某些样条空间奇异性和插值适定性问题研究
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摘要
众所周知，样条作为计算几何中表示和逼近几何对象的基本工具，在很多工程领域有着重要而广泛的应用，多项式函数的某些特例早已出现于一些数学研究工作中。鉴于客观事物的复杂多样性，开展多元样条函数的研究，无论是理论上还是应用上都有着重要意义，虽然多元样条函数与一元样条函数有着一定的联系，但它绝不是一元样条函数的简单推广，两者之间存在着本质的差别，所以有关它的研究成果不像一元样条那样完美，有些问题还值得进一步研究。本文的主要工作如下：
在第二章中，定义了类似于△_(MS)剖分的△_(MS)~μ剖分，利用罗钟铉教授提出的模中生成基方法得到了对于任意的μ，S_(μ+1)~μ(△_(MS)~μ)空间奇异的代数型条件，王仁宏教授在1975年将样条函数的结构等价地转化为相应的代数问题，2001年罗钟铉教授通过定义模中的约化准则给出了求解模中生成基的机械化方法。由于该方法获得的一个内网点处的协调方程的生成基在一般情况下由若干个一次和零次的模中多项式向量所构成，因此对于研究多元样条函数空间带来很好的便利条件，从杜宏的文中我们不难发现在△_(MS)~μ剖分样条空间奇异性和代数曲线的内蕴性质之间存在着一定的等价关系，我们还给出了当μ=2时，S_(μ+1)~μ(△_(MS)~μ)样条空间奇异时等价的几何性条件，为了更好的说明我们的结论，一些具体的例子在该章中也给出。这些结果对于以后研究代数曲线的分类和参数化将有极大的帮助。
第三章，利用多元样条对散乱数据插值是多元样条一个重要的应用领域，二元样条空间在数值逼近、曲面拟合、散乱数据插值、多元数值积分、有限元方法、偏微分方程数值解、计算机辅助几何设计和计算机图形学等方面有着广阔的应用，显然，要了解多元样条空间并将其应用于实际，最首要的问题是弄清它的代数结构。对于次数d相对光滑度r较大的情形，已经有了许多的结论，如d≥3r+2的情形。但实际应用中，由于低次样条计算简单和稳定，人们对低次样条空间更感兴趣。例如r=1时，d=2，3，4的情况。而S_3~1(△)的情形则至今悬而未决，人们既不能给出其维数也不知道其维数是否依赖于剖分的几何形状，确立任何三角剖分下样条函数空间S_3~1(△)的维数遇到了难以想象的困难，成为多元样条函数研究领域的一个公开问题。但空间S_3~1(△)却是一个特别重要的空间，之所以特别重要，除了二元三次样条函数的计算简单和稳定的优点外，还在于它是维数(尽管我们目前还不能确切说明)超过其三角剖分顶点数的所有样条函数空间次数最低的。换句话说，空间S_3~1(△)是可以在其三角剖分的所有顶点上考虑插值的次数最低的二元C~1样条函数空间。
由于确立任意三角剖分空间S_3~1(△)维数具有难以想象的困难，所以可以先考虑某些特殊的剖分。显然，寻找一般的三角剖分，并给出其相应的维数，具有十分重要的意义，在本章中，讨论了满足一定的条件的一类三角剖分，研究了在其上的S_3~1(△)空间，我们先对该三角剖分进行分解，然后递归地在该三角剖分上建立了S_3~1(△)空间的容许集和Lagrange插值集合，从而明确地确定了S_3~1(△)的维数，因此能够确定这类三角剖分的非奇异性。在本章的最后，还给出了一种在平面散乱点集上构建三角剖分的方法，使得生成的三角剖分正好在我们所考虑的这类三角剖分内。
第四章，众所周知，二元多项式空间P_d的自由度个数是(?)，那么，分片连续的多项式—二元样条空间的维数是多少?这一问题对于研究样条的插值适定性等许多其它的问题都具有重要的意义，与多项式空间维数相比，样条空间的维数研究异常困难，至今仍有许多与之相关的公开问题，三角剖分是实际中较常用的剖分，三角剖分下二元样条函数维数的问题也最令人关注，二元样条空间的维数研究问题，最早始于strang给出的关于维数的猜想，最有代表性的是L.L.Schumaker给出关于一般三角剖分下二元样条函数维数的下界和上界，对于一个一般的三角剖分，很难给出一个通用的维数公式，因为样条空间的维数不仅依赖于三角剖分的拓扑性质，即剖分的顶点数，边数和三角形的个数，而且很强烈地依赖于剖分的几何性质，在本章中，我们利用对三角剖分的顶点进行编号和光滑余因子方法，对一般的三角剖分的上界进行了重新估计，改进了以前的结论。特别是对含有奇异网点和贯穿线较多的三角剖分上的样条空间效果更明显，并且给出一些比较方便地判断样条空间维数的推论。
第五章，T-网格，从本质上来说就是一个容许T结点的矩形网格，T样条，是定义在T网格上的PB样条(Point-Based Spline)，邓建松等人在Sederberg等人引入的T样条的基础上，限制样条在T网格的每个剖腔上是一个张量积多项式并且内网线处满足一定的光滑性，提出了T网格上样条函数空间的概念，利用B网方法，他们得到当光滑阶小于多项式一半时规则T网格上样条空间维数公式。在本章中我们利用对内线的协调方程重新编序的技巧，使得协调方程组所对应的系数矩阵正好为一个准上三角矩阵，从而能够方便地给出任意规则T网格样条空间的维数公式，并且这个公式对一般T网格样条空间，诸如：T网格上的周期样条空间、组合T网格上样条空间和有洞T网格上样条空间的维数同样适用，因而我们的结论更具一般性。
It is well known that spline is an important approximation tool in computationalgeometry, and it is widely used in many engineering fields. Some special examples ofpolynomial functions appeared in mathematics researches already. For the complexityof object, the research of the multivariate spline has very important significance in thetheoretics and applications. Multivariate splines have connection with univariate splinesin some sense, but they are different in essence, it is not the simple extension of theunivariate splines. Hence, the research results of multivariate splines are not perfect asthose of univariate splines, it still need to be further studied. The thesis is organized asfollows:
