μ基的应用—空间曲线奇异点的计算及有理曲面的隐式化
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摘要
μ基是新近出现在几何造型领域中研究曲线和曲面性质与计算的一种代数工具,它提供了一种联系曲线和曲面的参数表示与隐式表示之间的桥梁。基于μ基的隐式化方法,表示紧凑且效率大大提高。此外,μ基在求曲线的奇异点,计算直纹面的自交线等几何问题上有很好的应用,展示了不同寻常的优势。
本文将在已有的研究结果的基础上,完善曲线与曲面μ基的计算与应用。对于曲线而言,本文利用μ基理论计算了空间有理曲线上的奇异点,并通过构造多元结式,对奇异点进行了理论分析。对于曲面,本文研究了旋转曲面的μ基的计算及隐式方程求解的问题,并将相应结果推广到了双准线有理曲面的情况。
在第一章中,我们回顾了本学科的发展简史。简述了曲线曲面参数化与隐式化问题的研究成果,并综述了贯穿全文的基础-μ基的研究进展。
在第二章中,我们介绍了本文所需要的基础知识,并介绍了平面有理曲线,空间有理曲线及一般有理曲面的μ基的已知理论。
在第三章中,我们给出了计算空间有理曲线奇异点的两种方法。第一种方法是利用随机技巧将空间有理曲线奇异点的计算化为平面有理曲线奇异点的计算；第二种方法是基于投影的思想,将空间有理曲线投影为平面有理曲线处理。两种方法都可行高效。
在第四章中,我们继续分析空间曲线奇异点的问题。基于构造的多元稀疏结式,我们直接对空间有理曲线的μ基所派生出的三个二元多项式进行公共根分析。利用相应结式,完成了空间有理曲线奇异点的计算,并给出了关于奇异点重数的一些相关理论分析。
在第五章中,我们对旋转曲面的μ基及隐式化问题进行研究。针对旋转曲面的特性,我们利用准线的μ基构造出了相应旋转曲面的μ基。再通过构造Sylvester型矩阵,Bezout型矩阵,得到旋转曲面的隐式方程。
在第六章中,我们将第五章的结果推广到由垂直双准线生成的曲面,得到了相应曲面的μ基和隐式方程。
The μ-bases of rational curves/surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of rational curves/surfaces. They provide efficient algorithms to implicitize rational curves/surfaces as well as algorithms to compute singularities of rational planar curves/ruled surraces.
In this thesis, based on the existing research results, we present more appli-cations of μ-bases in rational space curves and surfaces of revolution. For rational curves, we apply μ-bases to compute the singularities of rational space curves, and based on a new constructing bihomogeneous resultant matrix to analyze the mul-tiplicities of all singular points. For rational surfaces, we construct a μ-basis for surface of revolution, and a Sylvester-type matrix whose determinant gives the im-plicit equation of surface of revolution. And we extend the result to more general surfaces which has a pair of orthogonal directrices.
In Chapter1, we first give a brief history of Computer Aided Geometric de-sign. We mention parametrization and implicitization are two important problems in CAGD, and summarize recent developments of μ-basis of rational curves and surfaces and its applications.
In Chapter2, we provide some preliminary knowledge that we will use in the following chapters, including the definitions and known applications of μ-bases for rational planar curves, rational space curves and rational surfaces.
In Chapter3, we discuss the singularities of rational space curves. Two methods are provided to compute the singularities of arbitrary degree curves. These methods are a generalization of the paper (Chen, Wang and Liu, Computing singular points of plane rational curves. Journal of Symbolic Computation43,92-117,2008), which are based on the μ-basis of the rational space curve and on random technique. The comparison between our two methods and a generalized resultants method are provided. Examples are provided to illustrate the effectiveness of our methods.
In Chapter4, we provide a different technique to detect the singularities of rational space curves. From a μ-basis for a space curve, we generate three planar algebraic curves of different bidegrees whose intersection points correspond to the parameters of the singularities. To find these intersection points, a new sparse resultant matrix is constructed. We can get the information of the singularities through this resultant matrix by Gaussian eliminations.
