非代数多项式空间曲线性质的研究
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摘要
为了克服代数多项式空间曲线曲面造型的不足,很多学者提出了基于非代数多项式空间的其它形式的曲线曲面造型方法。本文在他们研究的基础上,主要做了以下工作:
第一,在研究四次C-曲线性质的基础上,讨论了与给定多边形相切的分段四次C-Bézier曲线和四次C-B样条闭曲线和开曲线。所构造的C-Bézier曲线是C 1连续的,且对切线多边形是保形的。四次C-B样条闭曲线和开曲线是C 3连续的,且对切线多边形也是保形的。所构造曲线段的控制顶点由切线多边形的顶点直接计算产生。最后以实例表明,本文的方法是有效的。
第二,在讨论三次H-Bézier曲线性质的基础上,提出了三次H-Bézier曲线的任意分割算法,即对三次H-Bézier曲线上任意一点,求该点把曲线分成的两个子曲线段的控制参数和控制顶点;给出了三次H-Bézier曲线与三次Bézier曲线的拼接条件,以及三次H-Bézier曲线在曲面造型中应用的例子。采用本文方法所得结果简单、直观,有效地增强了三次H-Bézier方法控制及表达曲线形状的能力。
第三,给出了五次H-Bézier曲线的细分公式,并且证明了细分过程产生的控制多边形序列收敛于原曲线。在收敛性的基础上,证明了五次H-Bézier曲线的两个重要性质:保凸性和变差缩减性。
In order to overcome the shortcomings of curves and surfaces modeling in algebraic polynomial space, many scholars propose other forms of curves and surfaces in the non-algebraic polynomial space. Based on the study of the scholars, this thesis does some study as follows:
Firstly, through analysis of the properties of quartic C-curves, we present an approach of constructing planar piecewise quartic C-Bézier curves and quartic C-B spline curves with all edges tangent to a given control polygon. The C-Bézier curve segments are joined together with C 1continuity and the Quartic C-B spline closed curves and open curves are C 3 continuous. All curves are shape preserving to their tangent polygons. All control points of the curve segments can be calculated simply by the vertices of the given tangent polygon. Finally some numerical examples illustrate that the method given in this paper is effective.
Secondly, based on the analysis of the properties of cubic H-Bézier curves, a subdivision algorithm is proposed, to compute the control parameters and control points of the two subcurves subdivided by any point of cubic H-Bézier curves. The connection conditions between cubic H-Bézier curves and cubic Bézier curves are derived and the applications of cubic H-Bézier curves in the surface modeling are given. The obtained results, which are simple and intuitionistic, can effectively improve the shape representation and control of cubic H-Bézier curves.
Thirdly, an effective subdivision formula for H-Bézier curves of degree five is presented. Furthermore, it is proved that the control polygons generated by the subdivision converge to the original H-Bézier curves of degree five. Two important properties, the variation diminishing (V-D) property and convexity preserving property, are proved for H-Bézier curves of degree five.
第一,在研究四次C-曲线性质的基础上,讨论了与给定多边形相切的分段四次C-Bézier曲线和四次C-B样条闭曲线和开曲线。所构造的C-Bézier曲线是C 1连续的,且对切线多边形是保形的。四次C-B样条闭曲线和开曲线是C 3连续的,且对切线多边形也是保形的。所构造曲线段的控制顶点由切线多边形的顶点直接计算产生。最后以实例表明,本文的方法是有效的。
第二,在讨论三次H-Bézier曲线性质的基础上,提出了三次H-Bézier曲线的任意分割算法,即对三次H-Bézier曲线上任意一点,求该点把曲线分成的两个子曲线段的控制参数和控制顶点;给出了三次H-Bézier曲线与三次Bézier曲线的拼接条件,以及三次H-Bézier曲线在曲面造型中应用的例子。采用本文方法所得结果简单、直观,有效地增强了三次H-Bézier方法控制及表达曲线形状的能力。
第三,给出了五次H-Bézier曲线的细分公式,并且证明了细分过程产生的控制多边形序列收敛于原曲线。在收敛性的基础上,证明了五次H-Bézier曲线的两个重要性质:保凸性和变差缩减性。
In order to overcome the shortcomings of curves and surfaces modeling in algebraic polynomial space, many scholars propose other forms of curves and surfaces in the non-algebraic polynomial space. Based on the study of the scholars, this thesis does some study as follows:
Firstly, through analysis of the properties of quartic C-curves, we present an approach of constructing planar piecewise quartic C-Bézier curves and quartic C-B spline curves with all edges tangent to a given control polygon. The C-Bézier curve segments are joined together with C 1continuity and the Quartic C-B spline closed curves and open curves are C 3 continuous. All curves are shape preserving to their tangent polygons. All control points of the curve segments can be calculated simply by the vertices of the given tangent polygon. Finally some numerical examples illustrate that the method given in this paper is effective.
