Said-Ball曲线曲面的降阶逼近
|
摘要
本文主要讨论了Said-Ball曲线、曲面的降阶逼近，共分为四章。
第一章绪论介绍了本文所讨论问题的实际背景及研究的意义。
第二章介绍了本文的预备知识。作者首先介绍了Said-Ball奇函数的定义及其基本性质并通过Said-Ball曲线的升阶公式得到了Said-Ball曲线的降一阶的矩阵表示，通过对端点求导，得到了端点任意阶导数的一般表示式。
第三章作者给出了两种Said-Ball曲线降多阶方法，分别利用广义逆理论及两Said-Ball曲线在最小二乘范数下距离函数取最小值，将给定的n次Said-Ball曲线一次降为m次Said-Ball曲线。两种方法均考虑了不带端点插值条件和具有端点高阶插值条件的情形，并给出了降阶Said-Ball曲线控制顶点的显示表示式。
在第四章作者将Said-Ball曲线一次降n-m阶的两种方法推广到张量积Said-Ball曲面，得到了n×m阶张量积Said-Ball曲面一次降为n_1×m_1阶张量积Said-Ball曲面，在降阶过程中作者分别考虑了不带角点插值条件和具有角点插值条件的情形。
文中给出了数值实例，来显示所给出的两种方法的效果。
This paper is concerned about the degree reduction of the Said-Ball curves and surfaces. There are totally four chapters in this paper.
In Chapter one the author surveys the significances and background of the degree reduction techniques for curves and surfaces.
Chapter two is the preparation knowledge of this paper, which firstly covers the current definition of the Said-Ball curves as well as their properties, and secondly finds for the first time the matrix representation of such curves one degree lower based on the degree elevation formula of the Said-Ball curves, and finally offers the general formulae of derivatives of arbitrary orders at endpoints by differentiation.
In Chapter three the author offers two methods of multi-degree reduction of the Said-Ball curves, one of which is proposed by using the theory of generalized inverse matrix theory, another of which is proposed by minimizing L2 norm distance function between two Said-Ball curves, each of which reduces a given degree n Said-Ball curve to a degree m Said-Ball curve. In the process of degree reduction, two cases are considered respectively, one of which is about the case without conditions of endpoint interpolations, the other of which is about the case with conditions of endpoint interpolations of higher orders. The explicit formulas of control points of the reduced Said-Ball curves are obtained.
In the last chapter, the above two methods are generalized to the tensor product of the Said-Ball surfaces, each of which reduces a given degree nxm tensor product Said-Ball surface to an n1 xm1 one. In the process of degree reduction for the tensor product Said-Ball surfaces, we not only discuss the case with constraints of corner interpolations, but also the case without that.
In this paper, some numerical examples are given to demonstrate the effects of our methods.
第一章绪论介绍了本文所讨论问题的实际背景及研究的意义。
第二章介绍了本文的预备知识。作者首先介绍了Said-Ball奇函数的定义及其基本性质并通过Said-Ball曲线的升阶公式得到了Said-Ball曲线的降一阶的矩阵表示，通过对端点求导，得到了端点任意阶导数的一般表示式。
第三章作者给出了两种Said-Ball曲线降多阶方法，分别利用广义逆理论及两Said-Ball曲线在最小二乘范数下距离函数取最小值，将给定的n次Said-Ball曲线一次降为m次Said-Ball曲线。两种方法均考虑了不带端点插值条件和具有端点高阶插值条件的情形，并给出了降阶Said-Ball曲线控制顶点的显示表示式。
在第四章作者将Said-Ball曲线一次降n-m阶的两种方法推广到张量积Said-Ball曲面，得到了n×m阶张量积Said-Ball曲面一次降为n_1×m_1阶张量积Said-Ball曲面，在降阶过程中作者分别考虑了不带角点插值条件和具有角点插值条件的情形。
文中给出了数值实例，来显示所给出的两种方法的效果。
This paper is concerned about the degree reduction of the Said-Ball curves and surfaces. There are totally four chapters in this paper.
In Chapter one the author surveys the significances and background of the degree reduction techniques for curves and surfaces.
Chapter two is the preparation knowledge of this paper, which firstly covers the current definition of the Said-Ball curves as well as their properties, and secondly finds for the first time the matrix representation of such curves one degree lower based on the degree elevation formula of the Said-Ball curves, and finally offers the general formulae of derivatives of arbitrary orders at endpoints by differentiation.
