G^2保凸的分段四次插值样条曲线

本构造了一种Ｇ＾２连续的四次保凸插值样条函数,该曲线计算简单,且可以局部修改。     

一种平面保凸插值构造光滑曲线的方法

Ａ．Ｍｅｈａｕｔｅ和Ｆ．Ｕｔｒｅｒａｓ（１９９４）给出了一种平面函数型保凸插值构造光滑曲线的方法（以下简称为Ｍ－Ｕ方法）．本文在利用其方法本质的基础上，给出了一种平面上参数型保凸插值构造光滑曲线的方法，同Ｍｅｈａｕｔｅ和Ｕｔｒｅｒａｓ的方法一样，这里的方法也有局部性．另外这种方法还可以构造平面上的封闭曲线．     

保凸C^1分段二次多项式插值方法

本导出了二次多项式保凸的充要条件,通过插值部分新节点,得到了一种新的保凸Ｃ＾１分段二镒多项式插值函数。     

一种保形C~1三次样条插值方法

本文得到了构造一个保形Ｃ１三次插值样条函数的充要条件，并给出了一种构造保形Ｃ１三次插值样条函数的方法．     

保凸分段三次Bézier插值样条曲线

本文讨论分段三次 Bézier曲线的保凸插值 ,对给定的凸数据点列在相邻两型值点之间构造两个三次 Bézier曲线子段 ,两段之间 G2连续的 ,所构造的曲线插值所有型值点且是 G1的和保凸的     

一种有理三次保形插值样条

一种有理三次保形插值样条王艳春，许有信（南京理工大学）（南京航空航天大学）ＡＳＨＡＰＥＰＲＥＳＥＲＶＩＮＧＲＡＴＩＯＮＡＬＣＵＢＩＣＩＮＴＥＲＰＯＬＡＴＩＯＮＳＰＬＩＮＥ￥ＷａｎｇＹａｎ－ｃｈｕｎ（ＮａｎｊｉｎｇＵｎｉｖｅｒｓｉｔｙｏｆＳｃｉｅｎｃｅ...     

基样条插值的极限性质

本文证明了当m→∞时‖s^(k)mf-f^(k)‖p→0（1＜p＜∞,k=0,1,2）的充要条件是f∈Bπ,p,其中Bπ,p＝Bπ∩Lp(R),Bπ表示指数π型的整函数在R上限制为有界函数所构成的集合,而Smf是在整数点对f 插值的唯一确定的m-1次基样条,进而得一了函数类Bπ,p的三个等价的特征刻划。     

一类保凸的细分格式

提出了一类新的具有高阶连续性和保凸性的5-点二重细分格式.可以证明在参数的某些取值范围内,极限曲线是$C^k~(k=0,1,\ldots,7)$连续的.本文还给出了极限曲线保凸时的参数的取值范围.数值例子表明该格式是灵活有效的.     

一类二次保形拟插值函数的研究

通过讨论一种保形拟插值的基函数与二次规范B-样条函数之间的关系,提出了一类二次保形拟插值样条函数,得到了这类保形拟插值函数在具有线性再生性质,并保持原有数据点列的单调性和凸性时分别应满足的条件,并给出几个应用实例.     

Mixed interpolating-smoothing splines and the ν-spline

In their monograph, Bezhaev and Vasilenko have characterized the “mixed interpolating-smoothing spline” in the abstract setting of a Hilbert space. In this paper, we derive a similar characterization under slightly more general conditions. This is specialized to the finite-dimensional case, and applied to a few well-known problems, including the ν-spline (a piecewise polynomial spline in tension) and near-interpolation, as well as interpolation and smoothing. In particular, one of the main objectives in this paper is to show that the ν-spline is actually a mixed spline, an observation that we believe was not known prior to this work. We also show that the ν-spline is a limiting case of smoothing splines as certain weights increase to infinity, and a limiting case of near-interpolants as certain tolerances decrease to zero. We conclude with an iteration used to construct curvature-bounded ν-spline curves.     

二元线性样条函数插值

二元样条函数插值在计算几何与计算机辅助几何设计中有着重要的作用.本文给出了一种矩形剖分上二元线性样条函数进行Lagrange插值时插值适定结点组所满足的拓扑与几何性质,这种性质依赖于二元线性样条函数所决定的分片线性代数曲线.     

Convexity Preserving Interpolatory Subdivision Schemes

We construct local subdivision schemes that interpolate functional univariate data and that preserve convexity. The resulting limit function of these schemes is continuous and convex for arbitrary convex data. Moreover this class of schemes is restricted to a subdivision scheme that generates a limit function that is convex and continuously differentiable for strictly convex data. The approximation order of this scheme is four. Some generalizations, such as tension control and piecewise convexity preservation, are briefly discussed. November 29, 1996. Date revised: May 28, 1997.     

Morse Theory And Invariant Concepts of Convexity

In optimization, the literature on invariant definitions of convexity has largely centered on the notion of invexity. However, the connection with Morse theory and related geometric trends has not been recognized and results do not appear to be as impressive as those in topology. Our purpose is to improve the situation in optimization, and to clarify the relationship between these two trends. The main result is that the problem can be reduced to the case of functions of one variable.     

A method of adding–removing knots for solving smoothing problems with obstacles

We study a method of adding–removing knots that has been proposed in the literature for solving the smoothing problem with obstacles. The method uses the coefficients of natural splines in the expansion by radial basis functions. We present examples of cycling and counterexamples to possible use of some ideas. We also give some sufficient conditions for finiteness of the method.     

Convexity properties of gradient maps

We consider the action of a real reductive group G on a Kähler manifold Z which is the restriction of a holomorphic action of a complex reductive group H. We assume that the action of a maximal compact subgroup U of H is Hamiltonian and that G is compatible with a Cartan decomposition of H. We have an associated gradient map μp:Z→p where g=k⊕p is the Cartan decomposition of g. For a G-stable subset Y of Z we consider convexity properties of the intersection of μp(Y) with a closed Weyl chamber in a maximal abelian subspace a of p. Our main result is a Convexity Theorem for real semi-algebraic subsets Y of Z=P(V) where V is a unitary representation of U.     

The Convexity Number of a Graph

 For two vertices u and v of a connected graph G, the set I[u,v] consists of all those vertices lying on a u−v shortest path in G, while for a set S of vertices of G, the set I[S] is the union of all sets I[u,v] for u,v∈S. A set S is convex if I[S]=S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. The clique number ω(G) is the maximum cardinality of a clique in G. If G is a connected graph of order n that is not complete, then n≥3 and 2≤ω(G)≤con(G)≤n−1. It is shown that for every triple l,k,n of integers with n≥3 and 2≤l≤k≤n−1, there exists a noncomplete connected graph G of order n with ω(G)=l and con(G)=k. Other results on convex numbers are also presented. Received: August 19, 1998 Final version received: May 17, 2000     

Variational Interpolation of Subsets

We consider the problem of the variational interpolation of subsets of Euclidean spaces by curves such that the L² norm of the second derivative is minimized. It is well-known that the resulting curves are cubic spline curves. We study geometric boundary conditions arising for various types of subsets such as subspaces, polyhedra, and submanifolds, and we indicate how solutions can be computed in the case of convex polyhedra.