In Chapter 2, we define△_(MS)~μpartition is similar to△_(MS) partition. For anyμ, the algebraic singularity condition of S_(μ+1)~μ(△_(MS)~μ) is obtained by the Generator bases inmodules method presented by Professor Z. X. Luo. Wang transformed the structureof spline to corresponding algebraic problem in 1975. In 2001 a mechanical method isproposed to solve the generator bases in modules through the reduced rules by ProfessorZ. X. Luo. It becomes very convenient to study the multivariate spline spaces becausethe generator basis of the conformality equations over an inner vertex are composed ofsome vectors of degree 1 and 0 in general, which are obtained from this method. FromDu's paper, it is easy to find some equivalent relations between the singularity of splinespace over△_(MS)~μpartition and the intrinsic properties of algebraic curves. Whenμ=2, the geometric singularity condition of S_(μ+1)~μ(△_(MS)~μ) is proposed. Also, in order to illustrateour conclusions clearly, some examples are given out. All these results are very helpfulfor the further research on the classification and parametrization of algebraic curves andso on.
In Chapter 3, using multivariate splines to interpolate scattered points is an important applied field. Bivariate spline spaces have many applications, such as numericalapproximation, surface fitting, scattered points interpolation, multivariate numerical integration, finite element method, numerical solution of partial differential equation, CAGDand computer graphic etc.. Obviously, in order to better understand rnultivariate spline space and to apply it, the first problem is to understand its algebraic structure. We alreadyget perfect results for the case of degree d is much larger corresponding to smoothness r, such as d＞3r+2. However, splines with lower degree are more interested for people inapplications. For example, when r=1, the case of d=2, 3, 4 respectively. But the caseof S_3~1(△) is still unknown to us, people can not give its dimension and do not know itsdimension whether depends on the geometric condition of the partition. Hence, it is anopen problem to determine the dimension of S_3~1(△) spline space. But S_3~1(△) spline spaceis very important to us. Since its simple computation and stability, S_3~1(△) is the lowestdegree space in those spaces satisfying the dimensions excess the number of the verticesof triangulations until now. In other words, S_3~1(△) space is the lowest degree C~1 spacewhich can interpolate all the vertices in triangulations.
