平面点集凸包快速构建算法的研究

文章提出了一种提高构建凸包速度的新方法。该算法生成一个网格来管理离散点,在淘汰明显不位于凸包上的点时,将对离散点的取舍转换为对格的取舍,计算工作量只与离散点的范围及网格的密度有关,与离散点的数目无关;同时对点集也进行了初略的排序。在求取剩余点集的凸包时,采用了一种先分段求取凸包边界,最后将这些边界合并成凸包的方法,该方法充分利用了剩余点集所具有的有序性。     

一种平面点集凸壳的快速算法

文章提出了一种对平面离散点集凸壳的快速算法,该算法首先对离散点进行扫描线方式排序,构造初始凸壳,然后把剩下的离散点加入到已有的凸壳中生成新的凸壳.实验表明该算法具有很好的效率.     

基于栅格划分构建平面点集凸壳的算法研究

提出了一个构建平面点集凸壳的新算法。该算法用栅格阵列将待处理点集划分成若干个子集,这样凸壳可以由部分位于点集边缘的子集确定;然后按逆时针顺序逐步处理这些子集,得到一个包含待处理点集的简单多边形,删除凹顶点后就得到待处理点集的凸壳。由于只对点集边缘的点进行局部处理,从而提高了构建凸壳的效率。在最坏情况下该算法的时间复杂度为O(NlogN)。     

An enhanced convex-hull edge method for flatness tolerance evaluation

Flatness is one of the elementary geometric forms to be tested for ensuring the acceptable quality of machined components. The minimum-zone evaluation problem for flatness is to determine the minimum tolerance zone enclosing all the points measured from a surface of the machined components. Although a variety of attempts has been made for the past two decades to solve the problem, methods guaranteeing an exact minimum-zone solution are very limited. One category of the exact methods is certainly the computational-geometry-based approaches such as the so-called convex-hull edge method. This paper presents a modified convex-hull edge method which significantly enhances its computational efficiency such that large-sized data sets with thousands of points can be solved within a negligible computation time using a personal computer. To test the validity of the method, a number of test data sets including those published in the literature and new large data sets generated in this paper were solved. The obtained solutions were compared with those produced by the existing methods. The results show the effectiveness, efficiency, and robustness of the method in that an exact minimum tolerance is always obtained for each of the test data sets solved in less than tens of CPU milliseconds.     

基于凸壳技术的Delaunay三角网生成算法

该文提出了一种针对散乱点集的快速构建Delaunay的算法。该算法首先对散乱点按有向角进行排序,以排序后的点顺序为基础,利用凸壳特性快速将散乱点联结成三角网,最后利用拓扑结构快速将其优化为Delaunay三角网。在联网过程中,充分利用有序点子集的凸壳特性,避免了所有的交点测试,从而保证了对散乱点集生成Delaunay三角网的效率。     

Efficient convexity and domination algorithms for fine- and medium-grain hypercube computers

This paper gives hypercube algorithms for some simple problems involving geometric properties of sets of points. The properties considered emphasize aspects of convexity and domination. Efficient algorithms are given for both fine- and medium-grain hypercube computers, including a discussion of implementation, running times and results on an Intel iPSC hypercube, as well as theoretical results. For both serial and parallel computers, sorting plays an important role in geometric algorithms for determining simple properties, often being the dominant component of the running time. Since the time required to sort data on a hypercube computer is still not fully understood, the running times of some of our algorithms for unsorted data are not completely determined. For both the fine- and medium-grain models, we show that faster expected-case running time algorithms are possible for point sets generated randomly. Our algorithms are developed for sets of planar points, with several of them extending to sets of points in spaces of higher dimension.The research of E. Cohen, R. Miller, and E. M. Sarraf was partially supported by National Science Foundation Grant ASC-8705104. R. Miller was also partially supported by National Science Foundation Grants DCR-8608640 and IRI-8800514. Q. F. Stout's research was partially supported by National Science Foundation Grant DCR-85-07851, and an Incentives for Excellence Grant from the Digital Equipment Corporation.     