Secondly, based on the analysis of the properties of cubic H-Bézier curves, a subdivision algorithm is proposed, to compute the control parameters and control points of the two subcurves subdivided by any point of cubic H-Bézier curves. The connection conditions between cubic H-Bézier curves and cubic Bézier curves are derived and the applications of cubic H-Bézier curves in the surface modeling are given. The obtained results, which are simple and intuitionistic, can effectively improve the shape representation and control of cubic H-Bézier curves.
Thirdly, an effective subdivision formula for H-Bézier curves of degree five is presented. Furthermore, it is proved that the control polygons generated by the subdivision converge to the original H-Bézier curves of degree five. Two important properties, the variation diminishing (V-D) property and convexity preserving property, are proved for H-Bézier curves of degree five.
引文
[1] Ferguson J., Multivariable curve interpolation. Report D2-22504, The Boeing Co. Seattle, Washington, 1963.
[2] Ferguson J., Multivariable curve interpolation. Journal of the ACM, 1964, 11(2): 221-228.
[3] Coons, S.A., Surfaces for computer aided design of space Figures. MIT Project MAC-TR-255, June, 1964.
[4] Coons, S.A., Surfaces for computer aided design of space forms, MIT Project MAC-TR-41, June, 1967.
[5] Bézier, P., Numerical Control: Mathematics and Applications. John Wiley and Sons, New York, 1972.
[6] Bézier, P., Mathematical and practical possibilities of UNISURF. Computer Aided Geometric Design, in: Barnhill,R.E. and Riesenfield, R.E., eds., Academic Press, New York, 1974: 127-152.
[7] Bézier, P., The Mathematical basis of the UNISURF CAD system. Butterworths, England, 1986.
[8] Forrest, A.R., Interactive interpolation and approximation by Bézier curve. The Computer Journal, 1972 (15): 71-79.
[9] Farin, G.E., Bézier polynomials over triangles and the construction of piecewise C r polynomials. TR/91, Department of Mathematics, Brunel University, Uxbridge, U.K., 1980.
[10] Farin, G.E., Triangular Bernstein- Bézier patches. Computer Aided Geometric Design, 1986, 3(2): 83-127.
[11] Gordon, W.J. and Riesenfeld, R.F., Bernstein Bézier methods for the Computer Aided Geometric Design of free-form curves and surfaces. Journal of ACM, 1974 (21): 293-310.
[12] Farin, G., Curves and Surfaces for Computer Aided Geometric Design. Second Edition, Academic Press, 1990.
[13] Gordon, W.J. and Riesenfeld, R.F., B-spline curves and surfaces. in: Barnhill, R.E. and Riesenfeld, R.F. eds., Computer Aided Geometric Design, Academic Press, New York, 1974: 95-126.
[14] Cox M., The numerical evaluation of B-spline. Journal of Applied Mathematics , 1972(10): 134-149.
[15] de Boor, C., On Calculation with B-spline. Journal of Approximation Theory, 1972,6(1): 50-62.
[16] Boehm W., Inserting new knots into B-spline curve. Computer Aided Design, 1980, 12(4): 199-201.
[17] Cohen, E., Lyche, T., Riesenfeld, R.F., Discrete B-spline and subdivision techniques in computer aided geometric design and computer graphics. Computer Graphics and Image Processing, 1980, 14(2): 87-111.
