近似隐式化和分片代数簇某些问题的研究
|
摘要
开展隐式曲线曲面的研究在计算机辅助几何设计和几何造型中既有深刻的理论价值又有广泛的应用前景。本文主要针对参数曲线曲面的近似隐式化和分片代数簇的某些问题展开研究。主要工作如下：
第二章我们讨论了参数曲线曲面的近似隐式化。参数曲线曲面和隐式曲线曲面是计算机辅助几何设计和几何造型中两种常见的表示形式。将参数形式的曲线曲面转化为代数形式的曲线曲面的过程称为隐式化。然而，精确隐式化(尤其是曲面的精确隐式化)在几何造型中没有得到广泛的应用。这主要是参数曲线曲面的精确隐式化过程很复杂而且不一定可以实现，再加上隐式曲线曲面的阶数很高并且具有不希望的奇异点和多余分支，从而会引起计算的不稳定性和几何造型中拓扑结构的不一致性，这就大大限制了精确隐式化在实际中的运用。
为了解决上述问题，我们提出了如下三种近似隐式化算法：首先，我们利用二阶二次代数样条曲线对参数曲线进行近似隐式化，所得到的代数样条曲线不会产生多余分支和自交点，并且具有良好的误差估计和逼近性质。其次，我们利用Multiquadric拟插值和径向基神经网络，提出了参数曲线近似隐式化的另一种方法。该方法具有保形性好，光滑度高，逼近性能好，样本节点数据少等优点。最后，考虑到很难将上述两种方法直接推广到参数曲面的近似隐式化，我们利用紧支集径向基函数作多元散乱数据插值的技巧提出了参数曲面近似隐式化的一种算法。
第三章我们讨论了分片代数簇的某些问题。分片代数簇作为多元样条的公共零点集合，是经典代数簇的推广，它不仅和许多实际问题如多元样条插值，CAD和CAGD等有关，而且还为研究经典代数几何提供理论依据。因此，研究分片代数簇是很重要的。首先，讨论了分片代数簇的维数性质，通过引入分片代数簇完全相交的概念讨论了分片代数簇的维数与其定义方程组个数的关系．其次，简要讨论了分片代数曲线的奇点性质。再次，为了有效计算分片代数簇，我们讨论了凸多面体内任意维代数簇的计算问题。通过添加超平面技巧将Groebner基方法应用到凸多面体内任意维代数簇的计算上，从而把凸多面体内的代数簇转化为另外一组多项式方程组的正解，并且得到了该代数簇在凸多面体内的极小分解．最后，基于B-样条系数的Descartes符号准则和Bézier曲线的de Casteljau算法，我们给出了一元样条实根分离算法，也就是计算出一列不相交区间，使得每个区间恰好只包含此样条函数的一个实根。
第四章我们主要构造了一类具有紧支集的无穷次可微径向函数。众所周知，Gauss分布函数是一类广泛应用于多元插值和径向基网络的正定径向函数。它具有非常好的逼近效果和指数衰减性质。对Gauss分布函数离中心远处截断，可以马上得到紧支集径向基函数。然而，这样得到的紧支集径向基函数用于多元插值和函数逼近显然是不连续的。因此，结合Gauss函数的特点对其进行改进，我们构造了一类具有可控自由参数的紧支集无限次可微函数。在对自由参数一定的约束条件下，此类函数能够有效地应用到多元函数逼近和多元散乱数据插值中。
Implicit surfaces play an important role in many theoretic fields and applied fieldsin Computer Aided Geometry Design and Geometry Modeling. In this thesis, we mainlystudy some problems on approximate implicitization and piecewise algebraic varieties.Our primary work is organized as follows:
In chapter 2, we discuss the problem of approximate implicitization of parametriccurves/surfaces. It is well known that parametric curves/surfaces and implicit curves/-surfaces are two important topics in Computer Aided Geometry Design and GeometricModeling. The procedure of transform the parametric form into algebraic form is calledimplicitization. However, accurate implicitization (especially surface implicitization) hasnot been popular in practice. This is due to the fact that exact implicitization of para-metric curves/surfaces always involves complicated computation and its exact implicitform usually cannot be computed. Moreover, the degree of implicit curves and surfacesis higher and implicit curves and surfaces may have singular points and unexpectedcomponents, which lead to computational instability and topological inconsistency ingeometric modeling.
In order to solve this problem, we propose the following three algorithms to deal withapproximate implicitization. Firstly, we use quadratic algebraic spline with smoothnesstwo to tackle approximate implicitization of parametric curves. The resulting approxi-mate curves not only don't have unwanted components and self-intersections, but alsohave good error estimate and approximation behavior. Secondly, we propose a new ap-proach to solve approximate implicitization of parametric curves based on radial basisfunction networks and multiquadric quasi-interpolation. This approach possesses theadvantages of shape preserving, better smoothness, good approximation behavior andrelatively less data etc. Lastly, we propose a method to solve approximate implicitizationof parametric surfaces based on multivariate interpolation by using compactly supportedradial basis functions.
In chapter 3, we discuss several problems on piecewise algebraic varieties. As thezeros of multivariate splines, the piecewise algebraic variety is a generalization of theclassical algebraic variety. It is important to study the interpolation by multivariatesplines and algebraic geometry etc. Firstly, we discuss the relationship between thedimension of piecewise algebraic varieties and the number of their defining equations, aswell as several dimension properties by introducing the concept of complete intersectionof piecewise algebraic varieties. Secondly, several singular point properties of piecewisealgebraic curves are discussed. Thirdly, the intersection problem of piecewise curves andsurfaces boils down to the computation of piecewise algebraic varieties. In order to solvepiecewise algebraic varieties, we propose a new method to compute an algebraic varietyon a convex polyhedron by adding hyperplanes with the method of Groebner bases. Thus,the algebraic variety on the convex polyhedron is transformed to the positive solutionsof a system of polynomials. Besides, the minimal decomposition is also obtained. Lastly,we present an algorithm to isolate real roots of a univariate spline, i.e., computing asequence of disjoint intervals such that each of them contains exactly one real root of agiven spline function, which is primarily based on the use of Descartes' rule of signs withits B-spline coefficients and de Casteljau algorithm of B(?)zier curve.