Finding the convex hull of a simple polygon in linear time

Though linear algorithms for finding the convex hull of a simply-connected polygon have been reported, not all are short and correct. A compact version based on Sklansky's original idea⁽⁷⁾ and Bykat's counter-example⁽⁸⁾ is given. Its complexity and correctness are also shown.     

一种在计算机上生成凸包的算法

在计算机上实现生成凸包的算法很多,笔者设计的这一算法主要是利用了在计算机表示中,任意两个点之间必定是有一定距离的,而数学上两个点之间的距离可以是无限小这一特点。我们总可以在计算机上任意两个点的位置之间插入一个数学上的点,这个点计算机可能无法表示,但它是可以被计算的。利用这一特点设计了本算法。     

关于求平面点集凸包的一个O(n)时间算法的商榷

王志强等于1998年提出了一个计算平面点集凸包的新算法,并且声称该算法的最坏时间复杂度为O(n),从而为张性时间排序提供了可能性,该文对王志强等提出的求平面点集凸包算法的时间分析提出了不同观点,进一步明确了平面集凸包算法和排序算法的时间下界为Ω(nlogn).     

一个改进的简单多边形凸包算法

凸包问题是计算几何的基本问题之一,在许多领域均有应用。该文通过给出反例,证明文献[4]提出的简单多边形凸包的双动线检测算法不能正确求出任意多边形的凸包,并分析了其缺点,提出了一个改进的算法。改进的算法解决了线性算法所不能解决的自交问题,且实现简单。     

Matching Convex Shapes with Respect to the Symmetric Difference

This paper deals with questions from convex geometry related to shape matching. In particular, we consider the problem of moving one convex figure over another, minimizing the area of their symmetric difference. We show that if we just let the two centers of gravity coincide, the resulting symmetric difference is within a factor of 11/3 of the optimum. This leads to efficient approximate matching algorithms for convex figures. Received November 1996; revised March 1997.     

On the ultimate convex hull algorithm in practice

Kirkpatrick and Seidel [13,14] recently proposed an algorithm for computing the convex hull of n points in the plane that runs in O(n log h) worst case time, where h denotes the number of points on the convex hull of the set. Here a modification of their algorithm is proposed that is believed to run in O(n) expected time for many reasonable distributions of points. The above O(n log h) algorithmsare experimentally compared to the O(n log n) ‘throw-away’ algorithms of Akl, Devroye and Toussaint [2, 8, 20]. The results suggest that although the O(n Log h) algorithms may be the ‘ultimate’ ones in theory, they are of little practical value from the point of view of running time.     

Parallel algorithms for some functions of two convex polygons

Let P andQ be two convex,n-vertex polygons. We consider the problem of computing, in parallel, some functions ofP andQ whenP andQ are disjoint. The model of parallel computation we consider is the CREW-PRAM, i.e., it is the synchronous shared-memory model where concurrent reads are allowed but no two processors can simultaneously attempt to write in the same memory location (even if they are trying to write the same thing). We show that a CREW-PRAM havingn ^1/k processors can compute the following functions in O(k¹⁺) time: (i) the common tangents betweenP andQ, and (ii) the distance betweenP andQ (and hence a straight line separating them). The positive constant can be made arbitrarily close to zero. Even with a linear number of processors, it was not previously known how to achieve constant time performance for computing these functions. The algorithm for problem (ii) is easily modified to detect the case of zero distance as well.This research was supported by the Office of Naval Research under Grants N00014-84-K-0502 and N00014-86-K-0689, and the National Science Foundation under Grant DCR-8451393, with matching funds from AT&T.     