[18] Prautzsch, H. and Piper, B., A fast algorithm to raise the degree of spline curves. Computer Aided Geometric Design, 1991, 8(4): 253-265.
[19] Rowin, M.S., Conic, Cubic and T-conic segments. D2-23252, The Beoing Company, Seattle, WA98124, 1964.
[20] Ball, A.A., ConSurf, Part 1: Introduction to conic lofting title. Computer Aided Design, 1974 (6): 243-249.
[21] Forrest, A.R., The twisted cubic curve: A computer aided geometric design approach. Computer Aided Design, 1980 (12): 165-172.
[22] Farin, G., Algorithms for rational Bézier curves. Computer Aided Design, 1983 (15): 73-79.
[23] Pigel, L., Representation of rational Bézier curves. Computer Aided Geometric Design, 1985 (2): 151-155.
[24]刘鼎元,有理Bézier曲线.应用数学学报, 1985 (8): 70-82.
[25]汪国昭,沈金福,有理Bézier曲线的离散和几何性质.浙江大学学报, 1985 (19): 123-130.
[26] Vergeest, S.M., CAD Surface data exchange using STEP. Computer Aided Design, 1991, 23(4): 269-281.
[27] Versprille, K.J., Computer aided design application of rational B-spline approximation form. Ph.D. thesis, Syracuse University, Syracuse, N.Y., 1975.
[28] Piegl, L. and Tiller, W., Curve and surface constructions using rational B-splines. Computer Aided Design, 1987 (19): 485-498.
[29] Piegl, L. and Tiller, W., A menagerie of rational B-spline circles. IEEEE Computer Graphics and its Application, 1989 (9): 48-56.
[30] Piegl, L., Modifying the shape of rational B-splines, part 1:curves. Computer Aided Design, 1989 (21): 509-518.
[31] Piegl, L., Modifying the shape of rational B-splines, part 2:surfaces. Computer Aided Design, 1989 (21): 538-546.
[32] Piegl, L., On NURBS: a survey. IEEE Computer Graphics and Application, 1991 (10): 55-71.
[33] Tiller, W., Rational B-splines for curve and surface representation. IEEE ComputerGraphics and its Application, 1983 (3): 61-69.
[34] Tiller, W., Geometric modeling using non-uniform rational B-splines: mathematical techniques. SIGGRAH tutorial notes, ACM, New York, 1986.
[35] Tiller, W., Knot-remove algorithms for NURBS curves and surfaces. Computer Aided Design, 1992 (24): 445-453.
[36] Choi, B.K., Yoo, W.S. and Lee, C.S., Matrix representation for NURBS curves and surfaces. Computer Aided Design, 1990 (22): 637-642.
[37] Grabowski, H. and Li, X., Coefficient formula and matrix of nonuniform B-spline functions. Computer Aided Design, 1992 (24): 637-642.
[38] Pottmann H., Wagner M.G., Helix splines as example of affine Tchebycheffian splines. Advances in Computational Mathematics, 1994, 2: 123-142.
[39] Zhang Jiwen, C-curves: an extension of cubic curves. CAGD, 1996, 13: 199-217.
[40] Zhang Jiwen, C-Bézier Curves and Surfaces. Graphical Models and Image Processing, 1999, 61: 2-15.
[41]张纪文,罗国明, C曲线及其性质.计算机辅助工程,1996,2: 53-61.
[42]张纪文,罗国明,三次样条曲线的拓广-C曲线.计算机辅助工程,1996,3: 12-20.
[43] Jiwen Zhang, Two different forms of C-B-splines. CAGD, 1997, 14: 31-41.
[44] Peńa J.M., Shape preserving representations for trigonometric polynomial curves. CAGD, 1999, 14: 5-11.
[45] Sánchez-Reyes J., Harmonic rational Bézier curves, p- Bézier curves and trigonometric polynomial. CAGD, 1998, 15: 909-923.
[46] Mainar E., Peńa J M., Sánchez-Reyes J., Shape preserving alternatives to the rational Bézier model. CAGD, 2001, 18: 37-60.