In chapter 4, a kind of multivariate compactly supported infinitely differentiablefunctions is constructed. It is well known that the Gauss distribution function is awidespread used positive definite radial basis function in multivariate scattered datainterpolation. Moreover, it possesses good global approximation behavior and decaysexponentially. It is clear that we can generate the radial functions with compact supportby cutting off Gauss function at large distances from the centers. However the resultingradial function interpolation (approximation) is obviously discontinuous. Thus, we con-struct a kind of compactly supported radial functions which have infinite differentiableproperty for any space dimension with two free parameters. Under certain conditions onparameters, they can be applied to function approximation and scattered data interpo-lation.
第二章我们讨论了参数曲线曲面的近似隐式化。参数曲线曲面和隐式曲线曲面是计算机辅助几何设计和几何造型中两种常见的表示形式。将参数形式的曲线曲面转化为代数形式的曲线曲面的过程称为隐式化。然而，精确隐式化(尤其是曲面的精确隐式化)在几何造型中没有得到广泛的应用。这主要是参数曲线曲面的精确隐式化过程很复杂而且不一定可以实现，再加上隐式曲线曲面的阶数很高并且具有不希望的奇异点和多余分支，从而会引起计算的不稳定性和几何造型中拓扑结构的不一致性，这就大大限制了精确隐式化在实际中的运用。
为了解决上述问题，我们提出了如下三种近似隐式化算法：首先，我们利用二阶二次代数样条曲线对参数曲线进行近似隐式化，所得到的代数样条曲线不会产生多余分支和自交点，并且具有良好的误差估计和逼近性质。其次，我们利用Multiquadric拟插值和径向基神经网络，提出了参数曲线近似隐式化的另一种方法。该方法具有保形性好，光滑度高，逼近性能好，样本节点数据少等优点。最后，考虑到很难将上述两种方法直接推广到参数曲面的近似隐式化，我们利用紧支集径向基函数作多元散乱数据插值的技巧提出了参数曲面近似隐式化的一种算法。
第三章我们讨论了分片代数簇的某些问题。分片代数簇作为多元样条的公共零点集合，是经典代数簇的推广，它不仅和许多实际问题如多元样条插值，CAD和CAGD等有关，而且还为研究经典代数几何提供理论依据。因此，研究分片代数簇是很重要的。首先，讨论了分片代数簇的维数性质，通过引入分片代数簇完全相交的概念讨论了分片代数簇的维数与其定义方程组个数的关系．其次，简要讨论了分片代数曲线的奇点性质。再次，为了有效计算分片代数簇，我们讨论了凸多面体内任意维代数簇的计算问题。通过添加超平面技巧将Groebner基方法应用到凸多面体内任意维代数簇的计算上，从而把凸多面体内的代数簇转化为另外一组多项式方程组的正解，并且得到了该代数簇在凸多面体内的极小分解．最后，基于B-样条系数的Descartes符号准则和Bézier曲线的de Casteljau算法，我们给出了一元样条实根分离算法，也就是计算出一列不相交区间，使得每个区间恰好只包含此样条函数的一个实根。
第四章我们主要构造了一类具有紧支集的无穷次可微径向函数。众所周知，Gauss分布函数是一类广泛应用于多元插值和径向基网络的正定径向函数。它具有非常好的逼近效果和指数衰减性质。对Gauss分布函数离中心远处截断，可以马上得到紧支集径向基函数。然而，这样得到的紧支集径向基函数用于多元插值和函数逼近显然是不连续的。因此，结合Gauss函数的特点对其进行改进，我们构造了一类具有可控自由参数的紧支集无限次可微函数。在对自由参数一定的约束条件下，此类函数能够有效地应用到多元函数逼近和多元散乱数据插值中。
Implicit surfaces play an important role in many theoretic fields and applied fieldsin Computer Aided Geometry Design and Geometry Modeling. In this thesis, we mainlystudy some problems on approximate implicitization and piecewise algebraic varieties.Our primary work is organized as follows:
In chapter 2, we discuss the problem of approximate implicitization of parametriccurves/surfaces. It is well known that parametric curves/surfaces and implicit curves/-surfaces are two important topics in Computer Aided Geometry Design and GeometricModeling. The procedure of transform the parametric form into algebraic form is calledimplicitization. However, accurate implicitization (especially surface implicitization) hasnot been popular in practice. This is due to the fact that exact implicitization of para-metric curves/surfaces always involves complicated computation and its exact implicitform usually cannot be computed. Moreover, the degree of implicit curves and surfacesis higher and implicit curves and surfaces may have singular points and unexpectedcomponents, which lead to computational instability and topological inconsistency ingeometric modeling.
In order to solve this problem, we propose the following three algorithms to deal withapproximate implicitization. Firstly, we use quadratic algebraic spline with smoothnesstwo to tackle approximate implicitization of parametric curves. The resulting approxi-mate curves not only don't have unwanted components and self-intersections, but alsohave good error estimate and approximation behavior. Secondly, we propose a new ap-proach to solve approximate implicitization of parametric curves based on radial basisfunction networks and multiquadric quasi-interpolation. This approach possesses theadvantages of shape preserving, better smoothness, good approximation behavior andrelatively less data etc. Lastly, we propose a method to solve approximate implicitizationof parametric surfaces based on multivariate interpolation by using compactly supportedradial basis functions.