基于凸壳算法的建模算法的研究

基于对当前三维建模方法的研究,提出了基于凸壳算法的物体建模算法。该算法是随机搭建模型产生不规则物体,凸显建模的随机性,对搭建得到不规则物体的顶点集采用凸壳算法进行三角连网,完成随机不规则物体建模,并对模型进行纹理贴图,增强模型的质感。实验证明该算法有良好的操作性。     

Randomized quickhull

This paper contains a simple, randomized algorithm for constructing the convex hull of a set ofn points in the plane with expected running timeO(nlogh) whereh is the number of points on the convex hull. Supported in part by NSA Grant MDA904-93-H-3026 and by the NSF Regional Geometry Institute (Smith College, July 1993) Grant DMS-90 13220.     

海量数据凸壳快速优化算法研究

分析描述加速凸壳算法的基本思想.在分析传统的加速凸壳算法的基础上,根据加速算法剔除内点的时机将加速算法分成静态加速算法和动态加算法.同时阐述了动态加速算法的应用条件,并将动态加速算法应于金字塔凸壳算法之中.通过大量实验数据对比说明动态加速算法对提高平面海量散乱点集的生成速度非常有效。     

Linear Data Structures for Fast Ray-Shooting amidst Convex Polyhedra

We consider the problem of ray shooting in a three-dimensional scene consisting of k (possibly intersecting) convex polyhedra with a total of n facets. That is, we want to preprocess them into a data structure, so that the first intersection point of a query ray and the given polyhedra can be determined quickly. We describe data structures that require preprocessing time and storage (where the notation hides polylogarithmic factors), and have polylogarithmic query time, for several special instances of the problem. These include the case when the ray origins are restricted to lie on a fixed line ℓ ₀, but the directions of the rays are arbitrary, the more general case when the supporting lines of the rays pass through ℓ ₀, and the case of rays orthogonal to some fixed line with arbitrary origins and orientations. We also present a simpler solution for the case of vertical ray-shooting with arbitrary origins. In all cases, this is a significant improvement over previously known techniques (which require Ω(n ²) storage, even when k ≪ n). Work by Haim Kaplan and Natan Rubin has been supported by Grant 975/06 from the Israel Science Fund. Work by Micha Sharir and Natan Rubin was partially supported by NSF Grant CCF-05-14079, by a grant from the U.S.–Israeli Binational Science Foundation, by grant 155/05 from the Israel Science Fund, Israeli Academy of Sciences, by a grant from the AFIRST French–Israeli program, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. A preliminary version of this paper appeared in Proc. 15th Annu. Europ. Sympos. Alg. (2007), 287–298.     

基于凸包求解的简单多边形方向判断新算法

Graham ScanA求解简单多边形凸包算法简洁高效,但是对于未确定方向的简单多边形,该算法需设定一个方向求解其凸包。提出一种新的算法,该算法通过利用凸包求解的Graham ScanA算法来判断简单多边形的方向。算法取得了较好的实用效果。     

Optimal parallel algorithms for point-set and polygon problems

In this paper we give parallel algorithms for a number of problems defined on point sets and polygons. All our algorithms have optimalT(n) * P(n) products, whereT(n) is the time complexity andP(n) is the number of processors used, and are for the EREW PRAM or CREW PRAM models. Our algorithms provide parallel analogues to well-known phenomena from sequential computational geometry, such as the fact that problems for polygons can oftentimes be solved more efficiently than point-set problems, and that nearest-neighbor problems can be solved without explicitly constructing a Voronoi diagram.The research of R. Cole was supported in part by NSF Grants CCR-8702271, CCR-8902221, and CCR-8906949, by ONR Grant N00014-85-K-0046, and by a John Simon Guggenheim Memorial Foundation fellowship. M. T. Goodrich's research was supported by the National Science Foundation under Grant CCR-8810568 and by the National Science Foundation and DARPA under Grant CCR-8908092.     

Capturing crossings: Convex hulls of segment and plane intersections

We give a simple O(nlogn) algorithm to compute the convex hull of the (possibly Θ(n²)) intersection points in an arrangement of n line segments in the plane. We also show an arrangement of dn hyperplanes in d-dimensions whose arrangement has Θ(nd−1) intersection points on the convex hull.