[47] Chen Qinyu, Wang Guozhao, A class of Bézier-like curves. CAGD, 2003, 20: 29-39.
[48] Wang Guozhao, Chen Qinyu, Zhou Minghua, NUATB B-spline curves. CAGD, 2004, 21: 193-205.
[49]王成伟,带有给定切线多边形的C 2连续的C-B样条曲线.计算机辅助设计与图形学学报,2001,13(12): 1133-1136.
[50]王成伟,带有给定切线多边形的C-Bézier闭曲线和B-型样条闭曲线.数值计算与计算机应用,2002,4: 303-309.
[51]单开佳,汪国昭,低阶C- Bézier曲线的升阶.高等应用数学学报A辑,2002,17(4) : 441-445.
[52]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学报,2002,14(5): 421-424.
[53]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.
[54]樊建华,罗国明,王卫民, C- Bézier曲面分割、拼接及其应用.工程图学学报,2002,3: 133-138.
[55]樊建华,罗国明,林兴,王卫民, C- Bézier构造工程曲面.机械工程学报,2003,39(1): 89-92.
[56]樊建华, C- Bézier曲线曲面造型理论及算法研究.浙江大学硕士学位论文,2001.
[57]陈秦玉,汪国昭,圆弧的C- Bézier曲线表示.软件学报,2002,13(11): 2155-2161.
[58]陈秦玉,杨勋年,汪国昭,四次C-曲线的性质及其应用.高等应用数学学报A辑,2003,18(1): 45-50.
[59]金义明,五次C-曲线的生成及其性质.浙江大学硕士学位论文,2001.
[60]金义明,汪国昭,五次C-Bézier曲线的生成及其性质.高校应用数学学报A辑,2005,20(1): 91-96.
[61]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,2004.
[62] Yang Qinmin, Wang Guozhao, Inflection points and singularities on C-curve. CAGD, 2004, 21: 207-213.
[63]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.
[64]汪仁泰,陈建兰,空间曲线的C-Bézier插值.浙江工业大学学报2004,32(2): 225-233.
[65]王刘强,刘旭敏, C-曲线间G 1拼接条件及在曲面造型中的应用.计算机工程与应用,2004,43(18): 45-46.
[66]林上华,汪国昭, C-曲线定义区间的扩展.计算机辅助设计与图形学学报,2005,17(10): 2281-2285.
[67]叶正麟,吴荣军,平面C-Bézier曲线的奇拐点分析.计算数学,2005,27(1): 63-70.
[68]戴中寅,杨松林, C- Bézier曲线与NURBS曲线的光滑拼接.江苏大学学报,2005,26(6A): 95-97.
[69]赵玉林,高阶C- Bézier曲线曲面性质研究及其应用.浙江大学硕士学位论文,2005.
[70]吴荣军,叶正麟,罗卫民,有理C-Bézier曲线的形状分析.计算机学报,2007,30(11): 2055-2059.
[71]周贵宾, C曲线和H曲线的Path.浙江大学硕士学位论文,2007.
[72]林新辉, C-Bézier曲线降阶逼近.浙江大学硕士学位论文,2007.
[73] Lü, Y G, Uniform hyperbolic polynomial B-spline curves. CAGD, 2002, 19: 379-393.
[74] Wang Guozhao, Yang Qinmin, Planar cubic hybrid hyperbolic polynomial curve and its shape classification. Progress in Natural Science, 2004, 14(1): 41-46.
[75]杨勤民,三次混合多项式曲线和区间曲面的研究.浙江大学硕士学位论文,2002.
[76]王媛,康宝生,代数双曲混合H-Bézier函数及其性质.西北大学学报,2006, 36(5): 693-697.
[77]王媛, H-Bézier曲线的理论及应用研究.西北大学硕士学位论文,2006.
[78]胡晴峰,双曲混合多项式曲线及其性质.浙江大学硕士学位论文,2003.
[79]钱江,唐月红, H Bézier-Like曲线在工程中的应用.数值计算与计算机应用,2007,28(3): 167-178.
[80]吴荣军,平面三次H-Bézier曲线的形状分析.应用数学学报, 2007, 30(5) : 816-821.
