曲面上数据的C~1有理插值

1.引言本文考虑如下问题：对于一给定的点集P＝｛（P；，人）EIR4｝仁l且P；End在一光＄双面曲面o上（称其为定义域曲面），构造一分片C‘连续的函数f（称其为曲面上的函数）使得人P；）一人，i二l，…，M．在物体表面（定义域曲面）上构造函数的问题产生于许多应用领域．如刻划地球上的降怀量；机翼上的压力；人体表的温度等．自该问题于1985年由Barnhill［3］提出以来，已发ffi出若干解决它的方法（综述文章参见［4，11］）．其中大部分方法寻求全局解（见卜，14，16］且假设定义域曲面为平面或球面．少数方法寻找局部解（见15，6，7…     

一种细分格式生成极限曲面的法向量的求法

在Levin给出的三角域上生成极限曲面的法向量求法基础上,给了同拟蝴蝶形细分在矩形域上生成极限曲面的情况,并得到了两个自由度,可以对法向量进行优化选取,这对讨论曲面的等距面有广泛的实际意义。     

对非对称有理细分矩阵的改进

在图形图像显示中,运算次数的多少直接影响图像的显示速度.通过对非对称细分矩阵的运算量较大的因素进行分析,并在分析的基础上对细分矩阵加以改进,简化细分矩阵结构减少一些不必要的运算与重复运算,构造出一种结构比较简单、运算量比较少的细分矩阵.新构造的细分矩阵可以有效的提高运算速度.应用新构造的细分矩阵生成细分曲线,对所生成的细分曲线进行比较,得出改进细分矩阵使得运算量明显减少,提高图形的显示速度.     

五次完全幂的少位数三进制展开*

本文讨论了Michael Bennett [Bennett M, Bugeaud Y, Mignotte M. Perfect powers with few binary digits and related Diophantine problem [J]. Ann Sc Norm Super Pisa Cl Sci, 2013,XII:941–953.]中提出的一类丢番图方程, 即五次完全幂的少位数三进制展开. 作者证明了丢番图方程 3^a + 3^b + 2 = n⁵, a > b > 0 有唯一的正整数解$(a,b,n)=(3,1,2).     

Catmull-Clark曲面控制网格的收敛性质

文献[4]给出Catmull—Clark细分曲面控制网格的收敛速率和一个误差计算公式．本文在这基础上提出一个新的算法，并借助此新算法得到关于Catmull—Clark细分曲面控制网络的收敛速率的更精确的估计和给出更好的误差计算公式．     

用边界曲线构造C~1 Coons曲面确定扭矢的方法

本文讨论了由四条边界曲线构造Ｃ１Ｃｏｏｎｓ曲面的问题，给出了确定角点扭矢的新方法．该方法沿四边形两对角线方向构造两条四次多项式曲线，每个角点处的扭矢，由一条四次曲线和两条边界曲线确定．跨界切矢由三次埃尔米特插值方法定义．文中还给出了一个用新方法构造曲面的实例．     

射影空间中代数曲线与超曲面的一些新的性质

发现了代数曲线的新的不变量一特征数,并得到了Pascal定理的不同于3次曲线的Cllasles定理和高次曲线中的Cayley-Bacharach定理等形式的高次推广.进一步研究了平面代数曲线的一些性质.通过定义m次Pascal超曲面,将Pascal定理推广到n维射影空间的m次超曲面中,证明了n-单纯形上的Pascal点位于一个m次Pascal超曲面的充要条件是其每个2维面上的Pascal点分别位于m次平面Pascal空间的一条代数曲线上.进一步,给出了一定条件下m次Pascal超曲面与m-1次Pascal超曲面之间的内在关系.     

插值细分曲线有理参数点的精确求值

本文提出了求值插值细分曲线上任意有理参数的算法.通过构造与细分格式相关的矩阵,m进制分解给定有理数以及特征分解循环节对应算子乘积,计算得到控制顶点权值,实现对称型静态均匀插值细分曲线的求值.本文给出了四点细分和四点Ternary细分曲线的求值实例.算法可以推广到求值其他非多项式细分格式中.     

广义Nekrasov矩阵的一组细分迭代判定条件

利用细分和迭代的思想,细分了矩阵的指标集,构造了迭代系数,给出了广义Nekrasov矩阵的一组细分迭代判定条件,并用数值算例说明了判定条件的有效性.     

一类双k次B样条曲面的G1连续性条件

本文针对两个k×k次B样条曲面的节点向量为端点插值、内部是单节点的情形 ,给出它们之间的G1光滑拼接条件 ,同时得到它们的公共边界曲线的控制顶点所要满足的本征方程 .其中本征方程是B样条曲面片所独有的现象 .     

Shape controlled interpolatory ternary subdivision

Ternary subdivision schemes compare favorably with their binary analogues because they are able to generate limit functions with the same (or higher) smoothness but smaller support.In this work we consider the two issues of local tension control and conics reproduction in univariate interpolating ternary refinements. We show that both these features can be included in a unique interpolating 4-point subdivision method by means of non-stationary insertion rules that do not affect the improved smoothness and locality of ternary schemes. This is realized by exploiting local shape parameters associated with the initial polyline edges.     