In Chapter5, we focus on μ-bases of surfaces of revolution. We compute a μ-basis for the surface of revolution from a μ-basis of its directrix and a rational parametrization of the circle. Then we construct a sparse Sylvester style resultant matrix and a Bezout style resultant matrix for three bivariate polynomials of bide-grees (1,μ),(1, n-μ),(2,0) which are derived from the μ-basis. Both determinants provide compact representations for the implicit equation of a rational surface of revolution.
In Chapter6, we extend the result in Chapter5to surfaces generated by two orthogonal directrices. We compute a μ-basis from the μ-basis of its directrices and construct a Sylvester style resultant matrix whose determinant gives the implicit equation of the surface. But the constructing of the Bezout style resultant is hard work, which is an open problem now.
本文将在已有的研究结果的基础上,完善曲线与曲面μ基的计算与应用。对于曲线而言,本文利用μ基理论计算了空间有理曲线上的奇异点,并通过构造多元结式,对奇异点进行了理论分析。对于曲面,本文研究了旋转曲面的μ基的计算及隐式方程求解的问题,并将相应结果推广到了双准线有理曲面的情况。
在第一章中,我们回顾了本学科的发展简史。简述了曲线曲面参数化与隐式化问题的研究成果,并综述了贯穿全文的基础-μ基的研究进展。
在第二章中,我们介绍了本文所需要的基础知识,并介绍了平面有理曲线,空间有理曲线及一般有理曲面的μ基的已知理论。
在第三章中,我们给出了计算空间有理曲线奇异点的两种方法。第一种方法是利用随机技巧将空间有理曲线奇异点的计算化为平面有理曲线奇异点的计算；第二种方法是基于投影的思想,将空间有理曲线投影为平面有理曲线处理。两种方法都可行高效。
在第四章中,我们继续分析空间曲线奇异点的问题。基于构造的多元稀疏结式,我们直接对空间有理曲线的μ基所派生出的三个二元多项式进行公共根分析。利用相应结式,完成了空间有理曲线奇异点的计算,并给出了关于奇异点重数的一些相关理论分析。
在第五章中,我们对旋转曲面的μ基及隐式化问题进行研究。针对旋转曲面的特性,我们利用准线的μ基构造出了相应旋转曲面的μ基。再通过构造Sylvester型矩阵,Bezout型矩阵,得到旋转曲面的隐式方程。
在第六章中,我们将第五章的结果推广到由垂直双准线生成的曲面,得到了相应曲面的μ基和隐式方程。
The μ-bases of rational curves/surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of rational curves/surfaces. They provide efficient algorithms to implicitize rational curves/surfaces as well as algorithms to compute singularities of rational planar curves/ruled surraces.
In this thesis, based on the existing research results, we present more appli-cations of μ-bases in rational space curves and surfaces of revolution. For rational curves, we apply μ-bases to compute the singularities of rational space curves, and based on a new constructing bihomogeneous resultant matrix to analyze the mul-tiplicities of all singular points. For rational surfaces, we construct a μ-basis for surface of revolution, and a Sylvester-type matrix whose determinant gives the im-plicit equation of surface of revolution. And we extend the result to more general surfaces which has a pair of orthogonal directrices.
In Chapter1, we first give a brief history of Computer Aided Geometric de-sign. We mention parametrization and implicitization are two important problems in CAGD, and summarize recent developments of μ-basis of rational curves and surfaces and its applications.
In Chapter2, we provide some preliminary knowledge that we will use in the following chapters, including the definitions and known applications of μ-bases for rational planar curves, rational space curves and rational surfaces.
In Chapter3, we discuss the singularities of rational space curves. Two methods are provided to compute the singularities of arbitrary degree curves. These methods are a generalization of the paper (Chen, Wang and Liu, Computing singular points of plane rational curves. Journal of Symbolic Computation43,92-117,2008), which are based on the μ-basis of the rational space curve and on random technique. The comparison between our two methods and a generalized resultants method are provided. Examples are provided to illustrate the effectiveness of our methods.