[81] Hering L., Closed( C 2- and C 3-Continuous) Bézier and B-spline curve with given tangent polygons. CAD, 1983, 15(1): 3-6.
[82]方逵,与给定的多边形相切的闭( G 2-连续)Bézier曲线.计算数学,1991,13(2): 34-37.
[83]方逵,蔡放,谭建荣,带有给定切线多边形的C 2和C 3连续Bézier闭样条曲线.计算机辅助设计与图形学学报,2000,12(5): 330-332.
[84]施法中,计算机辅助几何设计与非均匀有理B样条.北京:高等教育出版社,2001: 92-100.
[85]姜卫,服装CAD.上海:中国纺织大学出版社,1999.
[86] Pottmann H., The geometry of Tchebycheffian spline. Computer Aided Geometric Design, 1993, 10(2): 181-210.
[87] Mazure M.L., Chebyshev-Bernstein bases. Computer Aided Geometric Design, 1999, 16(7): 649-669.
[2] Ferguson J., Multivariable curve interpolation. Journal of the ACM, 1964, 11(2): 221-228.
[3] Coons, S.A., Surfaces for computer aided design of space Figures. MIT Project MAC-TR-255, June, 1964.
[4] Coons, S.A., Surfaces for computer aided design of space forms, MIT Project MAC-TR-41, June, 1967.
[5] Bézier, P., Numerical Control: Mathematics and Applications. John Wiley and Sons, New York, 1972.
[6] Bézier, P., Mathematical and practical possibilities of UNISURF. Computer Aided Geometric Design, in: Barnhill,R.E. and Riesenfield, R.E., eds., Academic Press, New York, 1974: 127-152.
[7] Bézier, P., The Mathematical basis of the UNISURF CAD system. Butterworths, England, 1986.
[8] Forrest, A.R., Interactive interpolation and approximation by Bézier curve. The Computer Journal, 1972 (15): 71-79.
[9] Farin, G.E., Bézier polynomials over triangles and the construction of piecewise C r polynomials. TR/91, Department of Mathematics, Brunel University, Uxbridge, U.K., 1980.
[10] Farin, G.E., Triangular Bernstein- Bézier patches. Computer Aided Geometric Design, 1986, 3(2): 83-127.
[11] Gordon, W.J. and Riesenfeld, R.F., Bernstein Bézier methods for the Computer Aided Geometric Design of free-form curves and surfaces. Journal of ACM, 1974 (21): 293-310.
[12] Farin, G., Curves and Surfaces for Computer Aided Geometric Design. Second Edition, Academic Press, 1990.
[13] Gordon, W.J. and Riesenfeld, R.F., B-spline curves and surfaces. in: Barnhill, R.E. and Riesenfeld, R.F. eds., Computer Aided Geometric Design, Academic Press, New York, 1974: 95-126.
[14] Cox M., The numerical evaluation of B-spline. Journal of Applied Mathematics , 1972(10): 134-149.
[15] de Boor, C., On Calculation with B-spline. Journal of Approximation Theory, 1972,6(1): 50-62.
[16] Boehm W., Inserting new knots into B-spline curve. Computer Aided Design, 1980, 12(4): 199-201.
[17] Cohen, E., Lyche, T., Riesenfeld, R.F., Discrete B-spline and subdivision techniques in computer aided geometric design and computer graphics. Computer Graphics and Image Processing, 1980, 14(2): 87-111.
[18] Prautzsch, H. and Piper, B., A fast algorithm to raise the degree of spline curves. Computer Aided Geometric Design, 1991, 8(4): 253-265.
[19] Rowin, M.S., Conic, Cubic and T-conic segments. D2-23252, The Beoing Company, Seattle, WA98124, 1964.
[20] Ball, A.A., ConSurf, Part 1: Introduction to conic lofting title. Computer Aided Design, 1974 (6): 243-249.
[21] Forrest, A.R., The twisted cubic curve: A computer aided geometric design approach. Computer Aided Design, 1980 (12): 165-172.
[22] Farin, G., Algorithms for rational Bézier curves. Computer Aided Design, 1983 (15): 73-79.
