关于圆的椭圆逼近显示的最优算法

本给出了一个逼近显示圆的新算法,该算法是通过相交多形而不是内接多边形逼近圆,由于构造构交多连形时其面积等于圆面积,因此新算法的最优逼近,同时还推广到椭圆。     

基于刘徽割圆术的等距线逼近算法

给出了一种基于刘徽割圆术的平面NURBS曲线的等距线的逼近算法。利用正多边形代替圆所扫掠出的区域边界来近似等距曲线，所得到的逼近曲线是与基曲线同次的NURBS曲线，并且可以达到任意的精度。     

怎样用八段圆弧画一个与给定椭圆偏差最小的曲线?

通过分析传统的八心圆椭圆近似作图的误差,从曲线最佳逼近的角度出发,给出一种偏差接近最小、且容易用尺规作图完成的新画法.研究结果,可以应用到数控机床加工之中.     

单位圆域上Bessel级数的Fejer和的一致逼近

讨论了单位圆域上Bcasel级数的Fejer和的—致逼近．给出了它的饱和阶和饱和类．     

具边界相交交界面的拟线性椭圆方程组的挠射问题

研究有界区域上的n维拟线性椭圆方程组的挠射问题,其中方程的系数在交界面上允许间断,而且交界面允许与区域外边界相交.通过构造一个交界面与外边界不相交的近似挠射问题,利用估计和逼近方法,并讨论弱解的正则性,得到了问题解的存在性,并将这些结果应用到两种群Lotka-Volterra互惠模型.     

任意次有理Bezier曲线／面对其控制多边形／网格的整体或局部逼近

本文给出一种利用权因子构造整体或局部逼近控制多边形／网格的有理Ｂｅｚｉｅｒ曲线／面的方法,该法适用于任意次数的有理Ｂｅｚｉｅｒ曲线／面、任意的控制多边形／网络,权因子的选择和逼近度的估计都只依赖于一个参数ｗ．当ｗ→＋∞时,相应的曲线／面可按预定要求整体或局部地逼近其控制多边形／网络,逼近阶为ｏ（１／ｗ）。     

带有给定凸切线多边形的保形五次样条逼近

本讨论带有给定切线多边形的保形逼近问题．给出了一条与给定切线多边形相切的保形五次参数祥条曲线。     

一类有理交错样条曲线的逼近性及形状修改

本文提出一类Ｃ＾２连续的四阶有理交错样条曲线，它能整体或局部逼近于控制多边形。     

两圆方程相减后所得方程的几何意义

1两圆相交 2两圆相切(内切或外切) 3两圆相离 4范例     

子空间单位球的逼近紧性

设X是Banach空间,Y为X的子空间,B_X,B_Y分别是X和Y的闭单位球.本文研究B_Y的逼近紧性,证明了B_Y在X中是逼近紧的当且仅当对Y的每个与B_X相交的平移Y_T,Y_T∩B_X在Y_T中都是逼近紧的.还得到B_Y逼近紧的稳定性结果.     

单位圆到任意曲线保角变换的近似计算方法

本文讨论了将单位圆内部映射成由任意曲线(包括任意曲线割缝)边界围成的单连通域内部或外部的保角变换问题.以多边形逼近单连通域的边界,采用Schwartz-Christoffel积分建立单位圆与该多边形的映射函数.给出了确定Schwartz-Christoffel积分中未知参数的数值计算方法.     

An inequality on the edge lengths of triangular meshes

We give a short proof of the following geometric inequality: for any two triangular meshes A and B of the same polygon C, if the number of vertices in A is at most the number of vertices in B, then the maximum length of an edge in A is at least the minimum distance between two vertices in B. Here the vertices in each triangular mesh include the vertices of the polygon and possibly additional Steiner points. The polygon must not be self-intersecting but may be non-convex and may even have holes. This inequality is useful for many purposes, especially in proving performance guarantees of mesh generation algorithms. For example, a weaker corollary of the inequality confirms a conjecture of Aurenhammer et al. [Theoretical Computer Science 289 (2002) 879-895] concerning triangular meshes of convex polygons, and improves the approximation ratios of their mesh generation algorithm for minimizing the maximum edge length and the maximum triangle perimeter of a triangular mesh.     

Scheduling two irregular polygons

Let A and B be irregular polygons with m and n vertices (m ⩾ n) on some circle line. How should polygon A be moved relative to polygon B in such a way that the maximum (minimum) distance between adjacent vertices on the circle line is minimized (maximized)? O(m logm) algorithms are given which solve these problems.     

Optimization of nested polygons

The optimization problem under consideration requires to find the largest regular polygon withk sides to be fitted into a regular polygon withk – 1 sides. If the sequence of these maximal polygons is started with an equilateral triangle, then the final nested polygon, a circle, possesses a radiusr=0.3414r ₃, wherer ₃ is the radius of the inscribed circle of the equilateral triangle. Lower bounds for the ratior/r ₃ are also obtained.     

Discrete linearized least-squares rational approximation on the unit circle

We already generalized the Rutishauser—Gragg—Harrod—Reichel algorithm for discrete least-squares polynomial approximation on the real axis to the rational case. In this paper, a new method for discrete least-squares linearized rational approximation on the unit circle is presented. It generalizes the algorithms of Reichel—Ammar—Gragg for discrete least-squares polynomial approximation on the unit circle to the rationale case. The algorithm is fast in the sense that it requires order m computation time where m is the number of data points and is the degree of the approximant. We describe how this algorithm can be implemented in parallel. Examples illustrate the numerical behavior of the algorithm.     

Generalization of the triangle and Ptolemy inequalities

The Euclidean triangle inequality generalizes to an alternating inequality for any oddsided polygon that can be inscribed in a circle; there is equality in the even cases. A generalization of Ptolemy's theorem follows by inversion. The results have Minkowskian analogues.     

On the numerical approximation of the rotation number

The starting point of this paper is a polygonal approximation of an invariant curve of a map. Using this polygonal approximation an approximation for the circle map (the restriction of the map to the invariant curve) is obtained. The rotation number of the circle map is then approximated by the rotation number of the approximated circle map. The error in the obtained approximate rotation number is discussed, and related to the error in the polygonal approximation of the invariant curve. Simple algorithms for the approximation of the rotation number are described. A numerical example illustrates the theory.     

Holding a regular pyramid by a circle

Among other things, we prove the following (1) If a regular pyramid whose base is a regular polygon of circum-radius 1 has height at least 1, then the pyramid can be held by a circle, while for every ${0 < \varepsilon < 1}$ , there is a regular pyramid with height ${1 - \varepsilon}$ and base polygon of circum-radius 1 that cannot be held by any circle. (2) A regular pyramid of height h whose base is an equilateral triangle of circum-radius 1 can be held by a circle if and only if h > 0.479 . . . (which complements a theorem by Tanoue). (3) A regular pyramid with square base of unit circum-radius can be held by a circle if and only if its height is greater than 0.828 . . ..     

Hierarchical Decompositions and Circular Ray Shooting in Simple Polygons

A hierarchical decomposition of a simple polygon is introduced. The hierarchy has logarithmic depth, linear size, and its regions have at most three neighbors. Using this hierarchy, circular ray shooting queries in a simple polygon on n vertices can be answered in O(log² n) query time and O(n log n) space. If the radius of the circle is fixed, the query time can be improved to O(log n) and the space to O(n).     

Hyperbolic polygons of minimal perimeter with given angles

We prove that, among all convex hyperbolic polygons with given angles, the perimeter is minimized by the unique polygon with an inscribed circle. The proof relies on work of Schlenker (Trans Am Math Soc 359(5): 2155–2189, 2007).