任意阶均匀拟Beta样条方法
|
摘要
拟Beta样条函数是基于ECT组E = {1, x , x~2 ,......, x~(k-3) , x~(k-2),φ( x ), -( x )}( k≥2)在每个节点处由一个关联矩阵连接而产生的,若每个关联矩阵都是非奇异、下三角、全正的矩阵,则存在非负的、具有最小支撑基和归一的拟Beta样条函数.拟Beta样条函数有许多类似于代数多项式B样条的性质.本文着重研究一般均匀拟Beta样条的显式表示和性质.主要研究内容和完成结果如下:
一、首先由空间? = spanE引出拟Beta样条空间S (U , A, X ),并给出它的一组样条基的显式表示,同时证明了样条基的一些重要性质;最后证明了样条空间S (U , A, X )的均匀插值问题的解的存在唯一性.
二、通过适当地取φ( x ), -φ( x),给出了四种特殊的均匀拟Beta样条的显示表示,它们是均匀分划下的多项式Beta样条;代数三角Beta样条;代数双曲Beta样条和有理Beta样条.当关联矩阵为单位矩阵时,它们又分别成为均匀分划下的多项式B样条,代数三角B样条,代数双曲B样条和有理B样条.
三、以均匀拟Beta样条函数为调配函数,构造了一类拟Beta样条曲线和曲面,并导出了它们的一些性质.
四、提出两种均匀拟Beta样条曲线的节点插入算法:1、基于递归公式的节点插入算法;2、基于广义Pólya多项式的节点插入算法.
Beta-like spline functions are generated from one ECT system which is shifted by integer translation via one connection matrix. If the matrix is nonsingular, lower triangular and totally positive, there exists Beta-like splines having minimal compact supports. The Beta-like splines share most properties of polynomial B splines. In this paper, the uniform Beta-like splines are only discussed .From this a new kind of curves and surfaces are constructed. The research achievements and main contents are as follows:
1. The uniform Beta-like spline space S (U , A, X ) is set up from the space (?) = spanE. The uniform Beta-like spline basis on the space S (U , A, X )is explicitly represented by extending the cardinal ECT B splines to the case for equal knots, from which some important properties of the basis are derived and the existence and uniqueness of the uniform interpolating functions on the space S (U , A, X ) are proved as well.
2. By taking the different four pairs of functionsφ( x ), ? ( x), the uniform Beta-like splines above become the polynomial Beta splines, the algebraic trigonometric Beta splines, the algebraic hyperbolic Beta splines and the rational polynomial Beta splines for equal knots respectively, which again reduce to the polynomial B , the algebraic hyperbolic B and the rational polynomial B splines for equal nodes respectively when each connection matrix is identity matrix. Their explicit representations are given in the paper, in particular for the cases of the splines of low order.
3. The generation of a kind of curves and tensor product surfaces by the new splines are given, and some their properties are shown in the paper.
4. Two algorithms for the knot insertion into the uniform Beta-like spline curves are provided. One is based on recursive formulas,The other is from the generalized Pólya polynomials.
一、首先由空间? = spanE引出拟Beta样条空间S (U , A, X ),并给出它的一组样条基的显式表示,同时证明了样条基的一些重要性质;最后证明了样条空间S (U , A, X )的均匀插值问题的解的存在唯一性.
二、通过适当地取φ( x ), -φ( x),给出了四种特殊的均匀拟Beta样条的显示表示,它们是均匀分划下的多项式Beta样条;代数三角Beta样条;代数双曲Beta样条和有理Beta样条.当关联矩阵为单位矩阵时,它们又分别成为均匀分划下的多项式B样条,代数三角B样条,代数双曲B样条和有理B样条.
三、以均匀拟Beta样条函数为调配函数,构造了一类拟Beta样条曲线和曲面,并导出了它们的一些性质.
四、提出两种均匀拟Beta样条曲线的节点插入算法:1、基于递归公式的节点插入算法;2、基于广义Pólya多项式的节点插入算法.