In chapter 3, we discuss several problems on piecewise algebraic varieties. As thezeros of multivariate splines, the piecewise algebraic variety is a generalization of theclassical algebraic variety. It is important to study the interpolation by multivariatesplines and algebraic geometry etc. Firstly, we discuss the relationship between thedimension of piecewise algebraic varieties and the number of their defining equations, aswell as several dimension properties by introducing the concept of complete intersectionof piecewise algebraic varieties. Secondly, several singular point properties of piecewisealgebraic curves are discussed. Thirdly, the intersection problem of piecewise curves andsurfaces boils down to the computation of piecewise algebraic varieties. In order to solvepiecewise algebraic varieties, we propose a new method to compute an algebraic varietyon a convex polyhedron by adding hyperplanes with the method of Groebner bases. Thus,the algebraic variety on the convex polyhedron is transformed to the positive solutionsof a system of polynomials. Besides, the minimal decomposition is also obtained. Lastly,we present an algorithm to isolate real roots of a univariate spline, i.e., computing asequence of disjoint intervals such that each of them contains exactly one real root of agiven spline function, which is primarily based on the use of Descartes' rule of signs withits B-spline coefficients and de Casteljau algorithm of B(?)zier curve.
In chapter 4, a kind of multivariate compactly supported infinitely differentiablefunctions is constructed. It is well known that the Gauss distribution function is awidespread used positive definite radial basis function in multivariate scattered datainterpolation. Moreover, it possesses good global approximation behavior and decaysexponentially. It is clear that we can generate the radial functions with compact supportby cutting off Gauss function at large distances from the centers. However the resultingradial function interpolation (approximation) is obviously discontinuous. Thus, we con-struct a kind of compactly supported radial functions which have infinite differentiableproperty for any space dimension with two free parameters. Under certain conditions onparameters, they can be applied to function approximation and scattered data interpo-lation.
引文
[1] Schoenberg I J. Contribution to the problem of application of equidistant data by analytic functions. Quart.Appl.Math., 1946, 4: 45-99;112-141.
[2] 王仁宏．多元齿的结构与插值．数学学报，1975，18：91-106．
[3] 王仁宏，施锡泉，罗钟铉，苏志勋．多元样条函数及其应用．北京：科学出版社，1994．
[4] 王仁宏．任意剖分下的多元样条分析．中国科学，数学专辑Ⅰ，1979，215-226．
[5] 王仁宏，梁学章．多元函数逼近．北京：科学出版社，1988．
[6] 王仁宏，数值逼近．北京：高等教育出版社，1999．
[7] 吴宗敏．函数的径向表示．数学进展，1998，27(3)：202-208．
[8] 陈荣华．径向基函数拟插值理论及其在微分方程数值解中的应用：(博士学位论文)．上海：复旦大学博士论文，2005．
[9] Buhmann M D. Radial Basis Functions: Theory and Implementations. Cambridge University Press, 2000.
[10] Wu Z M. Generalized Bochner's theorem for radia lfunction. Approx. Theory Appl., 1997, 13(3): 47-57.
[11] Micchelli C A. Interpolation of scattered data: distance matrix and conditionally positive definite functions. Constructive Approximation, 1986, 2: 11-22.
[12] Sun X. Conditionally positive definite functions and their application to multivariate interpolations. J.Approximation Theory, 1993, 74: 159-180.
[13] Franke R. Scattered data interpolation: test of some methods. Math. Comput., 1982, 38: 181-200.
[14] 吴宗敏．径向基函数，散乱数据拟合与无网格偏微分方程数值解．工程数学学报，2002，19(2)：1-12．
[15] 吴文俊．吴文俊论数学机械化．济南：山东教育出版社，1996．
[16] 王东明．消去法及其应用．北京：科学出版社，2002．
[17] 石赫．机械化数学引论．长沙：湖南教育出版社，1998．
[18] 杨路，张景中，侯晓荣．非线性方程组与定理机器证明．上海：上海科技教育出版社，1996．
[19] Cox D A, Little J and O'shea D. Ideals, Varieties and Algorithms. Berlin: Springer, 1992.
[20] Cox D A, Little J and O'shea D. Using Algebraic Geometry. Berlin: Springer, 1998.
[21] Walker R J. Algebraic Curves. Princeton: Princeton University Press, 1950.
[22] Bloomenthal J. Introduction to Implicit Surfaces. San Francisco: Morgan Kaufmann, 1997.
[23] 王东明．符号计算选讲．北京：清华大学出版社，2003．
[24] Sederberg T W, Arnon D S. Implicit equation for a parametric surface by Groebner basis. Proceedings of the 1984 Macsyma user's Conference. General Electric. Schenectady, New York, 431-435.
[25] Chionh E, Goldman R N. Using multivariate resultants to find the implicit equation of a rational surface. The Visual Computer, 1992, 8: 171-180.
[26] Chionh E, Goldman R N. Degree, multiplicity and inversion formulas for rational surfaces using u-resultants. CAGD, 1992, 9: 93-108.
[27] Manocha D, Canny J. Algorithm for implicitizing rational parametric surfaces. CAGD, 1992, 9: 25-50.
[28] Gao X S, Chou S C. Implicitization of rational parametric equations. Journal of Symbolic Computation, 1992, 14: 459-470.
[29] Sederberg T W, Anderson D C, Goldman R N. Implicit representation of parametric curves and surfaces. Computer Vision, Graphics and Image processing, 1994, 28: 72-84.
[30] Sederberg T W, Chen F L. Implicitization using moving curves and surfaces. Proc, of SIGGRAPH, 1995, 301-308.
[31] Gonzalez-Vega L. Implicitization of parametric curves and surfaces by using multidimensional Newton formulae. J. Sym. Comp., 1997, 23: 137-151.