In Chapter three the author offers two methods of multi-degree reduction of the Said-Ball curves, one of which is proposed by using the theory of generalized inverse matrix theory, another of which is proposed by minimizing L2 norm distance function between two Said-Ball curves, each of which reduces a given degree n Said-Ball curve to a degree m Said-Ball curve. In the process of degree reduction, two cases are considered respectively, one of which is about the case without conditions of endpoint interpolations, the other of which is about the case with conditions of endpoint interpolations of higher orders. The explicit formulas of control points of the reduced Said-Ball curves are obtained.
In the last chapter, the above two methods are generalized to the tensor product of the Said-Ball surfaces, each of which reduces a given degree nxm tensor product Said-Ball surface to an n1 xm1 one. In the process of degree reduction for the tensor product Said-Ball surfaces, we not only discuss the case with constraints of corner interpolations, but also the case without that.
In this paper, some numerical examples are given to demonstrate the effects of our methods.
引文
[1] Ball,A.A.,Consurf Par Ⅰ: Introduction of conic lofting title ,CAD,6(1974), 243 -249.
[2] Ball,A.A.,Consurf Par Ⅱ: Description of the algorithms, CAD,7 (1975), 237-242.
[3] Ball,A.A.,Consurf Par Ⅲ: How the program is used, CAD,9 (1977),9-12.
[4] 陈国栋，王国瑾．带端点插值条件的B(?)zier曲线降多阶逼近[J]．软件学报，2000，11(9)：1202—1206．
[5] Chen Guodong. The degree reduction and offset approximation in CAGD[D]. Hangzhou：Zhejiang University，2001(in chinese)(陈国栋。CAGD中的降阶变换和等距变换[博士学位论文]。杭州：浙江大学2001)
[6] Cheng Guo-dong, Wang Guo-jin..Multidegree reduction of tension product B(?)zier surfaces with conditions of comers interpolations. SCIENCE IN CHINA(Series F),2002,45(1): 51-58.
[7] Danneberg, L.,Nowacki, H.. Approximate conversion of surface representations with polynomial bases[J].Computer Aided Geometric Design, 1985,2(2): 123～131.
[8] 丁友东，李敏，广义Ball曲线的性质及其应用，应用数学学报，23(2000)，580-595。
[9] Eck, M.. Degree reduction of B(?)zier curves [J]. Computer Aided Geometric Design,1993,10(4):237～251.
[10] Falai Chen,Wenping Lou.Degree reduction of interval B(?)zier curves[J].CAD, 2000,32(3),571-582.
[11] Farin, G.. Algorithms for rational B(?)zier curves. Computer Aided Design, 1983,15(2): 73-77.
[12] Farin,G, NURBS for Curves and Surface Design, SIMA, Philadephia ,1991.
[13] Farin,G., Degree reduction faring of cubic B-Spline curves ,in :Bamhill, R.E.ed. Geometry Processing for Designing and Manufacturing, SIMA, Philadephia ,1992,3-34.
[14] Forrest, A.R. Interactive interpolation and approximation by B(?)zier curve. The Computer Journal, 1972,15(1):71-19.
[15] G.—D.Chen and G.—.J.Wang. Optimal multi-degree reduction of B(?)zier curves with constraints ofendpoints continuity[J].CAGD, 2002, 19(6): 365—377.
[16] Goodman, T.N.T. and Said ,H.B., Shape-preserving properties of generalized Ball basis,CAGD,8(1991 ), 115-121.
[17] Goodman, T.N.T. and Said ,H.B., Properties of generalized Ball basis and surfaces ,CAD,23 (1991), 554 - 560.
[18] Goodman, T.N.T. and Said ,H.B., Further properties of generalized Ball basis.Univ.of Dundee,Report AA/897,UK,1989.
[19] Guo Qingwei .The study of approximate conversion in CAGD. Hefei:University of Science and Technology of China,2003( in Chinese)(郭清伟。CAGD中近似转化的研究[博士学位论文]。合肥：中国科技大学2003)
[20] Graham, R.,Knuth,D., Patashnik,O.. Concrete Mathmatics, 3rd ed.. Rcading, Addison-Wesley, MA 1988.
[21] Hoschek, J.. Approximate conversion of spline curves [J]. Computer Aided Geometric Design,1987,4(1):59～66.
[22] Hu, S.M., and Jin, T.G, Degree reduetive approximation of B(?)zier curves, Proc. Symposium on Computational Geometry Hangzhou, China(1992), 110-126.