Beta-like spline functions are generated from one ECT system which is shifted by integer translation via one connection matrix. If the matrix is nonsingular, lower triangular and totally positive, there exists Beta-like splines having minimal compact supports. The Beta-like splines share most properties of polynomial B splines. In this paper, the uniform Beta-like splines are only discussed .From this a new kind of curves and surfaces are constructed. The research achievements and main contents are as follows:
1. The uniform Beta-like spline space S (U , A, X ) is set up from the space (?) = spanE. The uniform Beta-like spline basis on the space S (U , A, X )is explicitly represented by extending the cardinal ECT B splines to the case for equal knots, from which some important properties of the basis are derived and the existence and uniqueness of the uniform interpolating functions on the space S (U , A, X ) are proved as well.
2. By taking the different four pairs of functionsφ( x ), ? ( x), the uniform Beta-like splines above become the polynomial Beta splines, the algebraic trigonometric Beta splines, the algebraic hyperbolic Beta splines and the rational polynomial Beta splines for equal knots respectively, which again reduce to the polynomial B , the algebraic hyperbolic B and the rational polynomial B splines for equal nodes respectively when each connection matrix is identity matrix. Their explicit representations are given in the paper, in particular for the cases of the splines of low order.
3. The generation of a kind of curves and tensor product surfaces by the new splines are given, and some their properties are shown in the paper.
4. Two algorithms for the knot insertion into the uniform Beta-like spline curves are provided. One is based on recursive formulas,The other is from the generalized Pólya polynomials.
引文
[1] Piehl L, Tiller W. The NURBS Book.2nd ed.Berlin:Springer,1997
[2] Zhang J W. C-curves: an extension of cubic curves.CAGD,1996,13:199-127
[3] Zhang J W. Two different forms of C-B-splines.CAGD,1997,14:31-41
[4] Zhang J W. C-B é zier Curves and Surfaces. Graphical Models and Image Processing,1999,61:2-15
[5] Pottmann H, Wagner M G. Helix splines as example of affine Tchebycheffian splines. Adv.Comput. Math,1994,2:123-142
[6] Peńa J M. Shape preserving representations for trigonometric polynomial curves. CAGD,1997,14:5-11
[7] Sanchez-Reyes J. Harmonic rational Bézier curves:p-Bézier curves and trigonometric polyn-omial.CAGD,1998,15:909-923
[8] Mainar E,Peńa J M, Sanchez-Reyes J. Shape preserving alternatives to the rational Bézier model. CAGD,2001,18:37-60
[9] Mainar E, Peńa J M.A basis of C-Bézier splines with optimal properties. CAGD,2002,19:291-295
[10] Chen Q Y , Wang G Z.A class of Bézier-like curves. CAGD,2003,20:29-39
[11] Lü Y G, Wang G Z, Yang X N. Uniform hyperbolic polynomial B-spline curves. CAGD,2002,19:379-393
[12]吕勇刚,汪国昭,杨勋年.均匀三角多项式 B 样条曲线.中国科学(E 辑),2002,32:281-288
[13] Wang G Z , Chen Q Y , Zhou M H. NUAT B-spline curves.CAGD,2004,21: 193-205
[14]钱江.NUAH B 样条方法及其应用研究.南京航空航天大学硕士学位论文,南京,南京航空航天大学,2005
[15] 陈文喻,汪国昭.代数三角混合的样条曲线.计算机研究与发展,2006,43:679-687
[16]单 开 佳 , 汪 国 昭 . 低 阶 C-Bézier 曲 线 的 升 阶 . 高 校 应 用 数 学 学 报 A辑,2002,17(4):441-445