[32] Cox D A, Sederberg T W, Chen F L. The moving line ideal basis of planar rational curves. CAGD, 1998, 15: 803-827.
[33] Macro A, Martinez I J. Using polynomial interpolation for implicitizing algebraic curves. CAGD, 2001, 18: 309-319.
[34] Macro A, Martinez I J. Implicitization of rational surfaces by means of polynomial interpolation. CAGD, 2002, 19: 327-344.
[35] Vehlo L, Gomes J. Approximate conversion from parametric to implicit surfaces. Computer Graphics Forum, 1996, 15(5): 327-337.
[36] Sederberg T W, Zheng J, Klimaszewski K, Dokken T. Approximate implicit using monoid curves and surfaces. Graphical model and Image processing, 1999, 61: 177-198.
[37] Dokken T. Approximation implicitization. Mathematical Methods for Curves and Surfaces, Oslo 2000,T.Lyche and L.L.Schumaker(eds), Vanderbilt University Press, 2001, 81-102.
[38] Dokken T, Thomassen J B. Overview of approximate implicitization, in Topics in Algebraic Geometry anf Geometric Modeling. AMS Cont.Math., 2003, 334: 169-184.
[39] Chen F L, Deng D. Interval implicitization of rational curves. CAGD, 2004, 21: 401-415.
[40] Gao X S, Li M. Rational quadratic approximation to plane real algebraic curves. CAGD, 2004, 21: 805-828.
[41] Li M, Gao X S, Chou S C. Quadratic approximation to plane parametric curves and its application in approximate implicitization. Visual Computer, 2006, 22: 906-917.
[42] Wang R H, Wu J M. Approximation implicitization based on RBF networks and MQ quasi-interpolation. Journal of Computational Mathematics, 2007, 25(1): 97-103.
[43] Wang R H, Wu J M. Approximate implicitization based on interpolation using compactly supported radial basis functions. Manuscript.
[44] 王仁宏，吴金明．参数曲线的代数样条隐式化．Manuscript．
[45] Luo Z X, Wang R H. Structure and application of algebraic spline curve and surfaces. J.Math.Res. and Exp., 1992, 12: 579-582.
[46] 屈永明，孙建忠，陈发来．代数曲线的近似参数化．中国科技大学学报，1997，27(4)：382-388．
[47] Simon H. Neural Network: A Comprenhensive Foundation, second edition. Beijing, China Machine Press, 2004.
[48] 廖报通，陈发来．径向基函数神经网络在散乱数据插值中的应用．中国科学技术大学学报，2001，31(2)：135-142．
[49] Hardy R L. Multiquadric equations of topography and other irregular surfaces. J.Geophs Res., 1971, 176: 1905-1915.
[50] Buhmann M. Convergence of univariate quasi-interpolation using multiquadrics. IMA J.Numer.Anal., 1998, 8: 365-383.
[51] Beatson R K. Multiquadric B-splines. J. Approx. Theory, 1996, 87: 1-24.
[52] Beatson R K, Powell M. Univariate multiquadric approximation: quasi-interpolation to scattered data. DAMTP 1990/NA7, University of Cambridge.
[53] Wu Z M, Schaback R. Shape preserving interpolation with radial basis function. Acta Math.Appl.Sinica, 1994, 10: 441-446.
[54] Wu Z M. Dynamically knots setting in meshless method for solving time dependent propagations equation. Comput.Methods Appl.Mech.Energ., 2004, 193: 1221-1229.
[55] Wu Z M. Multivariate compactly supported positive definite radial functions. Adv.Comput.Math., 1995, 4: 283-292.
[56] Wendland H. Piecewise polynomial,positive definite and compactly supported radial basis functions of minimal degree. Adv,Comput.Math., 1995, 4: 389-396.
[57] Floater M S, Iske A. Multistep scattered data interpolation using compactly supported radial basis functions. J.Comput. and Appl. Math., 1996, 73: 65-78.
[58] Lazzaro D, Montefusco L B. Radial basis functions for the multivariate interpolation of large scattered data sets. J.Comput.and Appl.Math., 2002, 140: 521-536.
[59] Morse B S, Yoo T S, Rheingns P. Interpolating implicit surfaces from scattered surface data using CSRBF. In:Genova, Italy, SMI2001, International Conference on Shape Modeling and Applictions,2001, 89-98.
[60] Ohatake Y. 3D scattered data interpolation and approximation with multilevel CSRBF. Graphics model, 2005, 67: 150-160.
[61] Turk G, O'Brien J. Shape transformation using variational implicit surfaces, in Proceedings of ACM SIGGRAPH, 1999, 335-342.
[62] Turk G, O'Brien J. Modeling with implicit surfaces that interpolate. ACM Transaction on Graphics, 2002, 21(4): 855-873.
[63] Carr J, Beatson K. Reconstruction and representation of 3D objects with radial basis function. SIGGRAPH, 2001.
[64] Ohatake Y, Belyaev A, Alexa M, Turk G, Seidel H P. Multi-level partition of unity implicits. ACM Transaction on Graphics, 2003, 22: 463-470.
[65] Ohatake Y, Belyaev A, Seidel H P. A multi-scale approach to 3D scattered data interpolation with compactly supported basis functions, in: Shape Modelling International 2003, Seoul, Korea, 2003: 153-161.
[66] Ohatake Y, Belyaev A, Seidel H P. Sparse surface reconstruction with adaptive partition of unity and radial basis functions. Graphical Models, 2006, 68: 15-24.
[67] Osher S, Fedekiw R. Level Set Methods and Dynamic Implicit Surfaces. New York: Springer- Verlag, 2003.
[68] Bajaj C L, Xu G L. A-splines: local interpolation and approximation using G~k-continous piecewise real algebraic curves. CAGD, 1999, 16: 557-578.
