On the Problem of Instability in the Dimensions of Spline Spaces over T-Meshes with T-Cycles

The T-meshes are local modification of rectangular meshes which allow T-junctions. The splines over T-meshes are involved in many fields, such as finite element methods, CAGD etc. The dimension of a spline space is a basic problem for the theories and applications of splines. However, the problem of determining the dimension of a spline space is difficult since it heavily depends on the geometric properties of the partition. In many cases, the dimension is unstable. In this paper, we study the instability in the dimensions of spline spaces over T-meshes by using the smoothing cofactor-conformality method. The modified dimension formulas of spline spaces over T-meshes with T-cycles are also presented. Moreover, some examples are given to illustrate the instability in the dimensions of the spline spaces over some special meshes.     

Bivariate C1 Cubic Spline Spaces Over Even Stratified Triangulations

It is well-known that the basic properties of a bivariate spline space such as dimension and approximation order depend on the geometric structure of the partition. The dependence of geometric structure results in the fact that the dimension of a C ¹ cubic spline space over an arbitrary triangulation becomes a well-known open problem. In this paper, by employing a new group of smoothness conditions and conformality conditions, we determine the dimension of bivariate C ¹ cubic spline spaces over a so-called even stratified triangulation.     

B-spline bases for unequally smooth quadratic spline spaces on non-uniform criss-cross triangulations

In this paper, we investigate bivariate quadratic spline spaces on non-uniform criss-cross triangulations of a bounded domain with unequal smoothness across inner grid lines. We provide the dimension of the above spaces and we construct their local bases. Moreover, we propose a computational procedure to get such bases. Finally we introduce spline spaces with unequal smoothness also across oblique mesh segments.     

Dimensions of spline spaces over non-rectangular T-meshes

Spline spaces over rectangular T-meshes have been discussed in many papers. In this paper, we consider spline spaces over non-rectangular T-meshes. The dimension formulae of spline spaces over special simply connected T-meshes have been obtained. For T-meshes with holes, we discover a new type of dimension instability. We construct a relationship between the dimension of the spline space over a T-mesh \(\mathcal {T}\) with holes and the dimension of the spline space over a simply connected T-mesh associated with \(\mathcal {T}\).     

拟矩形剖分上的样条空间的维数

矩形剖分~(记为$\Delta_{QR}$)~是指在矩形剖分~(记为$\Delta_{R}$)的基础上进行局部修改后得到的剖分,通常包括T-剖分~(记为$\Delta_{T}$)~和L-剖分~(记为$\Delta_{L}$).本文利用光滑余因子协调方法讨论了该剖分上的二元样条空间$S^\mu_k(\Delta_{QR})$的维数.在满足一定约束条件下, 得到了仅依赖于样条空间的次数,光滑度和剖分拓扑结构的显式维数公式.     

A Lower Bound for the Dimension of Trivariate Spline Spaces

We consider trivariate C^r spline spaces of degree d defined on arbitrary tetrahedral partitions. A lower bound for the dimension of trivariate spline spaces over arbitrary tetrahedral partitions for d > r is computed. This is the first general lower bound known.     

On the dimensions of bivariate spline spaces and the stability of the dimensions

In this paper, the dimensions of bivariate spline spaces are studied using the Smoothing Cofactor-Conformality method. Based on the analysis on the conformality condition at one interior vertex, the stability (or singularity to the contrary) of the dimensions of general spline spaces is discussed in detail. By the aid of directed partition some new results on dimensions are obtained with the corresponding constraints depending on the degree, the smoothness order of the spline spaces and the structure of the partition as well.     

On the dimensions of bivariate spline spaces and the stability of the dimensions

In this paper, the dimensions of bivariate spline spaces are studied using the Smoothing Cofactor-Conformality method. Based on the analysis on the conformality condition at one interior vertex, the stability (or singularity to the contrary) of the dimensions of general spline spaces is discussed in detail. By the aid of directed partition some new results on dimensions are obtained with the corresponding constraints depending on the degree, the smoothness order of the spline spaces and the structure of the partition as well.     

样条空间S~μ_k（△）的维数

本文考虑了欧式空间Ｒ￣ｎ中任意单纯形剖分上的样条函数空间．证明了当ｋ≥（３μ＋１）２￣（ｎ－２）＋１时，计算任意单纯形剖分Δ上的ｋ次μ阶光滑样条空间的维数，可归结为计算每个σ－关联域（ｉ－单纯形σ∈Δ）Ｒ（σ）上的２￣（ｎ－ｉ－１）μ次μ阶光滑（ｉ≤ｎ－１）样条空间的维数。这里σ－关联域Ｒ（σ）是指Δ中所有包含σ的单纯形所成的单纯形剖分．     

Morgan-Scott型剖分上的样条空间的奇异性

Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.     

Upper and Lower Bounds on the Dimension of Superspline Spaces

Upper and lower bounds are provided on the dimension of bivariate polynomial superspline spaces which are defined by enforcing smoothness conditions across the interior edges of the underlying triangulation. The results generalize known bounds for classical spline spaces. As an example of the usefulness of such bounds, we show how they can be applied to analyze a new macroelement.     