Modified form of binary and ternary 3-point subdivision schemes

Binary 3-point scheme, developed by Hormann and Sabin [Hormann, K. and Sabin, Malcolm A., 2008, A family of subdivision schemes with cubic precision, Computer Aided Geometric Design, 25, 41-52], has been modified by introducing a tension parameter which generates a family of C¹ limiting curves for certain range of tension parameter. Ternary 3-point scheme, introduced by Siddiqi and Rehan [Siddiqi, Shahid S. and Rehan, K., 2009, A ternary three point scheme for curve designing, International Journal of Computer Mathematics, In Press, DOI: 10.1080/00207160802428220], has also been modified by introducing a tension parameter which generates family of C¹ and C² limiting curves for certain range of tension parameter. Laurent polynomial method is used to investigate the continuity of the subdivision schemes. The performance of modified schemes has been demonstrated by considering different examples along with its comparison with the established subdivision schemes.     

A degree estimate for subdivision surfaces of higher regularity

Subdivision algorithms can be used to construct smooth surfaces from control meshes of arbitrary topological structure. In contrast to tangent plane continuity, which is well understood, very little is known about the generation of subdivision surfaces of higher regularity. This work presents a degree estimate for piecewise polynomial subdivision surfaces saying that curvature continuity is possible only if the bi-degree of the patches satisfies , where is the order of smoothness on the regular part of the surface. This result applies to any stationary or non-stationary scheme consisting of masks of arbitrary size provided that some generic symmetry and regularity assumptions are fulfilled.

     

On the regularity analysis of interpolatory Hermite subdivision schemes

It is well known that the critical Hölder regularity of a subdivision schemes can typically be expressed in terms of the joint-spectral radius (JSR) of two operators restricted to a common finite-dimensional invariant subspace. In this article, we investigate interpolatory Hermite subdivision schemes in dimension one and specifically those with optimal accuracy orders. The latter include as special cases the well-known Lagrange interpolatory subdivision schemes by Deslauriers and Dubuc. We first show how to express the critical Hölder regularity of such a scheme in terms of the joint-spectral radius of a matrix pair {F₀,F₁} given in a very explicit form. While the so-called finiteness conjecture for JSR is known to be not true in general, we conjecture that for such matrix pairs arising from Hermite interpolatory schemes of optimal accuracy orders a “strong finiteness conjecture” holds: ρ(F₀,F₁)=ρ(F₀)=ρ(F₁). We prove that this conjecture is a consequence of another conjectured property of Hermite interpolatory schemes which, in turn, is connected to a kind of positivity property of matrix polynomials. We also prove these conjectures in certain new cases using both time and frequency domain arguments; our study here strongly suggests the existence of a notion of “positive definiteness” for non-Hermitian matrices.     

The independence of initial vectors in the subdivision schemes

Starting with an initial vector λ = (λ(κ))κ∈z ∈ ep(Z), the subdivision scheme generates asequence (Snaλ)∞n=1 of vectors by the subdivision operator Saλ(κ) = ∑λ(j)a(k - 2j), k ∈ Z. j∈zSubdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting tounderstand under what conditions the sequence (Snaλ)∞n=1 converges to an Lp-function in an appropriate sense.This problem has been studied extensively. In this paper we show that the subdivision scheme converges forany initial vector in ep(Z) provided that it does for one nonzero vector in that space. Moreover, if the integertranslates of the refinable function are stable, the smoothness of the limit function corresponding to the vectorλ is also independent of λ.     

The distance of a subdivision surface to its control polyhedron

For standard subdivision algorithms and generic input data, near an extraordinary point the distance from the limit surface to the control polyhedron after m subdivision steps is shown to decay dominated by the mth power of the subsubdominant (third largest) eigenvalue. Conversely, for Loop subdivision we exhibit generic input data so that the Hausdorff distance at the mth step is greater than or equal to the mth power of the subsubdominant eigenvalue.In practice, it is important to closely predict the number of subdivision steps necessary so that the control polyhedron approximates the surface to within a fixed distance. Based on the above analysis, two such predictions are evaluated. The first is a popular heuristic that analyzes the distance only for control points and not for the facets of the control polyhedron. For a set of test polyhedra this prediction is remarkably close to the true distance. However, a concrete example shows that the prediction is not safe but can prescribe too few steps. The second approach is to first locally, per vertex neighborhood, subdivide the input net and then apply tabulated bounds on the eigenfunctions of the subdivision algorithm. This yields always safe predictions that are within one step for a set of typical test surfaces.     

Smoothness of subdivision surfaces at extraordinary points

A stationary subdivision scheme such as Catmull and Clark's is described by a matrix iteration around an extraordinary point. We show how higher order smoothness of a limiting surface obtained by a stationary subdivision algorithm for tri- or quadrilateral nets depends on the spectral properties of the matrix and give necessary and sufficient conditions. The results are also useful to construct subdivision algorithms for surfaces of any smoothness order at extraordinary points. This revised version was published online in August 2006 with corrections to the Cover Date.     

C^＊-代数中的等价

在算子理论和算子代数的研究中，等价是一个很重要的工具，它对K-理论的研究和理解起着很重要的作用．寻找等价关系是研究不变量的一种方法，为了使大家更好的认识和学习C^＊-代数，我们在本文研究了算子代数中经常用到的代数等价、酉等价、同伦等几种等价之间的关系．     

The template‐based procedure of biorthogonal ternary wavelets for curve multiresolution processing

Curve multiresolution processing techniques have been widely discussed in the study of subdivision schemes and many applications, such as surface progressive transmission and compression. The ternary subdivision scheme is the more appealing one because it can possess the symmetry, smaller topological support, and certain smoothness, simultaneously. So biorthogonal ternary wavelets are discussed in this paper, in which refinable functions are designed for cure and surface multiresolution processing of ternary subdivision schemes. Moreover, by the help of lifting techniques, the template‐based procedure is established for constructing ternary refinable systems with certain symmetry, and it also gives a clear geometric templates of corresponding multiresolution algorithms by several iterative steps. Some examples with certain smoothness are constructed.