For the difficulty of determining the dimension of S_3~1(△), people turn to considersome special triangulations. Obviously, it is more useful to find the dimension for general triangulation. In this chapter, we discuss a class of triangulations satisfying someconditions, and study S_3~1(△) space on it. First we decompose those triangulations, thenconstruct admissible sets and Lagrange sets recursively, hence their dimensions are determined clearly. Thus, we find these triangulations are nonsingular. In the last, we give amethod to construct triangulations on planar scattered points, such that the generatedtriangulations are exactly in our considered class of triangulations.
In Chapter 4, the dimension of bivariate polynomial space P_d is (?) as known.Then what is the dimension of piecewise continuous polynomial space? This question hasimportant significance in studying posed interpolation of spline and so on. Comparingto multivariate polynomial spaces, the dimension of spline space is more difficult. Upto now, many related questions are still open. Triangulation is the common partition inpractical, so its dimension is followed more concern. The study of dimension of bivariatespline space is originated from the guess of dimension of spline space of Strang. Itsrepresentative is L.L. Schumaker who gave the upper bound and the lower bound ofdimension of arbitrary triangulation. It is diffcult to give a general dimension formulafor any triangulation. Because the dimension of spline space not only depends on thetopology property of triangulation, such as the number of vertices, edges and trianglesof triangulation, but also heavily depends on geometric property of the triangulation.In this chapter, we improved the upper bound of dimension for general triangulationby numbering vertices and Smoothing Cofactor method. Moreover, some corollaries areproposed for deciding the dimension of spline space.
In Chapter 5, T-mesh is a rectangle mesh with T nodes in essence. T-spline is point based spline defined on T-mesh. Deng etc. presented the definition of T-mesh splinespace, on base of T-spline introduced by Sederberg etc.. T-spline is a tensor productpolynomial in every cell of T-mesh and satisfying smoothness in every interior edge. UsingB-net method, they derived the dimension of T-mesh confined that the smoothness is lessthan half of degree of the spline functions. In this chapter, we use Smoothing Cofactormethod and reordering skill to conformality equations of inner lines, then derive thedimension formula in regular T-mesh spline space. Moreover, the formula also holds forgeneral T-mesh spline space, such as periodic spline space on T-mesh, composite T-meshspline space and T-mesh with holes spline space. Thus our result is more general.
在第二章中，定义了类似于△_(MS)剖分的△_(MS)~μ剖分，利用罗钟铉教授提出的模中生成基方法得到了对于任意的μ，S_(μ+1)~μ(△_(MS)~μ)空间奇异的代数型条件，王仁宏教授在1975年将样条函数的结构等价地转化为相应的代数问题，2001年罗钟铉教授通过定义模中的约化准则给出了求解模中生成基的机械化方法。由于该方法获得的一个内网点处的协调方程的生成基在一般情况下由若干个一次和零次的模中多项式向量所构成，因此对于研究多元样条函数空间带来很好的便利条件，从杜宏的文中我们不难发现在△_(MS)~μ剖分样条空间奇异性和代数曲线的内蕴性质之间存在着一定的等价关系，我们还给出了当μ=2时，S_(μ+1)~μ(△_(MS)~μ)样条空间奇异时等价的几何性条件，为了更好的说明我们的结论，一些具体的例子在该章中也给出。这些结果对于以后研究代数曲线的分类和参数化将有极大的帮助。