B-splines on 3-D tetrahedron partition in four-directional mesh

It is more difficult to construct 3-D splines than in 2-D case. Some results in the three directional meshes of bivariate case have been extended to 3-D case and corresponding tetrahedron partition has been constructed. The support of related B-splines and their recurrent formulas on integration and differentiation-difference are obtained. The results of this paper can be extended into higher dimension spaces, and can be also used in wavelet analysis, because of the relationship between spline and wavelets.     

A normalized representation of super splines of arbitrary degree on Powell–Sabin triangulations

In the paper, a family of bivariate super spline spaces of arbitrary degree defined on a triangulation with Powell–Sabin refinement is introduced. It includes known spaces of arbitrary smoothness r and degree \(3r-1\) but provides also other choices of spline degree for the same r which, in particular, generalize a known space of \(\mathscr {C}^{1}\) cubic super splines. Minimal determining sets of the proposed super spline spaces of arbitrary degree are presented, and the interpolation problems that uniquely specify their elements are provided. Furthermore, a normalized representation of the discussed splines is considered. It is based on the definition of basis functions that have local supports, are nonnegative, and form a partition of unity. The basis functions share numerous similarities with classical univariate B-splines.     

Multi-Box Splines

We construct local generators, comprising r functions, for refinable spaces of bivariate C^n-1 spline functions of degree n on meshes comprising all lines through points of the integer lattice in the directions of n + r + 1 pairwise linearly independent vectors with integer components. The generators are characterised by their Fourier transforms. Their shifts are shown to form a Riesz basis if and only if at most r lines in the mesh intersect other than in the integer lattice, which can occur for n ≤ 2r - 1. The symmetry of these generators is studied and examples are given.     

On explicit basis of bivariate spline space

For a subdivision Δ of a region in d-dimensional Euclidean space, we consider computation of dimension and of basis function in spline space S _k ^r (Δ) consisting of all C piecewise polynomial functions over Δ of degree at most k. A computational scheme is presented for computing the dimension and bases of spline space S _k ^r (Δ). This scheme based on the Grobner basis algorithm and the smooth co-factor method for computing multivariate spline. For bivariate splines, explicit basis functions of S _k ^r (Δ) are obtained for any integer k and r when Δ is a cross-cut partition. The Project is partly supported by the Science and Technology New Star Plan of Beijing and Education Committee of Beijing.     

研究二元样条空间维数的Blossoming方法

本文综述了研究二元样条的Ｂｌｏｓｓｏｍｉｎｇ方法．成功地重建了平面上贯穿剖分的维数公式．而且利用这种方法，对定义在Ｍｏｒｇａｎ－Ｓｃｏｔ剖分上样条空间的维数取得了一些新的结果．     

Bivariate spline interpolation at grid points

Summary. We describe algorithms for constructing point sets at which interpolation by spaces of bivariate splines of arbitrary degree and smoothness is possible. The splines are defined on rectangular partitions adding one or two diagonals to each rectangle. The interpolation sets are selected in such a way that the grid points of the partition are contained in these sets, and no large linear systems have to be solved. Our method is to generate a net of line segments and to choose point sets in these segments which satisfy the Schoenberg-Whitney condition for certain univariate spline spaces such that a principle of degree reduction can be applied. In order to include the grid points in the interpolation sets, we give a sufficient Schoenberg-Whitney type condition for interpolation by bivariate splines supported in certain cones. This approach is completely different from the known interpolation methods for bivariate splines of degree at most three. Our method is illustrated by some numerical examples. Received October 5, 1992 / Revised version received May 13, 1994     

Nöther-type theorem of piecewise algebraic curves on quasi-cross-cut partition

Nöther’s theorem of algebraic curves plays an important role in classical algebraic geometry. As the zero set of a bivariate spline, the piecewise algebraic curve is a generalization of the classical algebraic curve. Nöther-type theorem of piecewise algebraic curves is very important to construct the Lagrange interpolation sets for bivariate spline spaces. In this paper, using the characteristics of quasi-cross-cut partition, properties of bivariate splines and results in algebraic geometry, the Nöther-type theorem of piecewise algebraic curves on the quasi-cross-cut is presented.     

Bivariate Polynomial Natural Spline Interpolation Algorithms with Local Basis for Scattered Data

Because of its importance in both theory and applications, multivariate splines have attracted special attention in many fields. Based on the theory of spline functions in Hilbert spaces, bivariate polynomial natural splines for interpolating, smoothing or generalized interpolating of scattered data over an arbitrary domain are constructed with one-sided functions. However, this method is not well suited for large scale numerical applications. In this paper, a new locally supported basis for the bivariate polynomial natural spline space is constructed. Some properties of this basis are also discussed. Methods to order scattered data are shown and algorithms for bivariate polynomial natural spline interpolating are constructed. The interpolating coefficient matrix is sparse, and thus, the algorithms can be easily implemented in a computer.     

二元样条的积分表示及分层三角剖分下二次样条空间的维数

本文通过引入一个积分协调条件，首次给出了二元样条的一个积分表示．文中还定义了平面单连通多边形区域的所谓分层三角剖分，并确定了此剖分下二次样条空间的维数．