二次C-曲线研究

给出具有相同控制顶点的二次C-曲线与二次有理Bézier曲线表示同一参数曲线段的充要条件,由此得到了二次C-曲线不能精确表示双曲线段的结论;另外,还给出了二次C-曲线在任意一点的细分公式.     

重心坐标系下平面上三次Bezier曲线奇拐点的分布

本文利用重心坐标，研究Bezier曲线奇拐点的分布，绘出了奇拐点存在的区域和充要条件，简化了判别Bezier曲线奇拐点的充要条件。并给出了Bezier曲线的重心坐标表示方法．     

重心坐标系下平面上三次Bézier曲线奇拐点的分布

本文利用重心坐标,研究Bzier曲线奇拐点的分布,绘出了奇拐点存在的区域和充要条件,简化了判别Bzier曲线奇拐点的充要条件,并给出了Bzier曲线的重心坐标表示方法.     

关于Bertrand曲线和Mannheim曲线推广的一个充要条件

给出了Bertrand曲线和Mannheim曲线推广的一个充要条件.做为其特例,得到了Bertrand曲线和Mannheim曲线的充要条件.     

四次带参Ball曲线的形状分析

基于包络理论与拓扑映射的方法对四次带参Ball曲线进行了形状分析,得出了曲线上含有奇点,拐点和曲线为局部凸或全局凸的充分必要条件,这些条件完全由控制多边形和形状参数所决定;并进一步讨论了形状参数对形状分布图的影响及其对曲线形状的调节能力.研究表明,四次带参Ball曲线的形状调控能力要优于四次带参Bezier曲线.     

关于正半群的生成条件

本文引进了一个集族O_(?),用它把Kato的方法推广到了局部凸的拓扑线性空间,得到了某些半序Frechet空间中生成一个正半群的充要条件,把这些条件用到半序Banach空间上时,可以得到一些新结果,同时,把Kato的定理推广到了局部凸拓扑线性空间。     

生长曲线模型中共同均值参数的线性估计的可容许性

本文讨论了生长曲线模型中共同均值参数的估计问题,针对矩阵损失函数,给出了六种不同形式的优良性准则,在较特殊的生长曲线模型下,证明了这六种容许性在特殊的齐次线性估计类中是一致的,并且得到了其共同容许估计的充要条件,但在中它们是不一致的,可分为两类,并且也分别得到了其可容许估计的充要条件,还进一步讨论了非齐次线性估计的容许性问题。     

一般生长曲线模型中共同均值参数线性估计的泛容许性

本文研究一般生长曲线模型中共同均值参数的估计问题,给出了共同均值参数线性估计的泛容许定义,并在较特殊的齐次与非齐次线性估计类中得到了泛容许的充要条件．     

两条Bézier样条的光滑连接

研究了用一条样条曲线把两条不相连接的样条曲线光滑连接起来的问题,给出了连接两条一元n次参数样条曲线为一条新的一元n次参数样条曲线的条件,适用于参数样条曲线添加控制顶点的情形,进一步得到了两条一次、二次、三次Bézier样条曲线在几何连续性下实现自然光滑连接的条件.     

一类直纹面的研究

本给出了过任意空间C^k(k≥3)类光滑曲线的直纹面是可展曲面的充要条件．同时得到了该空间曲线为相应直纹面的曲率线,测地线和渐近曲线的充要条件。     

关于平面四次Bézier曲线的拐点与奇点

在计算几何中,已给出了三次Bezier曲线的保凸性的充要条件,并进行了几何解释。本文则是导出形式简洁的拐点和奇点方程并对四次Bezier曲线的拐点和奇点的分布进行讨论。按Bezier曲线的拐点个数进行分类,还得到了四次Bezier曲线有奇点的充分必要条件,并给出几个数值实例,实例说明,不但非凸的单纯特征多角形可以有凸的Bezier曲线段,而且非单纯特征多角形也可以有凸的Bezier曲线段。四次Bezier曲线的奇点和拐点是可以共存的。     

由两点确定的具凸包性和保凸性的三次有理Bézier曲线

This paper shows that under a necessary and sufficient condition, there exists a cubic rational Bezier curve with convex hull property and convexity preserving property which passes two given points inside the control polygon.     