[69] Xu G L, Bajaj C L, Xue W. Regular algebraic curve segments(Ⅰ)-definitions and characteristics. CAGD, 2000, 17: 485-501.
[70] Xu G L, Bajaj C L, Chun C I. Regular algebraic curve segments(Ⅱ)-Interpolation and approximation. CAGD, 2000, 17: 503-519.
[71] Hartshorne R. Algebraic Geometry. New York: Springer-Verlag, 1977.
[72] Bochnak J, Coste M, Roy M F. Real Algebraic Geometry. Berlin: Springer, 1998.
[73] 苏志勋．分片代数曲线曲面及其在CAGD中的应用：(博士学位论文)．大连：大连理工大学，1993．
[74] 赖义生．分片代数曲线和分片代数簇的若干研究：(博士学位论文)．大连：大连理工大学，2002．
[75] 许志强．多元样条，分片代数曲线和线性丢番图方程组：(博士学位论文)．大连：大连理工大学，2003．
[76] 朱春钢．分片代数曲线，分片代数簇和分片半代数集若干问题的研究：(博士学位论文)．大连：大连理工大学，2005．
[77] Shi X Q, Wang R H. The Bezout number for piecewise algebraic curves. BIT, 1999, (39)2: 339-349.
[78] Wang R H. Multivariate spline and algebraic geometry. Jour. Comp. Appl. Math., 2000, 121: 153-163.
[79] Wang R H, Lai Y S. Piecewise algebraic curve. Journal of Computational and Applied Mathematics, 2002, 144: 277-289.
[80] Wang R H, Lai Y S. Real piecewise algebraic variety. Jour. Comp. Math., 2003, 21(4): 473-480.
[81] Wang R H, Zhu C G. Piecewise algebraic varieties. Progress in Natural Science (English Ed.), 2004, 14: 568-572.
[82] 王仁宏，许志强．分片代数曲线Bezout数的估计．中国科学A辑，2003，2：185-192．
[83] Wang R H, Wu J M. Computation of an algebraic variety on a convex polyhedron. Journal of Information and Computational Science, 2006, 3(4): 903-911.
[84] Wang R H, Wu J M. Spline function real root isolation. Journal of Computational Mathematics, accepted.
[85] Buchberger B, Winkler F. Groebner Bases and Applications. London Math. Soc. Lecture Notes, 251, 1998.
[86] 王文举，王仁宏，刘秀平．分片代数曲线交点的结式求法．数学研究与评论，1999，19(1)：125-129．
[87] 刘秀平，王仁宏．分片代数簇的机械化求解．辽宁师范大学学报，2000，23：230-233．
[88] 赖以生，王仁宏．多项式的B-网结式及其应用．数学研究与评论，2005，25(3)：511-514．
[89] Goodman T. New bounds on the zeros of spline functions. J.Approx.Theory, 1994, 76(1): 123-130.
[90] De Boor C. The multiplicity of a spline zero. Annals of Numer.Math., 1997, 4: 229-238.
[91] Grandine T A. Computing zeros of spline functions. CAGD, 1989, 6: 129-136.
[92] Uspensky J V. Theory of Equations. New York: McGraw-Hill, 1948.
[93] Collins G E, Akritas A G. Polynomial real root isolation using Descarte's rule of signs, in: SYMSAC, 1976, 272-275.
[94] Rouillier F, Zimmermann P. Efficient isolation of polynomial's real roots. J.Comput. Appl.Math., 2004, 162: 33-50.
[95] Xia B C, Yang L. An algorithm for isolating the real roots of semi-algebraic system. J.Symbolic Computation, 2002, 34: 461-477.
[96] De Boor C. A Practical Guide to Splines. New York: Springer, 1978.
[97] Schumaker L L. Spline Fanctions: Basic Theory. NewYork: Wiley, 1981.
[98] Farin G. Curves and Surfaces for Computer-Aided Geometry Design: A Practical Guide. Academic Press, 1997.
[99] 周蕴时，苏志勋，奚勇江，程少春．CAGD中的曲线曲面．长春：吉林大学出版社，1993．
[100] Dahmen W. Subdivision algorithm converge quadratically. J. Comp. Appl. Math., 1986,16: 145-158.
[101] Pedersen P. Multivariate Sturm Theory. Proceedings AAECC-9, Lecture Notes in Computer Science, 1991, 539: 318-332.
[102] Gonzalez-Vega L, Trujillo G. Multivariate Sturm-Habicht sequences: real root counting on n-rectange and triangles. Revista Mathematica, 1997, 10: 119-130.
[103] Yang L. Recent advances on determining the number of real roots of parametric polynomial. J.Symbolic Computation, 1999, 28: 225-242.
[104] Powell M J. Radial basis function for multivariable interpolation: a review. Numerical Analysis,(eds).D.Griffiths,G.Waston, Longman Scientific and Technical, 1987, 223-241.
[105] Dyn N. Interpolation of scattered data by radial functions. Topics in Multivariate Approximation, (eds).C.Chui, L.Schumaker F. Utreras, Academic Press, 1987, 47-61.
[106] Schaback R. Creating surfaces from scattered data using radial basis function. Mathematical methods for curves and surfaces, Vanderbilt University Press, 1995, 477-496.
[107] Schaback R. Error estimates and condition numbers for radial basis function. Adv. Comput. Math., 1995, 3: 251-264.
[108] Wu Z M. Hermite-Birkhoff interpolation of scattered data by radial basis functions. Approx.Theory and its Appl., 1992, 8: 1-10.
[109] Wu Z M, Schaback R. Local error estimates for radial basis function interpolation ofscattered data. IMA Journal of Numerical Analysis, 1993, 13: 13-27.
