平面散乱点集约束Delaunay三角形剖分切割算法

文章提出了一种基于切割的平面散乱点集约束Delaunay三角剖分算法。该算法的基本思路是首先对平面散乱点集作约束最大空圆凸多边形剖分,然后对多边形的内部再作约束Delaunay三角形剖分。文章还证明了平面散乱点集的约束最大空圆凸多边形剖分是唯一的以及约束Delaunay三角剖分的不唯一性仅仅体现在约束最大空圆凸多边形的内部。使用约束最大空圆凸多边形的概念消除了由于“退化”现象（三个以上的点共圆）带来的算法上的潜在错误。     

约束Delaunay三角剖分中强行嵌入约束边的多对角线交换算法

在不允许改变原有点集的场合,实现约束Ｄｅｌａｕｎａｙ 三角剖分的一种有效算法是：将边界点与内点一起进行标准Ｄｅｌａｕｎａｙ 三角剖分,然后强行嵌入不在剖分中的约束边,最后删除域外三角形．其中,任意一条待嵌入约束边所经三角形构成的多边形区域称为该约束边的影响域,影响域内部的每条边称为对角线．文中对一般形状影响域中对角线的可交换性进行了研究,并在此基础上,结合对已有算法的分析和借鉴,提出并证明了两种强行嵌入约束边的多对角线交换算法,即递减算法与循环算法．其中的循环算法具有编程简单和运算速度快的特点     

Algorithm for constrained delaunay triangulation

A direct algorithm for computing constrained Delaunay triangulation in 2-D is presented. The algorithm inserts points along the constrained edges (break lines) to maintain the Delaunay criterion. Since many different insertions are possible, the algorithm computes only those that are on the Delaunay circles of each intersected triangle. A shelling procedure is applied to put triangles together in such a way that completeness and correctness are guaranteed.     

Automatic mesh generation for complex three-dimensional regions using a constrained Delaunay triangulation

This paper describes a method for generating tetrahedral meshes. The algorithm, based on the Delaunay triangulation, can treat objects of essentially arbitrary complexity. In order to preserve the surface triangulation of solid objects, it is necessary to override the Delaunay property and redefine the triangulation when points are introduced that are close to solid boundaries. Details of this constrained Delaunay algorithm are presented and an efficient implementation of the triangulation method is described. Techniques for controlling the distribution of mesh points and tetrahedron quality are also discussed.     

通用点线面集Delaunay三角剖分与动态编辑

总结并提出了一种通用点线面集Delaunay三角剖分与动态编辑的统一算法。可以实现离散点的Delaunay三角剖分,约束线、面的Delaunay三角剖分,任意多边形内带特征约束(包括点、线、面)的三角剖分,一般Delaunay三角剖分的外边界都是其离散点集的凸包,且内岛屿一般没有挖掉,本算法实现了Delaunay三角剖分时内、外边界的保界处理。     

复杂地质体中多值面的网格生成算法

针对现有的网格生成算法无法处理在自然界中大量存在的多值面地质现象，基于分割－归并方法，提出一种分裂－重构算法。在生成初始约束Delaunay三角形网格之后，遵循连续折线的正负区测试准则，对网格中的局部顶点进行分裂，重构相关的三角形的点、边以及三角形的拓扑关系。实验表明，该算法能够有效地生成多值面的网格。     

Delaunay三角网动态更新算法的研究进展

通过对Delaunay三角网动态更新算法进行研究,综述了Delaunay三角网中插入和删除点、约束线算法以往研究.详细介绍点定位、LOP优化、对角线交换等关键技术的研究进展,并对比各种方法的优缺点,分析已解决的问题和仍存在的问题.最后对更新算法研究不足之处进行总结,并提出若干可能的研究方向.     