[23] Hu, S.M., Wang, G.J. and Jin, T.G, Degree reductive approximation of B(?)zier surface patches,Proe. CAD/Graphics'93 Beijing, China ,285-289.
[24] Hu,S.M., Wang, G.J. and Jin, T.G, Properties of two types of generalized Ball curves, CAD,28(1996), 125-133.
[25] 胡事民．CAD系统数据通讯中若干问题的研究．杭州：浙江大学博士学位论文．1996年4月．
[26] Hu Shimin, Sun Jiaguang, Jin Tongguang, Wang Guozhao. Approximate degree reduction of B(?)zier curves[J]. Tsinghua Science and Technology. 1998,3 (2):997～1000.
[27] 江平．广义Ball基函数的对偶基及其应用[博士学位论文]。合肥：合肥工业大学2003．
[28] 刘松涛，刘根洪．广义Ball样条曲线及三角域上曲面升阶公式和转换算法，应用数学学报，1996，19(2)：243-253．
[29] Lachance, M.A.. Chebyshev economization for parametric surfaces [J]. Computer Aided Geometric Design, 1988,5(3): 195～208
[30] Othman, W and Goldman, R.N.,The dual basis functions for the generalized Ball basis of odd degree, CAGD, 14(1997),571-582.
[31] Phien, H.N.,Dejumrong ,N..Efficient algorithms for B(?)zier curves. CAGD, 2000, 17(3):247-250.
[32] 孙红兵，陈效群．区间B(?)zier曲面的降阶[J]．中国科学技术大学学报．2002，32(20)，147-154。
[33] Said ,H.B.,Generalized Ball curve and its recursive algorithm ,ACM Transaction on Graphics, 8 (1989),360-371.
[34] 王国谨，汪国昭，郑建民．《计算机辅助几何设计》．高等教育出版社，施普林格
出版社，2001年7月：99-112；249-263．
[35] 王国谨，高次Ball曲线及其几何性质，高校应用数学学报，1987，2：126-140．
[36] Watkins,m.., Worsy, A.. Degree reduction for B(?)zier curves. Computer Aided Design,1988,20(7): 398-405.
[37] Zhou Dengwen, Liu Fang, Ju Tao. Sun Jiaguang. New method of approximate degree reduction of B(?)zier surface[J]. Journal of Computer Aided Design & Computer Graphics, 2002,14(6): 553～556。(in Chinese)(周登文，刘芳，居涛，孙家广。张量积B(?)zier曲面降阶逼近的新方法[J]。计算机辅助设计与图形学学报，2002，14(6)：553～556)
[38] 奚梅成，Ball基函数的对偶基及其应用．计算数学，1997．2：147-153．
[2] Ball,A.A.,Consurf Par Ⅱ: Description of the algorithms, CAD,7 (1975), 237-242.
[3] Ball,A.A.,Consurf Par Ⅲ: How the program is used, CAD,9 (1977),9-12.
[4] 陈国栋，王国瑾．带端点插值条件的B(?)zier曲线降多阶逼近[J]．软件学报，2000，11(9)：1202—1206．
[5] Chen Guodong. The degree reduction and offset approximation in CAGD[D]. Hangzhou：Zhejiang University，2001(in chinese)(陈国栋。CAGD中的降阶变换和等距变换[博士学位论文]。杭州：浙江大学2001)
[6] Cheng Guo-dong, Wang Guo-jin..Multidegree reduction of tension product B(?)zier surfaces with conditions of comers interpolations. SCIENCE IN CHINA(Series F),2002,45(1): 51-58.
[7] Danneberg, L.,Nowacki, H.. Approximate conversion of surface representations with polynomial bases[J].Computer Aided Geometric Design, 1985,2(2): 123～131.
[8] 丁友东，李敏，广义Ball曲线的性质及其应用，应用数学学报，23(2000)，580-595。
[9] Eck, M.. Degree reduction of B(?)zier curves [J]. Computer Aided Geometric Design,1993,10(4):237～251.
[10] Falai Chen,Wenping Lou.Degree reduction of interval B(?)zier curves[J].CAD, 2000,32(3),571-582.
[11] Farin, G.. Algorithms for rational B(?)zier curves. Computer Aided Design, 1983,15(2): 73-77.
[12] Farin,G, NURBS for Curves and Surface Design, SIMA, Philadephia ,1991.