[110] 吴宗敏．关于径向基函数插值的收敛性．数学年刊，1993，14：480-486．
[111] Beatson B K, Newsam G N. Fast evaluation of radial basis functions: part 1. Comput. Math. Appl., 1992, 24(12): 7-19.
[112] Powell M J. Truncated laurent expansions of fast evaluation of thin-plate spline. Numer.Algebra, 1993, 5(2): 99-120.
[113] Roussos G, Baxter B J C. Rapid evaluation of radial basis functions. J.Comput.Appl.Math., 2005, 180: 51-70.
[114] Buhmann M. A new class of radial basis functions with compact support. Math.Comput., 2001, 70: 307-318.
[115] Wang R H, Wu J M. A kind of multivariate compactly supported infinitely differentiable functions and its applications. Journal of Information and Computational Science, 2007, 4(1): 309-316.
[116] Schoenberg I J, Metric space and completely monotone functions. Ann.Math., 1938, 39: 811-841.
[117] Bozzini M, Lenarduzzi L, Schaback R. Adaptive interpolation by scaled multiquadrics. ACM, 2002, 16: 375-387.
[2] 王仁宏．多元齿的结构与插值．数学学报，1975，18：91-106．
[3] 王仁宏，施锡泉，罗钟铉，苏志勋．多元样条函数及其应用．北京：科学出版社，1994．
[4] 王仁宏．任意剖分下的多元样条分析．中国科学，数学专辑Ⅰ，1979，215-226．
[5] 王仁宏，梁学章．多元函数逼近．北京：科学出版社，1988．
[6] 王仁宏，数值逼近．北京：高等教育出版社，1999．
[7] 吴宗敏．函数的径向表示．数学进展，1998，27(3)：202-208．
[8] 陈荣华．径向基函数拟插值理论及其在微分方程数值解中的应用：(博士学位论文)．上海：复旦大学博士论文，2005．
[9] Buhmann M D. Radial Basis Functions: Theory and Implementations. Cambridge University Press, 2000.
[10] Wu Z M. Generalized Bochner's theorem for radia lfunction. Approx. Theory Appl., 1997, 13(3): 47-57.
[11] Micchelli C A. Interpolation of scattered data: distance matrix and conditionally positive definite functions. Constructive Approximation, 1986, 2: 11-22.
[12] Sun X. Conditionally positive definite functions and their application to multivariate interpolations. J.Approximation Theory, 1993, 74: 159-180.
[13] Franke R. Scattered data interpolation: test of some methods. Math. Comput., 1982, 38: 181-200.
[14] 吴宗敏．径向基函数，散乱数据拟合与无网格偏微分方程数值解．工程数学学报，2002，19(2)：1-12．
[15] 吴文俊．吴文俊论数学机械化．济南：山东教育出版社，1996．
[16] 王东明．消去法及其应用．北京：科学出版社，2002．
[17] 石赫．机械化数学引论．长沙：湖南教育出版社，1998．
[18] 杨路，张景中，侯晓荣．非线性方程组与定理机器证明．上海：上海科技教育出版社，1996．
[19] Cox D A, Little J and O'shea D. Ideals, Varieties and Algorithms. Berlin: Springer, 1992.
[20] Cox D A, Little J and O'shea D. Using Algebraic Geometry. Berlin: Springer, 1998.
[21] Walker R J. Algebraic Curves. Princeton: Princeton University Press, 1950.
[22] Bloomenthal J. Introduction to Implicit Surfaces. San Francisco: Morgan Kaufmann, 1997.
[23] 王东明．符号计算选讲．北京：清华大学出版社，2003．
[24] Sederberg T W, Arnon D S. Implicit equation for a parametric surface by Groebner basis. Proceedings of the 1984 Macsyma user's Conference. General Electric. Schenectady, New York, 431-435.
[25] Chionh E, Goldman R N. Using multivariate resultants to find the implicit equation of a rational surface. The Visual Computer, 1992, 8: 171-180.
[26] Chionh E, Goldman R N. Degree, multiplicity and inversion formulas for rational surfaces using u-resultants. CAGD, 1992, 9: 93-108.
[27] Manocha D, Canny J. Algorithm for implicitizing rational parametric surfaces. CAGD, 1992, 9: 25-50.
[28] Gao X S, Chou S C. Implicitization of rational parametric equations. Journal of Symbolic Computation, 1992, 14: 459-470.
[29] Sederberg T W, Anderson D C, Goldman R N. Implicit representation of parametric curves and surfaces. Computer Vision, Graphics and Image processing, 1994, 28: 72-84.
[30] Sederberg T W, Chen F L. Implicitization using moving curves and surfaces. Proc, of SIGGRAPH, 1995, 301-308.
[31] Gonzalez-Vega L. Implicitization of parametric curves and surfaces by using multidimensional Newton formulae. J. Sym. Comp., 1997, 23: 137-151.
[32] Cox D A, Sederberg T W, Chen F L. The moving line ideal basis of planar rational curves. CAGD, 1998, 15: 803-827.
[33] Macro A, Martinez I J. Using polynomial interpolation for implicitizing algebraic curves. CAGD, 2001, 18: 309-319.
[34] Macro A, Martinez I J. Implicitization of rational surfaces by means of polynomial interpolation. CAGD, 2002, 19: 327-344.
[35] Vehlo L, Gomes J. Approximate conversion from parametric to implicit surfaces. Computer Graphics Forum, 1996, 15(5): 327-337.
[36] Sederberg T W, Zheng J, Klimaszewski K, Dokken T. Approximate implicit using monoid curves and surfaces. Graphical model and Image processing, 1999, 61: 177-198.
[37] Dokken T. Approximation implicitization. Mathematical Methods for Curves and Surfaces, Oslo 2000,T.Lyche and L.L.Schumaker(eds), Vanderbilt University Press, 2001, 81-102.