三维约束Delaunay三角化的研究

概述了约束三角化的研究进展,着重分析了三维约束Ｄｅｌａｕｎａｙ三角化中存在的问题,提出并论证了边界边、边界面片在Ｄｅｌａｕｎａｙ三角化中存在的条件,讨论了存在性条件在实际工程中的应用范围,充实了三维约束Ｄｅｌａｕｎａｙ三角化的研究基础,为三维Ｄｅｌａｕｎａｙ三角化算法的设计提供了理论依据。     

三维约束Delaunay三角化的实现

分析了约束Delaunay三角化中存在的边界一致性问题,给出了约束Delaunay三角化的理论依据,重点探讨了三维约束Delaunay三角化的可行性条件和范围,同时,给出了三维有限域约束Delaunay三角化的实现方法及其在石油地质勘探数据和机械零件方面的网格剖分实例.这种算法在复杂对象的科学计算和工程分析中发挥了重要作用.     

3维任意域内点集的Delaunay四面体化研究

Delaunay空球准则广泛应用于3维四面体剖分算法,但标准的Delaunay四面体化只适用于点集的凸包区域,且要求不存在多点共球。为了将Delaunay四面体化更广泛地应用于网络剖分,通过引入局部优化三角形面代替Deluany严格的空球准则,提出了3维任意域内点集Deluanay四面体化(DTETAD)的概念,并首先通过若干关键定理的证明,研究了一个四面体划分是DETEAD的充要条件,然后建立了DTETAD的空球准则。该研究成果为拓展Delaunay算法在更广泛范围的应用提供了理论依据。     

三维物体表面重建的分支处理算法研究

根据平面点集Delaunay三角剖分的特性,将Delaunay三角剖分应用到分支问题上,改进和实现了一种分支问题处理算法。将相邻层轮廓线投影到同一个剖面上形成一个带约束边的平面点集,并将它们Delaunay三角化,根据这些三角形组来生成新的轮廓线,使轮廓线一一对应。实验结果表明该算法实现的效果较符合实际情况,能有效地处理各种不同情况。     

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.     

On shape Delaunay tessellations

Shape Delaunay tessellations are a generalization of the classical Delaunay triangulation of a finite set of points in the plane, where the empty circle condition is replaced by emptiness of an arbitrary convex compact shape. We present some new and basic properties of shape Delaunay tessellations, concerning flipping, subgraph structures, and recognition.     

二维复杂限定Delaunay三角化算法

针对包括曲线边界和内部带有曲线限定条件的二维Delaunay三角化问题,提出了一种细化算法.首先给出了曲线段的逼近边定义,以保证限定曲线在网格中的存在;然后证明了该算法的收敛性和最终曲线的逼近边集合与原曲线的拓扑一致性,并且生成的网格符合Delaunay优化准则;最后给出了算法的应用实例,验证了其有效性.     

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).     

Efficient minimum spanning tree construction without Delaunay triangulation

Given n points in a plane, a minimum spanning tree is a set of edges which connects all the points and has a minimum total length. A naive approach enumerates edges on all pairs of points and takes at least Ω(n2) time. More efficient approaches find a minimum spanning tree only among edges in the Delaunay triangulation of the points. However, Delaunay triangulation is not well defined in rectilinear distance. In this paper, we first establish a framework for minimum spanning tree construction which is based on a general concept of spanning graphs. A spanning graph is a natural definition and not necessarily a Delaunay triangulation. Based on this framework, we then design an O(nlogn) sweep-line algorithm to construct a rectilinear minimum spanning tree without using Delaunay triangulation.     

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.     

实现约束Delaunay三角剖分的健壮算法

相对于标准的Ｄｅｌａｕｎａｙ三角剖分，本文给出了复杂区域三角剖分所应满足的两个约束条件及相应的基于轨迹生成和边界裁剪的剖分算法，并证明了该算法符合约束圆准则，文中详细分析了退化及数值误差对剖分结果的影响，着重在提高算法健壮性方面，对该算法做了进一步完善，使它能够完全满足散乱据场网格剖分的分析。     

An intersection algorithm based on Delaunay triangulation

A robust method for finding points of intersection of line segments in a 2-D plane is presented. The plane is subdivided by Delaunay triangulation to localize areas where points of intersection exist and to guarantee the topological consistency of the resulting arrangement. The subdivision is refined by inserting midpoints recursively until the areas containing points of intersection are sufficiently localized. The method is robust in the sense that it does not miss points of intersection that are easily detectable when costly line-pair checking is performed. The algorithm is adaptive in the sense that most of the computational cost is incurred for the areas where finding points of intersection is difficult     

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.