[13] Farin,G., Degree reduction faring of cubic B-Spline curves ,in :Bamhill, R.E.ed. Geometry Processing for Designing and Manufacturing, SIMA, Philadephia ,1992,3-34.
[14] Forrest, A.R. Interactive interpolation and approximation by B(?)zier curve. The Computer Journal, 1972,15(1):71-19.
[15] G.—D.Chen and G.—.J.Wang. Optimal multi-degree reduction of B(?)zier curves with constraints ofendpoints continuity[J].CAGD, 2002, 19(6): 365—377.
[16] Goodman, T.N.T. and Said ,H.B., Shape-preserving properties of generalized Ball basis,CAGD,8(1991 ), 115-121.
[17] Goodman, T.N.T. and Said ,H.B., Properties of generalized Ball basis and surfaces ,CAD,23 (1991), 554 - 560.
[18] Goodman, T.N.T. and Said ,H.B., Further properties of generalized Ball basis.Univ.of Dundee,Report AA/897,UK,1989.
[19] Guo Qingwei .The study of approximate conversion in CAGD. Hefei:University of Science and Technology of China,2003( in Chinese)(郭清伟。CAGD中近似转化的研究[博士学位论文]。合肥：中国科技大学2003)
[20] Graham, R.,Knuth,D., Patashnik,O.. Concrete Mathmatics, 3rd ed.. Rcading, Addison-Wesley, MA 1988.
[21] Hoschek, J.. Approximate conversion of spline curves [J]. Computer Aided Geometric Design,1987,4(1):59～66.
[22] Hu, S.M., and Jin, T.G, Degree reduetive approximation of B(?)zier curves, Proc. Symposium on Computational Geometry Hangzhou, China(1992), 110-126.
[23] Hu, S.M., Wang, G.J. and Jin, T.G, Degree reductive approximation of B(?)zier surface patches,Proe. CAD/Graphics'93 Beijing, China ,285-289.
[24] Hu,S.M., Wang, G.J. and Jin, T.G, Properties of two types of generalized Ball curves, CAD,28(1996), 125-133.
[25] 胡事民．CAD系统数据通讯中若干问题的研究．杭州：浙江大学博士学位论文．1996年4月．
[26] Hu Shimin, Sun Jiaguang, Jin Tongguang, Wang Guozhao. Approximate degree reduction of B(?)zier curves[J]. Tsinghua Science and Technology. 1998,3 (2):997～1000.
[27] 江平．广义Ball基函数的对偶基及其应用[博士学位论文]。合肥：合肥工业大学2003．
[28] 刘松涛，刘根洪．广义Ball样条曲线及三角域上曲面升阶公式和转换算法，应用数学学报，1996，19(2)：243-253．
[29] Lachance, M.A.. Chebyshev economization for parametric surfaces [J]. Computer Aided Geometric Design, 1988,5(3): 195～208
[30] Othman, W and Goldman, R.N.,The dual basis functions for the generalized Ball basis of odd degree, CAGD, 14(1997),571-582.
[31] Phien, H.N.,Dejumrong ,N..Efficient algorithms for B(?)zier curves. CAGD, 2000, 17(3):247-250.
[32] 孙红兵，陈效群．区间B(?)zier曲面的降阶[J]．中国科学技术大学学报．2002，32(20)，147-154。
[33] Said ,H.B.,Generalized Ball curve and its recursive algorithm ,ACM Transaction on Graphics, 8 (1989),360-371.
[34] 王国谨，汪国昭，郑建民．《计算机辅助几何设计》．高等教育出版社，施普林格
出版社，2001年7月：99-112；249-263．
[35] 王国谨，高次Ball曲线及其几何性质，高校应用数学学报，1987，2：126-140．
[36] Watkins,m.., Worsy, A.. Degree reduction for B(?)zier curves. Computer Aided Design,1988,20(7): 398-405.
[37] Zhou Dengwen, Liu Fang, Ju Tao. Sun Jiaguang. New method of approximate degree reduction of B(?)zier surface[J]. Journal of Computer Aided Design & Computer Graphics, 2002,14(6): 553～556。(in Chinese)(周登文，刘芳，居涛，孙家广。张量积B(?)zier曲面降阶逼近的新方法[J]。计算机辅助设计与图形学学报，2002，14(6)：553～556)
[38] 奚梅成，Ball基函数的对偶基及其应用．计算数学，1997．2：147-153．