最佳逼近圆周的正多边形算法

本文定义了圆周的最佳逼近正n边形,在上述意义下,解决了圆周的最佳逼近正n边形半径的计算问题。     

最佳逼近圆周的正多边形算法(Ⅲ)

利用渐近平均距离定义了圆周的最佳逼近正多边形 ,在上述意义下 ,解决了圆周的最佳逼近正多边形半径R的计算问题     

最佳逼近圆周的正多边形算法(Ⅲ)

利用渐近平均距离定义了圆周的最佳逼近正多边形,在上述意义下,解决了圆周的最佳逼近正多边形半径R的计算问题.     

广义重心坐标的递推关系

从线性方程组解空间的角度理解广义重心坐标(GBCs),给出平面重心坐标从n 边形 到n 1边形的递推关系式。将构造重心坐标的问题转化为构造函数的问题,不需考虑坐标函数的 几何意义,选取满足约束条件的函数即可构造重心坐标。在推导过程中,n 1边形(n≥3)可看 作n边形与一顶点的组合,将该顶点用n边形的顶点线性表出,可将n 1边形上的重心坐标化为 n边形上的齐次坐标(homogeneous coordinates)。为第n 1个坐标函数施加一定限制条件,即得到 n 边形上一组重心坐标。     

逻辑函数多边形简化法

本文提出一种逻辑函数的多边形简化法。这种方法的要点是:将n 个逻辑变量C_1,C_2,……,C_n 及它们的补(?),(?),……,(?)构成一个2n 边形。将逻辑函数中的每一个最小项看成是2n 边形中的一个n 边形,然后按文中指出的法则读出重边或重点就得到了简化式。     

用广义von Koch曲线生成元构造D_m对称分形

为了生成具有m重旋转和m个反射的Dm对称分形,提出用含有正n边形的广义von Koch曲线生成元构造对称分形的简便方法.以长度为单位1的线段作为初始元,定义了关于初始元的垂直平分线对称的生成元,它用去掉底边的边长为1/3的正n边形替代初始元的三等分的中间线段而生成;基于生成元构造了从边界向内部生长的广义von Koch曲线的D3对称分形;进一步基于含有正n边形生成元构造了von Koch曲线的Dm对称分形.实验结果表明,本文提出的用广义von Koch曲线生成元构造分形的方法可以生成具有严格Dm对称的分形,这些分形从正m多边形的边界向内部生长并随n值增加,生长到超出正m多边形的其它边界.本文提出了用广义von Koch曲线生成元构造具有内部结构的对称分形的方法.     

基于n边形折叠的网格简化算法

提出在三角网格中利用多个三角形组合及检索n边形(n为正整数)的规则,并提出一种具有相似折叠规律的n边形折叠的网格简化算法,该算法以n边形折叠为基本简化操作,并以二次误差作为误差度量,每次n边形折叠操作可以减少,n-1个顶点以及2(n-1)个三角形,n越大达到某一简化目标所需的折叠次数越少,因此简化速度也可能越快.通过选取适当的n值及新顶点位置,新算法可以转化成顶点删除、边折叠及三角形折叠3种已知的几何元素删除算法,因此也可以视做为基于二次误差度量的几何元素删除简化算法的总括算法.最后分别对几种n取值情况列举实验数据,说明该算法的有效性.     

如何编程求任意均匀n边形板的重心

本次擂台赛的题目是任意给定一个n边形的均匀板,请编程求该板的重心。这里的n边形可能是凸多边形也可能是凹多边形,有部分参赛程序只能解决凸多边形问题。这次的擂主程序比较长,限于篇幅无法刊用,这里刊出本次评分第二名的参赛稿,供大家参考。     

塑造n边形曲面的外形

提出了一种高效基于物理性质的算法来动态的塑造n边形曲面的外形.这种算法是基于曲面的物理性质.通过极小化一个能量泛函,用户能够直接使用不同形式的外力作为虚拟的塑造工具来改变n边形曲面的外形.用户也能够定义必要的几何约束来进一步控制曲面外形.与通常移动控制点的方法比较,这种基于物理性质的方法更直观和有效.     