第三章，利用多元样条对散乱数据插值是多元样条一个重要的应用领域，二元样条空间在数值逼近、曲面拟合、散乱数据插值、多元数值积分、有限元方法、偏微分方程数值解、计算机辅助几何设计和计算机图形学等方面有着广阔的应用，显然，要了解多元样条空间并将其应用于实际，最首要的问题是弄清它的代数结构。对于次数d相对光滑度r较大的情形，已经有了许多的结论，如d≥3r+2的情形。但实际应用中，由于低次样条计算简单和稳定，人们对低次样条空间更感兴趣。例如r=1时，d=2，3，4的情况。而S_3~1(△)的情形则至今悬而未决，人们既不能给出其维数也不知道其维数是否依赖于剖分的几何形状，确立任何三角剖分下样条函数空间S_3~1(△)的维数遇到了难以想象的困难，成为多元样条函数研究领域的一个公开问题。但空间S_3~1(△)却是一个特别重要的空间，之所以特别重要，除了二元三次样条函数的计算简单和稳定的优点外，还在于它是维数(尽管我们目前还不能确切说明)超过其三角剖分顶点数的所有样条函数空间次数最低的。换句话说，空间S_3~1(△)是可以在其三角剖分的所有顶点上考虑插值的次数最低的二元C~1样条函数空间。
由于确立任意三角剖分空间S_3~1(△)维数具有难以想象的困难，所以可以先考虑某些特殊的剖分。显然，寻找一般的三角剖分，并给出其相应的维数，具有十分重要的意义，在本章中，讨论了满足一定的条件的一类三角剖分，研究了在其上的S_3~1(△)空间，我们先对该三角剖分进行分解，然后递归地在该三角剖分上建立了S_3~1(△)空间的容许集和Lagrange插值集合，从而明确地确定了S_3~1(△)的维数，因此能够确定这类三角剖分的非奇异性。在本章的最后，还给出了一种在平面散乱点集上构建三角剖分的方法，使得生成的三角剖分正好在我们所考虑的这类三角剖分内。
第四章，众所周知，二元多项式空间P_d的自由度个数是(?)，那么，分片连续的多项式—二元样条空间的维数是多少?这一问题对于研究样条的插值适定性等许多其它的问题都具有重要的意义，与多项式空间维数相比，样条空间的维数研究异常困难，至今仍有许多与之相关的公开问题，三角剖分是实际中较常用的剖分，三角剖分下二元样条函数维数的问题也最令人关注，二元样条空间的维数研究问题，最早始于strang给出的关于维数的猜想，最有代表性的是L.L.Schumaker给出关于一般三角剖分下二元样条函数维数的下界和上界，对于一个一般的三角剖分，很难给出一个通用的维数公式，因为样条空间的维数不仅依赖于三角剖分的拓扑性质，即剖分的顶点数，边数和三角形的个数，而且很强烈地依赖于剖分的几何性质，在本章中，我们利用对三角剖分的顶点进行编号和光滑余因子方法，对一般的三角剖分的上界进行了重新估计，改进了以前的结论。特别是对含有奇异网点和贯穿线较多的三角剖分上的样条空间效果更明显，并且给出一些比较方便地判断样条空间维数的推论。
第五章，T-网格，从本质上来说就是一个容许T结点的矩形网格，T样条，是定义在T网格上的PB样条(Point-Based Spline)，邓建松等人在Sederberg等人引入的T样条的基础上，限制样条在T网格的每个剖腔上是一个张量积多项式并且内网线处满足一定的光滑性，提出了T网格上样条函数空间的概念，利用B网方法，他们得到当光滑阶小于多项式一半时规则T网格上样条空间维数公式。在本章中我们利用对内线的协调方程重新编序的技巧，使得协调方程组所对应的系数矩阵正好为一个准上三角矩阵，从而能够方便地给出任意规则T网格样条空间的维数公式，并且这个公式对一般T网格样条空间，诸如：T网格上的周期样条空间、组合T网格上样条空间和有洞T网格上样条空间的维数同样适用，因而我们的结论更具一般性。
It is well known that spline is an important approximation tool in computationalgeometry, and it is widely used in many engineering fields. Some special examples ofpolynomial functions appeared in mathematics researches already. For the complexityof object, the research of the multivariate spline has very important significance in thetheoretics and applications. Multivariate splines have connection with univariate splinesin some sense, but they are different in essence, it is not the simple extension of theunivariate splines. Hence, the research results of multivariate splines are not perfect asthose of univariate splines, it still need to be further studied. The thesis is organized asfollows:
In Chapter 2, we define△_(MS)~μpartition is similar to△_(MS) partition. For anyμ, the algebraic singularity condition of S_(μ+1)~μ(△_(MS)~μ) is obtained by the Generator bases inmodules method presented by Professor Z. X. Luo. Wang transformed the structureof spline to corresponding algebraic problem in 1975. In 2001 a mechanical method isproposed to solve the generator bases in modules through the reduced rules by ProfessorZ. X. Luo. It becomes very convenient to study the multivariate spline spaces becausethe generator basis of the conformality equations over an inner vertex are composed ofsome vectors of degree 1 and 0 in general, which are obtained from this method. FromDu's paper, it is easy to find some equivalent relations between the singularity of splinespace over△_(MS)~μpartition and the intrinsic properties of algebraic curves. Whenμ=2, the geometric singularity condition of S_(μ+1)~μ(△_(MS)~μ) is proposed. Also, in order to illustrateour conclusions clearly, some examples are given out. All these results are very helpfulfor the further research on the classification and parametrization of algebraic curves andso on.