In Chapter4, we provide a different technique to detect the singularities of rational space curves. From a μ-basis for a space curve, we generate three planar algebraic curves of different bidegrees whose intersection points correspond to the parameters of the singularities. To find these intersection points, a new sparse resultant matrix is constructed. We can get the information of the singularities through this resultant matrix by Gaussian eliminations.
In Chapter5, we focus on μ-bases of surfaces of revolution. We compute a μ-basis for the surface of revolution from a μ-basis of its directrix and a rational parametrization of the circle. Then we construct a sparse Sylvester style resultant matrix and a Bezout style resultant matrix for three bivariate polynomials of bide-grees (1,μ),(1, n-μ),(2,0) which are derived from the μ-basis. Both determinants provide compact representations for the implicit equation of a rational surface of revolution.
In Chapter6, we extend the result in Chapter5to surfaces generated by two orthogonal directrices. We compute a μ-basis from the μ-basis of its directrices and construct a Sylvester style resultant matrix whose determinant gives the implicit equation of the surface. But the constructing of the Bezout style resultant is hard work, which is an open problem now.
引文
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[11]F. Chen, L. Shen, and J. Deng. Implicitization and parametrization of quadractic and cubic surfaces by μ-bases. Computing,79:131-142,2007.
[12]F. Chen and W. Wang. The μ-basis of a planar rational curve-properties and com-putation. Graphical Models,64:368-381,2003.
[13]F. Chen and W. Wang. Revisiting the μ-basis of a rational ruled surface. Journal of Symbolic Computation,36:699-716,2003.
[14]F. Chen, W. Wang, and Y. Liu. Computing singular points of plane rational curves. Journal of Symbolic Computation,43:92-117,2008.
[15]F. Chen and T. W.Sederberg. A new implicit representation of a planar rational curve with high order singularity. Computer Aided Geometric Design,19:151-167, 2002.
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[28]A. Dickensteina and I. Z. Emirisb. Multihomogeneous resultant resultant formulae by means of complexes. Journal of Symbolic Computation,36:317-342,2003.
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[31]S. Ganguli. The theory of plane curves. University of Calcutta,1931.
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[33]R. Goldman. Some applications of resultants to problems in computational geometry. The Visual Computer,1:101-107,1985.
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[2]S. Abhyankar and A. Sathaye. Geometric theory of algebraic space curves. Lecture Notes in Mathematics, Springer,1974.
[3]D. A.Cox, J. B.Little, and D. O'Shea. Ideals, varieties, and algorithms:an introduc-tion to computational algebraic geometry and commutative Algebra. Third Edition, Springer,2007.
[4]F. Aries and R. Senoussi. An implicitizationalgorithm for rationalsurfaces with nobasepoints. Journal of Symbolic Computation,31:357-365,2001.
[5]T. Becker, V. Weispfenning, and H. Kredel. Grobner bases:a computational approach to commutative algebra. Springer-Verlag,1993.
[6]T. Berrya and R. R.Patterson. Implicitization and parametrization of nonsingular cubic surfaces. Computer Aided Geometric Design,18:723-738,2001.
[7]L. Buse and T. L. Baa. Matrix-based implicit representations of rational algebraic curves and applications. Computer Aided Geometric Design,27:681-699,2010.
[8]B. Buchberger. Applications of Grobner bases in non-linear computational geometry. Lecture Notes in Computer Science,296:52-80,1988.
[9]F. Chen. Reparametrization of a rational ruled surface using the μ-basis. Computer Aided Geometric Design,20:11-17,2003.
[10]F. Chen, D. Cox and Y.Liu. the μ-basis of a rational parametric surface. Computer Aided Geometric Design,39:689-706,2005.
[11]F. Chen, L. Shen, and J. Deng. Implicitization and parametrization of quadractic and cubic surfaces by μ-bases. Computing,79:131-142,2007.