线性流形上一类矩阵方程的最佳逼近问题

本文研究矩阵方程AXB=C在线性流形Γ={X∈SR~(n×n)|||TX-Y||=min}上的最佳逼近问题,得到了问题解的表达式．此外,还给出了求解该问题的数值算法和算例．     

圆的等面积逼近和生成

本文描述用多边形等面积逼近和生成圆的算法，此算法是用一个与圆相交的多边形（而不是通常用的内接多边形）逼近一个圆，这个多边形的面积精确地等于圆的面积。因此，可以认为这种算法产生的多边形是对圆的一种等面积最佳逼近。     

On approximation behavior of the greedy triangulation for convex polygons

We prove that the greedy triangulation heuristic for minimum weight triangulation of convex polygons yields solutions within a constant factor from the optimum. For interesting classes of convex polygons, we derive small upper bounds on the constant approximation factor. Our results contrast with Kirkpatrick's (n) bound on the approximation factor of the Delaunay triangulation heuristic for minimum weight triangulation of convexn-vertex polygons. On the other hand, we present a straightforward implementation of the greedy triangulation heuristic for ann-vertex convex point set or a convex polygon takingO(n ²) time andO(n) space. To derive the latter result, we show that given a convex polygonP, one can find for all verticesv ofP a shortest diagonal ofP incident tov in linear time. Finally, we observe that the greedy triangulation for convex polygons having so-called semicircular property can be constructed in timeO(n logn).     

A new approximation algorithm for labeling points with circle pairs

We study the NP-hard problem of labeling points with maximum-radius circle pairs: given n point sites in the plane, find a placement for 2n interior-disjoint uniform circles, such that each site touches two circles and the circle radius is maximized. We present a new approximation algorithm for this problem that runs in time and O(n) space and achieves an approximation factor of (≈1.486+ε), which improves the previous best bound of 1.491+ε.     

A Linear-Time Approximation Scheme for Minimum Weight Triangulation of Convex Polygons

A linear-time heuristic for minimum weight triangulation of convex polygons is presented. This heuristic produces a triangulation of length within a factor 1 + ε from the optimum, where ε is an arbitrarily small positive constant. This is the first subcubic algorithm that guarantees such an approximation factor, and it has interesting applications. Received November 21, 1996; revised March 9, 1997.     

On approximation algorithms for the terminal Steiner tree problem

The terminal Steiner tree problem is a special version of the Steiner tree problem, where a Steiner minimum tree has to be found in which all terminals are leaves. We prove that no polynomial time approximation algorithm for the terminal Steiner tree problem can achieve an approximation ratio less than (1−o(1))lnn unless NP has slightly superpolynomial time algorithms. Moreover, we present a polynomial time approximation algorithm for the metric version of this problem with a performance ratio of 2ρ, where ρ denotes the best known approximation ratio for the Steiner tree problem. This improves the previously best known approximation ratio for the metric terminal Steiner tree problem of ρ+2.     

Approximating Euclidean circles by neighbourhood sequences in a hexagonal grid

In this paper the nodes of the hexagonal grid are used as points. There are three types of neighbours on this grid, therefore neighbourhood sequences contain values 1, 2, 3. The grid is coordinatized by three coordinates in a symmetric way. Digital circles are classified based on digital distances using neighbourhood sequences. They can be triangle, hexagon, enneagon and dodecagon. Their corners and side-lengths are computed, such as their perimeters and areas. The radius of a digital disk is usually not well-defined, i.e., the same disk can have various radii according to the neighbourhood sequence used. Therefore the non-compactness ratio is used to measure the quality of approximation of the Euclidean circles. The best approximating neighbourhood sequence is presented. It is shown that the approximation can be improved using two neighbourhood sequences in parallel. Comparisons to other approximations are also shown.     

Ring-coupled unicycles: Boundedness, convergence, and control

In cyclic pursuit a platoon of vehicles are coupled in a unidirectional ring at the interaction level according to some control scheme. In the paper, a new cyclic pursuit control law is proposed, where each vehicle’s linear speed and angular speed are proportional to the projection of its prey’s position on its forward direction and lateral direction respectively. Through these interactions a cooperative behavior emerges and vehicles in the platoon eventually move at a constant speed on a circle with constant inter-vehicle spacings. The control scheme ensures ultimate boundedness and leads to only two stable equilibrium polygons. This contrasts with other cyclic pursuit control schemes, where vehicles may diverge to infinity and there are more stable equilibrium polygons as the total number of vehicles increases. For this control scheme, ultimate boundedness is proved using the pseudo-linearization technique. Possible equilibrium polygons are analyzed and stability and convergence properties are established through root locus analysis of a complex characteristic polynomial. Design rules are discussed, showing how the radius of the circle they converge to is controlled by an appropriate choice of control parameters.     

移动最小二乘代理模型支持域半径的优化方法

移动最小二乘代理模型描述局部波动的能力优于一般的代理模型,但其精度受支持域半径的影响。在经验公式的基础上提出了一种针对移动最小二乘代理模型支持域半径的优化方法。对支持域内抽样点数寻优获取最佳半径值,提高近似精度进而达到减少抽样点的目的。数值实验结果表明,对于不同基函数阶次和权函数的情况,提出的方法大大提高了移动最小二乘代理模型的近似精度,与基于经验公式的移动最小二乘代理模型相比,其仅需较少的抽样点即可达到相同的近似精度。