[17] 陈秦玉,杨勋年,汪国昭.四次 C-曲线的性质及其应用.高校应用数学学报 A 辑,2003, 18(1):45-50
[18]陈秦玉,汪国昭.圆弧的 C-Bézier 曲线表示.软件学报,2002,13(11):2155-2161
[19]樊建华,邬义杰,林兴.C-Bézier 曲线分割算法及 G 1拼接条件.计算机辅助设计与图形学报,2002,14(5):421-424
[20]樊建华,张纪文,邬义杰.C-Bézier曲线的形状修改.软件学报,2002,13(11): 2194-2200
[21] 徐松.CAGD 中C-Bézier 曲线上曲面可展性研究.南京航空航天大学硕士学位论文,南京,南京航空航天大学,2004
[22] Yang Q M, Wang G Z. Inflection points and singularities on C-curve. CAGD 2004, 21:207-213
[23] 何玲娜,汪国昭.五次C-曲线的细分.浙江大学学报(理学版),2004,31(2):125-129
[24]赵玉林.高阶 C-Bézier 曲线曲面性质研究及其应用.南京航空航天大学硕士学位论文, 南京,南京航空航天大学,2005
[25] Barsky B. Computer Graphics and Geometric Modeling Using Beta-Splines. New York : Spring-Verlag ,1988
[26] Barry P J, Dyn N, Goldman R N, Micchelli C A. Identities for piecewise polynomial spaces determined by connection matrices. Equations Math,1991,42:123–136
[27] Dyn N, Micchelli C A. Piecewise polynomial spaces and geometric continuity of curves. Numer.Math,1988,54 :319-337
[28] Dyn N, Edelman A, Micchelli C A. On Locally Supported Basic Functions for the Representation of Geometrically Continuous Curves .Analysis,1987,7:313-341
[29] Gasca M, Micchelli C A. Total Positivity and its Applications. Dordrecht, Kluwer Academic Publishers, 1996
[30] Micchelli C A. Mathematical Aspects of Geometric Modeling. Philadelphia,SIAM, 1995:55-147
[31] Barsky B A, Derose T D. Geometric continuity of parametric curves: constructions of geometrically continuous splines.IEEE CG & A, 1990,10(1):60-68
[32] Tang Y H, Mühlbach G. Cardinal ECT-splines. Numerical Algorithems,2005, 38:259-283
[33] 蒋春娟.ECT B 样条曲线的节点插入算法.南京航空航天大学硕士学位论文,南京,南京航空航天大学,2006
[34] Karlin S, Studden W J. Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience Publishers, 1966
[35] Barry P J. De Boor-Fix dual functionals and algorithms for Tchebychevian B-splines curves.Construct. Approx.,1996,12:385-408
[36] M ü hlbach G. One sided Hermite interpolation by piecewise different generalized polynomials. Journal of Computational and Applied Mathematics, 2006,196:285-298
[37] M ü hlbach G. Hermite interpolation by piecewise different generalized polynomials. Adv.Comput. Math, to appear
[38] Mühlbach G. ECT B-splines defined by generalized divided differences. Adv.Comput. Math, 2006, 187:96–122
[39] Mühlbach G, Tang Y H. Computing ECT B-splines recursively. Numerical Algorithms,2006, 41:35-78
[40] Schumaker L L. Spline Functions: Basic Theory. New York: Wiley Interscience ,1981
[41] 王国瑾,汪国昭,郑建民.计算机辅助几何设计.北京:高等教育出版社,2001
[42] 苏步青,刘鼎元.计算几何.上海:上海科学技术出版社,1981
[43]施法中.计算机辅助几何设计与非均匀有理 B 样条.北京:北京航空航天大学出版社, 1994
[44]孙家昶.样条函数与计算几何.北京:科学出版社,1982
[2] Zhang J W. C-curves: an extension of cubic curves.CAGD,1996,13:199-127
[3] Zhang J W. Two different forms of C-B-splines.CAGD,1997,14:31-41
[4] Zhang J W. C-B é zier Curves and Surfaces. Graphical Models and Image Processing,1999,61:2-15
[5] Pottmann H, Wagner M G. Helix splines as example of affine Tchebycheffian splines. Adv.Comput. Math,1994,2:123-142
[6] Peńa J M. Shape preserving representations for trigonometric polynomial curves. CAGD,1997,14:5-11
[7] Sanchez-Reyes J. Harmonic rational Bézier curves:p-Bézier curves and trigonometric polyn-omial.CAGD,1998,15:909-923
[8] Mainar E,Peńa J M, Sanchez-Reyes J. Shape preserving alternatives to the rational Bézier model. CAGD,2001,18:37-60