[38] Dokken T, Thomassen J B. Overview of approximate implicitization, in Topics in Algebraic Geometry anf Geometric Modeling. AMS Cont.Math., 2003, 334: 169-184.
[39] Chen F L, Deng D. Interval implicitization of rational curves. CAGD, 2004, 21: 401-415.
[40] Gao X S, Li M. Rational quadratic approximation to plane real algebraic curves. CAGD, 2004, 21: 805-828.
[41] Li M, Gao X S, Chou S C. Quadratic approximation to plane parametric curves and its application in approximate implicitization. Visual Computer, 2006, 22: 906-917.
[42] Wang R H, Wu J M. Approximation implicitization based on RBF networks and MQ quasi-interpolation. Journal of Computational Mathematics, 2007, 25(1): 97-103.
[43] Wang R H, Wu J M. Approximate implicitization based on interpolation using compactly supported radial basis functions. Manuscript.
[44] 王仁宏，吴金明．参数曲线的代数样条隐式化．Manuscript．
[45] Luo Z X, Wang R H. Structure and application of algebraic spline curve and surfaces. J.Math.Res. and Exp., 1992, 12: 579-582.
[46] 屈永明，孙建忠，陈发来．代数曲线的近似参数化．中国科技大学学报，1997，27(4)：382-388．
[47] Simon H. Neural Network: A Comprenhensive Foundation, second edition. Beijing, China Machine Press, 2004.
[48] 廖报通，陈发来．径向基函数神经网络在散乱数据插值中的应用．中国科学技术大学学报，2001，31(2)：135-142．
[49] Hardy R L. Multiquadric equations of topography and other irregular surfaces. J.Geophs Res., 1971, 176: 1905-1915.
[50] Buhmann M. Convergence of univariate quasi-interpolation using multiquadrics. IMA J.Numer.Anal., 1998, 8: 365-383.
[51] Beatson R K. Multiquadric B-splines. J. Approx. Theory, 1996, 87: 1-24.
[52] Beatson R K, Powell M. Univariate multiquadric approximation: quasi-interpolation to scattered data. DAMTP 1990/NA7, University of Cambridge.
[53] Wu Z M, Schaback R. Shape preserving interpolation with radial basis function. Acta Math.Appl.Sinica, 1994, 10: 441-446.
[54] Wu Z M. Dynamically knots setting in meshless method for solving time dependent propagations equation. Comput.Methods Appl.Mech.Energ., 2004, 193: 1221-1229.
[55] Wu Z M. Multivariate compactly supported positive definite radial functions. Adv.Comput.Math., 1995, 4: 283-292.
[56] Wendland H. Piecewise polynomial,positive definite and compactly supported radial basis functions of minimal degree. Adv,Comput.Math., 1995, 4: 389-396.
[57] Floater M S, Iske A. Multistep scattered data interpolation using compactly supported radial basis functions. J.Comput. and Appl. Math., 1996, 73: 65-78.
[58] Lazzaro D, Montefusco L B. Radial basis functions for the multivariate interpolation of large scattered data sets. J.Comput.and Appl.Math., 2002, 140: 521-536.
[59] Morse B S, Yoo T S, Rheingns P. Interpolating implicit surfaces from scattered surface data using CSRBF. In:Genova, Italy, SMI2001, International Conference on Shape Modeling and Applictions,2001, 89-98.
[60] Ohatake Y. 3D scattered data interpolation and approximation with multilevel CSRBF. Graphics model, 2005, 67: 150-160.
[61] Turk G, O'Brien J. Shape transformation using variational implicit surfaces, in Proceedings of ACM SIGGRAPH, 1999, 335-342.
[62] Turk G, O'Brien J. Modeling with implicit surfaces that interpolate. ACM Transaction on Graphics, 2002, 21(4): 855-873.
[63] Carr J, Beatson K. Reconstruction and representation of 3D objects with radial basis function. SIGGRAPH, 2001.
[64] Ohatake Y, Belyaev A, Alexa M, Turk G, Seidel H P. Multi-level partition of unity implicits. ACM Transaction on Graphics, 2003, 22: 463-470.
[65] Ohatake Y, Belyaev A, Seidel H P. A multi-scale approach to 3D scattered data interpolation with compactly supported basis functions, in: Shape Modelling International 2003, Seoul, Korea, 2003: 153-161.
[66] Ohatake Y, Belyaev A, Seidel H P. Sparse surface reconstruction with adaptive partition of unity and radial basis functions. Graphical Models, 2006, 68: 15-24.
[67] Osher S, Fedekiw R. Level Set Methods and Dynamic Implicit Surfaces. New York: Springer- Verlag, 2003.
[68] Bajaj C L, Xu G L. A-splines: local interpolation and approximation using G~k-continous piecewise real algebraic curves. CAGD, 1999, 16: 557-578.
[69] Xu G L, Bajaj C L, Xue W. Regular algebraic curve segments(Ⅰ)-definitions and characteristics. CAGD, 2000, 17: 485-501.
[70] Xu G L, Bajaj C L, Chun C I. Regular algebraic curve segments(Ⅱ)-Interpolation and approximation. CAGD, 2000, 17: 503-519.
[71] Hartshorne R. Algebraic Geometry. New York: Springer-Verlag, 1977.
[72] Bochnak J, Coste M, Roy M F. Real Algebraic Geometry. Berlin: Springer, 1998.
[73] 苏志勋．分片代数曲线曲面及其在CAGD中的应用：(博士学位论文)．大连：大连理工大学，1993．
[74] 赖义生．分片代数曲线和分片代数簇的若干研究：(博士学位论文)．大连：大连理工大学，2002．
[75] 许志强．多元样条，分片代数曲线和线性丢番图方程组：(博士学位论文)．大连：大连理工大学，2003．
[76] 朱春钢．分片代数曲线，分片代数簇和分片半代数集若干问题的研究：(博士学位论文)．大连：大连理工大学，2005．
[77] Shi X Q, Wang R H. The Bezout number for piecewise algebraic curves. BIT, 1999, (39)2: 339-349.