由两点确定的具凸包性和保凸性的三次有理Bézier曲线

１．引言 在文［１］中,本文作者探讨了当控制多边形为凸时,过控制多边形内部任意给定两点的三次有理 Ｂéｚｉｅｒ曲线的存在唯一性问题,给出了这样的曲线存在的充要条件并证明了其若存在则是唯一的,还给出了其权因子的计算式．但由两点确定的三次有理 Ｂｚｉｅｒ曲线的权因子不一定非负,从而不能保证曲线具凸包性和保凸性,而无论从理论还是实用角度看,曲线的这两个性质都是很重要的． 本文从如下方面进一步深化［１］的论题：当凸控制多边形内部两点 ｐ１, ｐ２满足什么条件时,过Ｐ１,Ｐ２两点的三次有理 Ｂéｚｉｅｒ曲线不仅…     

Some properties of convex hulls of integer points contained in general convex sets

In this paper, we study properties of general closed convex sets that determine the closedness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron. This result generalizes the well-known result due to Meyer (Math Program 7:223–225, 1974). Under a simple technical assumption, we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron.     

Nowhere Vanishing Torsion Closed Curves Always Hide Twice

Convexity may imply points of vanishing torsion as the spatial 4-vertex theorem shows. We state here that for a simple closed curve to have nowhere vanishing torsion, it must violate convexity hiding at least twice inside its convex hull. Both the 4-vertex and Lsquo;hiding-twice results are generalized by obtaining a relation between the number of vanishing torsion points (vertices) of a closed space curve and the number of its components inside its convex hull. We also comment on elastic curves.     

平面三次H-Bézier曲线的形状分析

本文对平面三次H-Bézier曲线的形状进行分析,讨论其诸如奇点、拐点、局部凸和全局凸的几何特征,得出曲线上含有奇点、拐点和曲线为局部凸或全局凸的用控制多边形边向量相对位置表示的充分必要条件.     

Low-degree points on Hurwitz-Klein curves

We investigate low-degree points on the Fermat curve of degree 13, the Snyder quintic curve and the Klein quartic curve. We compute all quadratic points on these curves and use Coleman's effective Chabauty method to obtain bounds for the number of cubic points on each of the former two curves.

     

A weighted curvature flow for shape deformation

In this paper we shall discuss a weighted curvature flow for a regular curve in the 2D Euclidean space. The weighted curvature flow for planar curves is a generalization of the well-known curvature flow discussed by Gage, Hamilton and Grayson. Under a suitable weighted curvature flow, convex curves will remain convex in the deformation process. However, the curve may not converge to a round point for general weights. Indeed, for a nonnegative weight function ω(u) with k isolated zeros, a curve will converge to a limiting k-polygon. The weighted curvature flow will have many useful properties which have applications to image processing. We shall also present some numerical simulations to illustrate how curves deform under the weighted curvature flow with different weight functions ω(u). Moreover, our algorithm is very effective and stable. The approximation of higher derivatives in our new algorithm only involve in the neighboring points.     

Continuous selector of representative measures and the space of faces of a convex compactum

A sufficient condition for the existence of a continuous selector of representative measure, concentrated at the extreme points of a convex metrizable compactum, is considered. A necessary condition for the existence of such a selector is deduced. An example is given of a convex compactum with a closed set of extreme points, for which no continuous selector exists.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 897–906, December, 1977.In conclusion, the author expresses deep gratitude to A. M. Vershik for the formulation of the problem and for help in the preparation of this paper.     

The set of all nondominated solutions in linear cases and a multicriteria simplex method

In this note we are interested in the properties of, and methods for locating the set of all nondominated solutions of multiple linear criteria defined over a polyhedron. We first show that the set of all dominated solutions is convex and that the set of all nondominated solutions is a subset of the convex hull of the nondominated extreme points. When the domination cone is polyhedral, we derive a necessary and sufficient condition for a point to be nondominated. The condition is stronger than that of Ref. [1] and enables us to give a simple proof that the set of all nondominated extreme points indeed is connected. In order to locate the entire set of all nondominated extreme points, we derive a generalized version of simplex method—multicriteria simplex method. In addition to some useful results, a necessary and sufficient condition for an extreme point to be nondominated is derived. Examples and computer experience are also given. Finally, we focus on how to generate the entire set of all nondominated solutions through the set of all nondominated extreme points. A decomposition theorem and some necessary and sufficient conditions for a face to be nondominated are derived. We then describe a systematic way to identify the entire set of all nondominated solutions. Through examples, we show that in fact our procedure is quite efficient.