[9] Mainar E, Peńa J M.A basis of C-Bézier splines with optimal properties. CAGD,2002,19:291-295
[10] Chen Q Y , Wang G Z.A class of Bézier-like curves. CAGD,2003,20:29-39
[11] Lü Y G, Wang G Z, Yang X N. Uniform hyperbolic polynomial B-spline curves. CAGD,2002,19:379-393
[12]吕勇刚,汪国昭,杨勋年.均匀三角多项式 B 样条曲线.中国科学(E 辑),2002,32:281-288
[13] Wang G Z , Chen Q Y , Zhou M H. NUAT B-spline curves.CAGD,2004,21: 193-205
[14]钱江.NUAH B 样条方法及其应用研究.南京航空航天大学硕士学位论文,南京,南京航空航天大学,2005
[15] 陈文喻,汪国昭.代数三角混合的样条曲线.计算机研究与发展,2006,43:679-687
[16]单 开 佳 , 汪 国 昭 . 低 阶 C-Bézier 曲 线 的 升 阶 . 高 校 应 用 数 学 学 报 A辑,2002,17(4):441-445
[17] 陈秦玉,杨勋年,汪国昭.四次 C-曲线的性质及其应用.高校应用数学学报 A 辑,2003, 18(1):45-50
[18]陈秦玉,汪国昭.圆弧的 C-Bézier 曲线表示.软件学报,2002,13(11):2155-2161
[19]樊建华,邬义杰,林兴.C-Bézier 曲线分割算法及 G 1拼接条件.计算机辅助设计与图形学报,2002,14(5):421-424
[20]樊建华,张纪文,邬义杰.C-Bézier曲线的形状修改.软件学报,2002,13(11): 2194-2200
[21] 徐松.CAGD 中C-Bézier 曲线上曲面可展性研究.南京航空航天大学硕士学位论文,南京,南京航空航天大学,2004
[22] Yang Q M, Wang G Z. Inflection points and singularities on C-curve. CAGD 2004, 21:207-213
[23] 何玲娜,汪国昭.五次C-曲线的细分.浙江大学学报(理学版),2004,31(2):125-129
[24]赵玉林.高阶 C-Bézier 曲线曲面性质研究及其应用.南京航空航天大学硕士学位论文, 南京,南京航空航天大学,2005
[25] Barsky B. Computer Graphics and Geometric Modeling Using Beta-Splines. New York : Spring-Verlag ,1988
[26] Barry P J, Dyn N, Goldman R N, Micchelli C A. Identities for piecewise polynomial spaces determined by connection matrices. Equations Math,1991,42:123–136
[27] Dyn N, Micchelli C A. Piecewise polynomial spaces and geometric continuity of curves. Numer.Math,1988,54 :319-337
[28] Dyn N, Edelman A, Micchelli C A. On Locally Supported Basic Functions for the Representation of Geometrically Continuous Curves .Analysis,1987,7:313-341
[29] Gasca M, Micchelli C A. Total Positivity and its Applications. Dordrecht, Kluwer Academic Publishers, 1996
[30] Micchelli C A. Mathematical Aspects of Geometric Modeling. Philadelphia,SIAM, 1995:55-147
[31] Barsky B A, Derose T D. Geometric continuity of parametric curves: constructions of geometrically continuous splines.IEEE CG & A, 1990,10(1):60-68
[32] Tang Y H, Mühlbach G. Cardinal ECT-splines. Numerical Algorithems,2005, 38:259-283
[33] 蒋春娟.ECT B 样条曲线的节点插入算法.南京航空航天大学硕士学位论文,南京,南京航空航天大学,2006
[34] Karlin S, Studden W J. Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience Publishers, 1966
[35] Barry P J. De Boor-Fix dual functionals and algorithms for Tchebychevian B-splines curves.Construct. Approx.,1996,12:385-408
[36] M ü hlbach G. One sided Hermite interpolation by piecewise different generalized polynomials. Journal of Computational and Applied Mathematics, 2006,196:285-298
[37] M ü hlbach G. Hermite interpolation by piecewise different generalized polynomials. Adv.Comput. Math, to appear
[38] Mühlbach G. ECT B-splines defined by generalized divided differences. Adv.Comput. Math, 2006, 187:96–122
[39] Mühlbach G, Tang Y H. Computing ECT B-splines recursively. Numerical Algorithms,2006, 41:35-78
[40] Schumaker L L. Spline Functions: Basic Theory. New York: Wiley Interscience ,1981
[41] 王国瑾,汪国昭,郑建民.计算机辅助几何设计.北京:高等教育出版社,2001
[42] 苏步青,刘鼎元.计算几何.上海:上海科学技术出版社,1981
[43]施法中.计算机辅助几何设计与非均匀有理 B 样条.北京:北京航空航天大学出版社, 1994
[44]孙家昶.样条函数与计算几何.北京:科学出版社,1982