In Chapter 3, using multivariate splines to interpolate scattered points is an important applied field. Bivariate spline spaces have many applications, such as numericalapproximation, surface fitting, scattered points interpolation, multivariate numerical integration, finite element method, numerical solution of partial differential equation, CAGDand computer graphic etc.. Obviously, in order to better understand rnultivariate spline space and to apply it, the first problem is to understand its algebraic structure. We alreadyget perfect results for the case of degree d is much larger corresponding to smoothness r, such as d＞3r+2. However, splines with lower degree are more interested for people inapplications. For example, when r=1, the case of d=2, 3, 4 respectively. But the caseof S_3~1(△) is still unknown to us, people can not give its dimension and do not know itsdimension whether depends on the geometric condition of the partition. Hence, it is anopen problem to determine the dimension of S_3~1(△) spline space. But S_3~1(△) spline spaceis very important to us. Since its simple computation and stability, S_3~1(△) is the lowestdegree space in those spaces satisfying the dimensions excess the number of the verticesof triangulations until now. In other words, S_3~1(△) space is the lowest degree C~1 spacewhich can interpolate all the vertices in triangulations.
For the difficulty of determining the dimension of S_3~1(△), people turn to considersome special triangulations. Obviously, it is more useful to find the dimension for general triangulation. In this chapter, we discuss a class of triangulations satisfying someconditions, and study S_3~1(△) space on it. First we decompose those triangulations, thenconstruct admissible sets and Lagrange sets recursively, hence their dimensions are determined clearly. Thus, we find these triangulations are nonsingular. In the last, we give amethod to construct triangulations on planar scattered points, such that the generatedtriangulations are exactly in our considered class of triangulations.
In Chapter 4, the dimension of bivariate polynomial space P_d is (?) as known.Then what is the dimension of piecewise continuous polynomial space? This question hasimportant significance in studying posed interpolation of spline and so on. Comparingto multivariate polynomial spaces, the dimension of spline space is more difficult. Upto now, many related questions are still open. Triangulation is the common partition inpractical, so its dimension is followed more concern. The study of dimension of bivariatespline space is originated from the guess of dimension of spline space of Strang. Itsrepresentative is L.L. Schumaker who gave the upper bound and the lower bound ofdimension of arbitrary triangulation. It is diffcult to give a general dimension formulafor any triangulation. Because the dimension of spline space not only depends on thetopology property of triangulation, such as the number of vertices, edges and trianglesof triangulation, but also heavily depends on geometric property of the triangulation.In this chapter, we improved the upper bound of dimension for general triangulationby numbering vertices and Smoothing Cofactor method. Moreover, some corollaries areproposed for deciding the dimension of spline space.
In Chapter 5, T-mesh is a rectangle mesh with T nodes in essence. T-spline is point based spline defined on T-mesh. Deng etc. presented the definition of T-mesh splinespace, on base of T-spline introduced by Sederberg etc.. T-spline is a tensor productpolynomial in every cell of T-mesh and satisfying smoothness in every interior edge. UsingB-net method, they derived the dimension of T-mesh confined that the smoothness is lessthan half of degree of the spline functions. In this chapter, we use Smoothing Cofactormethod and reordering skill to conformality equations of inner lines, then derive thedimension formula in regular T-mesh spline space. Moreover, the formula also holds forgeneral T-mesh spline space, such as periodic spline space on T-mesh, composite T-meshspline space and T-mesh with holes spline space. Thus our result is more general.
