earlier version of the results presented here appeared in theProceedings of the Third Annual Symposium on Computational Geometry (1987).     

Duality of constrained Voronoi diagrams and Delaunay triangulations

We introduce theconstrained Voronoi diagram of a planar straight-line graph containingn vertices or sites where the line segments of the graph are regarded as obstacles, and show that an extended version of this diagram is the dual of theconstrained Delaunay triangulation. We briefly discussO(n logn) algorithms for constructing the extended constrained Voronoi diagram.This work was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.     

An adaptive spatial clustering algorithm based on delaunay triangulation

In this paper, an adaptive spatial clustering algorithm based on Delaunay triangulation (ASCDT for short) is proposed. The ASCDT algorithm employs both statistical features of the edges of Delaunay triangulation and a novel spatial proximity definition based upon Delaunay triangulation to detect spatial clusters. Normally, this algorithm can automatically discover clusters of complicated shapes, and non-homogeneous densities in a spatial database, without the need to set parameters or prior knowledge. The user can also modify the parameter to fit with special applications. In addition, the algorithm is robust to noise. Experiments on both simulated and real-world spatial databases (i.e. an earthquake dataset in China) are utilized to demonstrate the effectiveness and advantages of the ASCDT algorithm.     

Point placement algorithms for Delaunay triangulation of polygonal domains

In some applications of triangulation, such as finite-element mesh generation, the aim is to triangulate a given domain, not just a set of points. One approach to meeting this requirement, while maintaining the desirable properties of Delaunay triangulation, has been to enforce the empty circumcircle property of Delaunay triangulation, subject to the additional constraint that the edges of a polygon be covered by edges of the triangulation. In finite-element mesh generation it is usually necessary to include additional points besides the vertices of the domain boundary. This motivates us to ask whether it is possible to trinagulate a domain by introducing additional points in such a manner that the Delaunay triangulation of the points includes the edges of the domain boundary. We present algorithms that given a multiply connected polygonal domain withN vertices, placeK additional points on the boundary inO(N logN + K) time such that the polygon is covered by the edges of the Delaunay triangulation of theN + K points. Furthermore,K is the minimum number of additional points such that a circle, passing through the endpoints of each boundary edge segment, exists that does not contain in its interior any other part of the domain boundary. We also show that by adding only one more point per edge, certain degeneracies that may otherwise arise can be avoided.     

An almost distribution-independent incremental Delaunay triangulation algorithm

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.     

Triangle refinement in a constrained Delaunay triangulation skeleton

This paper presents an algorithm with the purpose of improving upon the already successful constrained Delaunay triangulation (CDT) skeletonisation technique. Using such a triangulation to construct a skeleton has proven very effective, that can sometimes, however, produce triangles that do not represent the true nature of the underlying shape. The contour pixels chosen for triangulation are of significant importance, as they determine the triangle edges that define the skeleton. The algorithm described in this paper deals with this problem by inserting new triangulation points in strategic locations in end, normal and junction triangles. Results show that the skeletons produced by this algorithm are accurate, robust against noise and, above all, comply much better with a human's perception of the image than the original triangulation method.     

Delaunay三角网剖分的约束边嵌入改进算法

重点研究约束边强行嵌入D-三角网的问题。约束边嵌入是解决D-三角网转变为CD-三角网的一种非常有效的方法,而CD-三角网才能真实地虚拟地形地貌。针对基于凸凹判定的对角线交换算法存在的缺陷,提出"分裂约束边"的思想完善算法的健壮性,并引入快速点定位算法以提高算法的执行效率。     

Two algorithms for constructing a Delaunay triangulation

This paper provides a unified discussion of the Delaunay triangulation. Its geometric properties are reviewed and several applications are discussed. Two algorithms are presented for constructing the triangulation over a planar set ofN points. The first algorithm uses a divide-and-conquer approach. It runs inO(N logN) time, which is asymptotically optimal. The second algorithm is iterative and requiresO(N ²) time in the worst case. However, its average case performance is comparable to that of the first algorithm.This work was supported in part by the National Science Foundation under grant MCS-76-17321 and the Joint Services Electronics Program under contract DAAB-07-72-0259.     

平面点集Delaunay三角剖分的分治算法

为发展图形网格化技术,研究了平面点集的三角剖分算法.根据经典算法中在实际应用中遇到的共性问题,提炼了3个工具算法;为了更好地表示平面区域划分的拓扑信息,引入了双链接边表(DCEL)的数据结构.在此基础上,设计并实现了平面集Delaunay三角剖分分治算法,并对特殊退化情况进行了处理,通过计算表明了该算法时间复杂度为0(N* logN).实验数据结果验证了该算法的正确性、健壮性.     