[12]F. Chen and W. Wang. The μ-basis of a planar rational curve-properties and com-putation. Graphical Models,64:368-381,2003.
[13]F. Chen and W. Wang. Revisiting the μ-basis of a rational ruled surface. Journal of Symbolic Computation,36:699-716,2003.
[14]F. Chen, W. Wang, and Y. Liu. Computing singular points of plane rational curves. Journal of Symbolic Computation,43:92-117,2008.
[15]F. Chen and T. W.Sederberg. A new implicit representation of a planar rational curve with high order singularity. Computer Aided Geometric Design,19:151-167, 2002.
[16]F. Chen, J. Zheng, and T. W.Sederberg. The μ-basis of a rational ruled surface. Computer Aided Geometric Design,18:61-72,2001.
[17]E.-W. Chionh. Shifting planes always implicitize a surface of revolution. Computer Aided Geometric Design,26:369-377,2009.
[18]E.-W. Chionh, R. Glodman, and M. Zhang. Hybrid Dixon resultants. Mathematics of Surfaces VIII, Information Geometers Ltd,1998.
[19]E.-W. Chionh and R. Goldman. Elimination and resultants, part 1:Elimination and bivariate reaultants. IEEE Computer Graphics and Applications,15(1):69-77,1995.
[20]E.-W. Chionh and R. Goldman. Elimination and resultants, part 2:Multivariate resultants. IEEE Computer Graphics and Applications,15(2):60-69,1995.
[21]E.-W. Chionh and T. W. Sederberg. On the minors of the implicitization bezout matrix for a rational plane curve. Computer Aided Geometric Design,18:21-36, 2001.
[22]D. Cox. Equations of parametric curves and surfaces via syzygies. Contemporary mathematics,286:1-20,2001.
[23]D. Cox. The moving curve ideal and the rees algebra. Theoretical Computer Science, 392:23-36,2008.
[24]D. Cox, R. Goldman, and M. Zhang. On the validity of implicitization by moving quadrics for rational surfaces with base points. Jouranl of Symbolic Computation, 29:419-440,2000.
[25]D. Cox, J. Little, and D. O'Shea. Using algebraic geometry. Springer, New York, 1998.
[26]D. Cox, T. W.Sederberg, and F. Chen. The moving line ideal basis of planar rational curves. Computer Aided Geometric Design,15:803-827,1998.
[27]J. Deng, F. Chen, and L. Shen. Computing μ-bases of rational curves and surfaces using polynomial matrix factorization. Proceedings of ISSAC 2005,132-139,2005.
[28]A. Dickensteina and I. Z. Emirisb. Multihomogeneous resultant resultant formulae by means of complexes. Journal of Symbolic Computation,36:317-342,2003.
[29]A. Dixon. The eliminant of three quantities in two independent variables. Proceedings of the. London Mathematical Society,7:50-69,473-492,1908.
[30]I. Z. Emirisb and V. Y. Pan. Symbolic and numeric methods for exploiting structure in constructing resultant matrices. Journal of Symbolic Computation,33:393-413, 2002.
[31]S. Ganguli. The theory of plane curves. University of Calcutta,1931.
[32]R. Goldman. Polar forms in geometric modeling and algebraic geometry. Topics in Algebraic Geometry and Geometric Modeling, AMS Contemporary Mathematics, 334:3-24,2003.
[33]R. Goldman. Some applications of resultants to problems in computational geometry. The Visual Computer,1:101-107,1985.
[34]R. Goldman, B. Hassett, and X. Shi. Finding singularities of parametrized space curves using μ-bases and resultants. in preparation.
[35]L. Gonzalez-Vega, I. Necula, S. Perez-Diaz, J. Sendra, and J. Sendra. Algebraic methods in computer aided geometric design:Theoretical and practical applications. Geometric Computation. Lecture Notes Series on Computing,11:1-33,2004.
[36]J. Harris. The genus of space curves. Mathematische Annalen,249:191-204,1980.
[37]R. Harshorne. Algebraic Geometry. Graduate Texts in Mathematics, Springer,1977.
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