引文
[1] 王仁宏．多元齿的结构与插值．数学学报，1975，18(2)：91-106．
[2] 罗钟铉．K[x]~m中模的生成基及其应用．数学学报，2001，44(6)：983-994．
[3] Du H. A geometric appraoch to dimS_2~1(△_(MS)). AMS/IP Studies in Advanced Mathematics, vol. 34. 2003:67-70.
[4] Strang G. Piecewise polynomials and the finite element method. Bull. Amer. Math. Soc., 1973, 79:1128-1137.
[5] Schumaker L L. Bounds on the dimension of spaces of multivariate piecewise polynomials. Rockey Mountain J. Math., 1984, 14: 251-264.
[6] Deng J S, Chen F L, Feng Y Y. Dimensions of spline spaces over T-meshes. Jour. Comp. Appl. Math., 2006, 194:267-283.
[7] Sederberg T W, Zheng J M, Sewell D. Non-uniform recursive subdivision surfaces. Computer Graphics, 1994: 387-394.
[8] Schoenberg I J. Contribution to the problem of application of equidistant data by analytic functions. Quart. Appl. Math., 1946, 4:45-99,112-141.
[9] 王仁宏，施锡泉，罗钟铉等．多元样条函数及其应用．北京：科学出版社，1994．
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[18] Shi X Q. Parameter-introducing skill in study of splines.大连理工大学学报, 1994, 34(2): 208-212.
[19] 李崇君．特殊三角剖分上的多元样条及其应用：博士学位论文．大连：大连理工大学．2004．
[20] Luo Z X, Wang R H. Structure and dimension of multivariate spline space of lower degree on arbitrary triangulation. J. Comp. and Appl. Math., 2006, 195(1-2):113-133.
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[27] Schenck H, Stillman M. Local cohomology of bivariate splines. Journal of Pure and Applied Algebra, 1997, 117&118:535-548.
[28] David C, Little J, Shea D. Ideals,Varieties and Algorithms. Springer-Verlag, 1992.
[29] Chen X, Chen Z B, Yan Y H. A survey on the dimension of bivariate spline space from the blossom approach. Journal of Information and Computational Sciences, 2004, 1(2): 181-186.
[30] Chen Z B, Feng Y Y, Jernej K. The blossom approach to the dimension of the bivariate spline space. Journal of Computational Mathematics, 2000, 18(2):183-198.
[31] 陈之兵，奚梅成，陈效群等．研究二元样条空间维数的Blossoming方法．高等应用数学学报，1998，13(13)：64-78．
[32] Zhan Y W. On dissecting triangulations under bivariate spline spaces.应用数学,1994, 7(1):112-118.
[33] 王仁宏．任意剖分下的多元样条分析(Ⅱ)—空间形式．高等学校计算数学学报，1980，1：78-81．
[34] Chou Y S, Su L Y, Wang R H. The dimension of bivariate spline spaces over triangulations. Tntl. Ser. Numer. Math., 1985, 75: 71-83.
[35] Chui C K, Wang R H. Bases of bivariate spline spaces with cross-cut grid partitions.数学研究与评论,1982,1:1-3.
[36] Chui C K, Wang R H. On smooth multivariate spline functions. Mathematics of Computation, 1983, 41 (163): 131-142.
[37] Chui C K, Wang R H. Multivariate spline spaces. Jour. Math. Anal. Appl., 1983, 94: 197-221.
[38] Chui C K, Wang R H. Spaces of bivariate cubic and quartic splines on Type-1 triangulations. Jour. Math. Anal. Appl., 1984, 101:540-554.
[39] 周蕴时，苏志勋，奚勇江等．CAGD中的曲线曲面．吉林大学出版社，1993．
[40] Chui C K, Du H, Jia R Q. Stability of optimal-order approximation by bivariate splines over arbitrary triangulations. Transactions of the American Mathematical society, 1995, 347(9):3301-3318.