On sorting triangles in a delaunay tessellation

In a two-dimensional Delaunay-triangulated domain, there exists a partial ordering of the triangles (with respect to a vertex) that is consistent with the two-dimensional visibility of the triangles from that vertex. An equivalent statement is that a polygon that is star-shaped with respect to a given vertex can be extended, one triangle at a time, until it includes the entire domain. Arbitrary planar triangulations do not possess this useful property which allows incremental processing of the triangles.This work was partially supported by the National Science Foundation's US-Italy Collaborative Research Program under Grant INT-8714578 and Information, Robotics, and Intelligent Research Grant IRI-8704781.     

约束Delaunay三角网生成算法的研究与应用

在研究现有约束Delaunay三角网生成算法后,针对传统合成算法中点定位过程效率低,搜索影响域算法复杂的缺点,对合成算法进行了两点改进:一是快速定位点所在的三角形;二是优化搜索影响域的算法.改进后的算法有效的缩短了构网过程中定位点和搜索影响域的时间,提高了构网效率.分别利用传统合成算法与改进后的算法结合有限元算法在高炉炉衬侵蚀曲线预测工程项目中进行了分组仿真实验.实验数据表明,改进后的算法更加稳定,剖分效率更高.     

基于Delaunay三角剖分生成Voronoi图算法

针对Delaunay三角网生长算法和间接生成Voronoi图算法构网效率不高的问题,提出了一种Delaunay三角网生长法间接生成Voronoi图的改进算法。该算法以点集凸壳上一边快速生成种子三角形,定义了半封闭边界点的概念,在三角形扩展过程中动态删除封闭点及半封闭边界点,加快Delaunay三角网生成速度。然后又定义了有序目标三角形的概念,该算法能迅速查找点的有序目标三角形,生成无射线的Voronoi图;考虑凸壳上点的特性,借助三个无穷点生成带射线的Voronoi图。通过实验结果分析表明,改进的算法执行效率有了很大提高。     

Computing the relative neighborhood graph in the L1 and L∞ metrics

The Relative Neighborhood Graph (RNG) of a set of nk-dimensional points connects “relatively close” neighbors: two points are connected by an edge if they are at least as close to each other as to any other point. Toussaint recently investigated the properties of the RNG in the Euclidean metric and proposed algorithms for its computation. This note examines one of the open problems listed by Toussaint: the extension of the analysis to non-Euclidean metrics. It is shown that Bentley's range query data structures may be used to improve the speed of the best known RNG algorithm in the L_∞ (for k ? 2) and L₁ (for k = 2) metrics.     

基于四叉树结构的数字地表模型快速生成算法设计

为了研究数字地表模型的快速生成方法，在总结传统Delaunay三角化算法的基础上，给出了一个基于四叉树结构的数字地表模型快速生成算法的详细设计，该算法的基本思想是首先利用四叉树结构来对离散点进行分割，然后对四叉树叶节点进行Delaunay三角化，再两两合并四叉树节点三角网的凸壳，以快速生成地表表格网模型，该算法是以四叉树为基本单位为实现限定边和限定多边形的快速嵌入，最后给出了算法在不同情况下的测试结果，并对测试结果进行了具体分析，给出了算法的时间效率分析和空间复杂性分析，实测数据结果表明，该算法有着较好的性能，而且也非常稳定，通过实测结果分析和算法的时间效率分析，可以得到算法的时间效率近似为O(nlog(n）），通过算法的空间复杂性分析可以看出，算法可以自动适应不同的点空间分布情况，而且采用四叉树结构也非常有利于限定边和限定多边形的嵌入。     

New results for the minimum weight triangulation problem

Given a finite set of points in a plane, a triangulation is a maximal set of nonintersecting line segments connecting the points. The weight of a triangulation is the sum of the Euclidean lengths of its line segments. No polynomial-time algorithm is known to find a triangulation of minimum weight, nor is the minimum weight triangulation problem known to be NP-hard. This paper proposes a new heuristic algorithm that triangulates a set ofn points inO(n ³⁾ time and that never produces a triangulation whose weight is greater than that of a greedy triangulation. The algorithm produces an optimal triangulation if the points are the vertices of a convex polygon. Experimental results indicate that this algorithm rarely produces a nonoptimal triangulation and performs much better than a seemingly similar heuristic of Lingas. In the direction of showing the minimum weight triangulation problem is NP-hard, two generalizations that are quite close to the minimum weight triangulation problem are shown to be NP-hard.This research was done while the second author was with the Department of Computer Science, Virginia Polytechnic Institute and State University.     

Minimal roughness property of the Delaunay triangulation

A set of scattered data in the plane consists of function values measured on a set of data points in R². A surface model of this set may be obtained by triangulating the set of data points and constructing the Piecewise Linear Interpolating Surface (PLIS) to the given function values. The PLIS is combined of planar triangular facets with vertices at the data points. The roughness measure of a PLIS is the L² norm squared of the gradient of the piecewise linear surface, integrated over the triangulated region and obviously depends on the specific triangulation. In this paper we prove that the Delaunay triangulation of the data points minimizes the roughness measure of a PLIS, for any fixed set of function values. This Theorem connects for the first time, as far as we know, the geometry of the Delaunay triangulation with the properties of the PLIS defined over it.