[78] Wang R H. Multivariate spline and algebraic geometry. Jour. Comp. Appl. Math., 2000, 121: 153-163.
[79] Wang R H, Lai Y S. Piecewise algebraic curve. Journal of Computational and Applied Mathematics, 2002, 144: 277-289.
[80] Wang R H, Lai Y S. Real piecewise algebraic variety. Jour. Comp. Math., 2003, 21(4): 473-480.
[81] Wang R H, Zhu C G. Piecewise algebraic varieties. Progress in Natural Science (English Ed.), 2004, 14: 568-572.
[82] 王仁宏，许志强．分片代数曲线Bezout数的估计．中国科学A辑，2003，2：185-192．
[83] Wang R H, Wu J M. Computation of an algebraic variety on a convex polyhedron. Journal of Information and Computational Science, 2006, 3(4): 903-911.
[84] Wang R H, Wu J M. Spline function real root isolation. Journal of Computational Mathematics, accepted.
[85] Buchberger B, Winkler F. Groebner Bases and Applications. London Math. Soc. Lecture Notes, 251, 1998.
[86] 王文举，王仁宏，刘秀平．分片代数曲线交点的结式求法．数学研究与评论，1999，19(1)：125-129．
[87] 刘秀平，王仁宏．分片代数簇的机械化求解．辽宁师范大学学报，2000，23：230-233．
[88] 赖以生，王仁宏．多项式的B-网结式及其应用．数学研究与评论，2005，25(3)：511-514．
[89] Goodman T. New bounds on the zeros of spline functions. J.Approx.Theory, 1994, 76(1): 123-130.
[90] De Boor C. The multiplicity of a spline zero. Annals of Numer.Math., 1997, 4: 229-238.
[91] Grandine T A. Computing zeros of spline functions. CAGD, 1989, 6: 129-136.
[92] Uspensky J V. Theory of Equations. New York: McGraw-Hill, 1948.
[93] Collins G E, Akritas A G. Polynomial real root isolation using Descarte's rule of signs, in: SYMSAC, 1976, 272-275.
[94] Rouillier F, Zimmermann P. Efficient isolation of polynomial's real roots. J.Comput. Appl.Math., 2004, 162: 33-50.
[95] Xia B C, Yang L. An algorithm for isolating the real roots of semi-algebraic system. J.Symbolic Computation, 2002, 34: 461-477.
[96] De Boor C. A Practical Guide to Splines. New York: Springer, 1978.
[97] Schumaker L L. Spline Fanctions: Basic Theory. NewYork: Wiley, 1981.
[98] Farin G. Curves and Surfaces for Computer-Aided Geometry Design: A Practical Guide. Academic Press, 1997.
[99] 周蕴时，苏志勋，奚勇江，程少春．CAGD中的曲线曲面．长春：吉林大学出版社，1993．
[100] Dahmen W. Subdivision algorithm converge quadratically. J. Comp. Appl. Math., 1986,16: 145-158.
[101] Pedersen P. Multivariate Sturm Theory. Proceedings AAECC-9, Lecture Notes in Computer Science, 1991, 539: 318-332.
[102] Gonzalez-Vega L, Trujillo G. Multivariate Sturm-Habicht sequences: real root counting on n-rectange and triangles. Revista Mathematica, 1997, 10: 119-130.
[103] Yang L. Recent advances on determining the number of real roots of parametric polynomial. J.Symbolic Computation, 1999, 28: 225-242.
[104] Powell M J. Radial basis function for multivariable interpolation: a review. Numerical Analysis,(eds).D.Griffiths,G.Waston, Longman Scientific and Technical, 1987, 223-241.
[105] Dyn N. Interpolation of scattered data by radial functions. Topics in Multivariate Approximation, (eds).C.Chui, L.Schumaker F. Utreras, Academic Press, 1987, 47-61.
[106] Schaback R. Creating surfaces from scattered data using radial basis function. Mathematical methods for curves and surfaces, Vanderbilt University Press, 1995, 477-496.
[107] Schaback R. Error estimates and condition numbers for radial basis function. Adv. Comput. Math., 1995, 3: 251-264.
[108] Wu Z M. Hermite-Birkhoff interpolation of scattered data by radial basis functions. Approx.Theory and its Appl., 1992, 8: 1-10.
[109] Wu Z M, Schaback R. Local error estimates for radial basis function interpolation ofscattered data. IMA Journal of Numerical Analysis, 1993, 13: 13-27.
[110] 吴宗敏．关于径向基函数插值的收敛性．数学年刊，1993，14：480-486．
[111] Beatson B K, Newsam G N. Fast evaluation of radial basis functions: part 1. Comput. Math. Appl., 1992, 24(12): 7-19.
[112] Powell M J. Truncated laurent expansions of fast evaluation of thin-plate spline. Numer.Algebra, 1993, 5(2): 99-120.
[113] Roussos G, Baxter B J C. Rapid evaluation of radial basis functions. J.Comput.Appl.Math., 2005, 180: 51-70.
[114] Buhmann M. A new class of radial basis functions with compact support. Math.Comput., 2001, 70: 307-318.
[115] Wang R H, Wu J M. A kind of multivariate compactly supported infinitely differentiable functions and its applications. Journal of Information and Computational Science, 2007, 4(1): 309-316.
[116] Schoenberg I J, Metric space and completely monotone functions. Ann.Math., 1938, 39: 811-841.
[117] Bozzini M, Lenarduzzi L, Schaback R. Adaptive interpolation by scaled multiquadrics. ACM, 2002, 16: 375-387.