[41] Chui C K, Lai M L. Topics in Multivariate Approximation. New York: Academic press, 1987, pp. 19-36.
[42] Cde Boor K Hollig, Riemenschneider S. Box Splines. New York: Springer-Verlag, 1993.
[43] Alfeld P, Schumaker L L. The dimension of bivariate spline spaces of smoothness r for degree d≥4r+1. Constr. Approx., 1987, 3: 189-197.
[44] Alfeld P, Schumaker L L. On the dimension of bivariate spline spaces of smootheness r and degree d=3r+1. Numer. Math., 1990, 57: 651-661.
[45] Schumaker L L. Dual bases for spline spaces on cells. CAGD, 1988, 5: 277-284.
[46] Alfeld P. Bivariate spline spaces and minimal determining sets. J. Comput. Appl. Math., 2000, 119: 13-27.
[47] Alfeld P, Piper B, Schumaker L L. Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree d≥4r+1. Comput. Aided Geom. Des., 1987, 4:105-123.
[48] Morgan J, Scott R. The dimension of the space of C~1 piecewise polynomials, 1975. Unpublished manuscript.
[49] Shi X Q. The singularity of Morgan-scott triangulation. Comput. Aided Geom. Des., 1991, 8: 201-206.
[50] Diener D. Instability in the dimension of spaces of bivariate piecewise polynomials of degree 2r and smoothness order r. SIAM J. Numer. Anal., 1990, 27(2):543-551.
[51] 许志强．多元样条、分片代数曲线及线性丢番图方程组：博士学位论文．大连：大连理工大学．2003．
[52] Xu Z Q, Wang R H. The instability degree in the dimension of spaces of bivariate spline. Approx. Theory and its Appl., 2002, 18(1):68-80.
[53] Deng J S, Feng Y Y, Jernej K. A note on the dimension of the bivariate spline space over the Morgan-Scott triangulation. SIAM J. Numer Anal., 2001, 37(3):1021-1028.
[54] 朱德祥．高等几何．高等教育出版社，1983．
[55] Luo Z X, Chen L J. The singularity of S_(μ+1)~μ(△_(MS)~μ). Journal of Information and Computational Science, 2005, 2(4):739-746.
[56] Luo Z X, Chen L J. The singularity of S_(μ+1)~μ(△_(MS)~μ). Advances in Information and Computational Science. 2005:154-158.
[57] Luo Z X, Chen L J, Liu Y. Intrinsic properties and invariant of algebraic curves in the projective plane. Submitted to Computational Geometry.
[58] 尹宝才．计算多元样条函数的Grobner基方法：博士学位论文．大连：大连理工大学，1993，
[59] Winkler F. Recursive method for computing a Grobner basis of a module in K[x_1,…, x_m]~m. A. AE. C. C-5,. Menorca,Spain, 1987.
[60] 王媛媛．样条函数的数学机械化计算：硕士学位论文．大连：大连理工大学，2003．
[61] 王丹．多元样条函数空间的奇异性条件及插值适定性：硕士学位论文：大连理工大学，2004．
[62] Luo Z X, Deng Y, Meng Z L. Some results on Lagrange interpolation set over Ⅰ or Ⅱ- triangulations. Journal of Information and Computational Science, 2004, 1(1): 111-116.
[63] 邓勇．某些三角剖分上样条函数空间的奇异性及插值适定性：硕士学位论文：大连理工大学，2005．
[64] 周兴和．高等几何．北京：科学出版社、2003．
[65] 沈纯理，陈咸平，黄荣培等．经典几何．科学出版社，2004．
[66] 梅向明，刘增贤，林向岩．高等几何．高等教育出版社，1983．
[67] 陈之兵，邓建松，奚梅成等．Morgan-Scott剖分上样条空间S_6~3(△_(ms))的维数．中国科技技术大学学报，1998，28(5)：505-512．
[68] 邓建松，奚梅成，冯玉瑜．Morgan-Scott剖分上样条空间S_8~4(△_(ms))的维数．高等应用数学学报，1998，13(13)：91-98．
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