[23] Pigel, L., Representation of rational Bézier curves. Computer Aided Geometric Design, 1985 (2): 151-155.
[24]刘鼎元,有理Bézier曲线.应用数学学报, 1985 (8): 70-82.
[25]汪国昭,沈金福,有理Bézier曲线的离散和几何性质.浙江大学学报, 1985 (19): 123-130.
[26] Vergeest, S.M., CAD Surface data exchange using STEP. Computer Aided Design, 1991, 23(4): 269-281.
[27] Versprille, K.J., Computer aided design application of rational B-spline approximation form. Ph.D. thesis, Syracuse University, Syracuse, N.Y., 1975.
[28] Piegl, L. and Tiller, W., Curve and surface constructions using rational B-splines. Computer Aided Design, 1987 (19): 485-498.
[29] Piegl, L. and Tiller, W., A menagerie of rational B-spline circles. IEEEE Computer Graphics and its Application, 1989 (9): 48-56.
[30] Piegl, L., Modifying the shape of rational B-splines, part 1:curves. Computer Aided Design, 1989 (21): 509-518.
[31] Piegl, L., Modifying the shape of rational B-splines, part 2:surfaces. Computer Aided Design, 1989 (21): 538-546.
[32] Piegl, L., On NURBS: a survey. IEEE Computer Graphics and Application, 1991 (10): 55-71.
[33] Tiller, W., Rational B-splines for curve and surface representation. IEEE ComputerGraphics and its Application, 1983 (3): 61-69.
[34] Tiller, W., Geometric modeling using non-uniform rational B-splines: mathematical techniques. SIGGRAH tutorial notes, ACM, New York, 1986.
[35] Tiller, W., Knot-remove algorithms for NURBS curves and surfaces. Computer Aided Design, 1992 (24): 445-453.
[36] Choi, B.K., Yoo, W.S. and Lee, C.S., Matrix representation for NURBS curves and surfaces. Computer Aided Design, 1990 (22): 637-642.
[37] Grabowski, H. and Li, X., Coefficient formula and matrix of nonuniform B-spline functions. Computer Aided Design, 1992 (24): 637-642.
[38] Pottmann H., Wagner M.G., Helix splines as example of affine Tchebycheffian splines. Advances in Computational Mathematics, 1994, 2: 123-142.
[39] Zhang Jiwen, C-curves: an extension of cubic curves. CAGD, 1996, 13: 199-217.
[40] Zhang Jiwen, C-Bézier Curves and Surfaces. Graphical Models and Image Processing, 1999, 61: 2-15.
[41]张纪文,罗国明, C曲线及其性质.计算机辅助工程,1996,2: 53-61.
[42]张纪文,罗国明,三次样条曲线的拓广-C曲线.计算机辅助工程,1996,3: 12-20.
[43] Jiwen Zhang, Two different forms of C-B-splines. CAGD, 1997, 14: 31-41.
[44] Peńa J.M., Shape preserving representations for trigonometric polynomial curves. CAGD, 1999, 14: 5-11.
[45] Sánchez-Reyes J., Harmonic rational Bézier curves, p- Bézier curves and trigonometric polynomial. CAGD, 1998, 15: 909-923.
[46] Mainar E., Peńa J M., Sánchez-Reyes J., Shape preserving alternatives to the rational Bézier model. CAGD, 2001, 18: 37-60.
[47] Chen Qinyu, Wang Guozhao, A class of Bézier-like curves. CAGD, 2003, 20: 29-39.
[48] Wang Guozhao, Chen Qinyu, Zhou Minghua, NUATB B-spline curves. CAGD, 2004, 21: 193-205.
[49]王成伟,带有给定切线多边形的C 2连续的C-B样条曲线.计算机辅助设计与图形学学报,2001,13(12): 1133-1136.
[50]王成伟,带有给定切线多边形的C-Bézier闭曲线和B-型样条闭曲线.数值计算与计算机应用,2002,4: 303-309.
[51]单开佳,汪国昭,低阶C- Bézier曲线的升阶.高等应用数学学报A辑,2002,17(4) : 441-445.
[52]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学报,2002,14(5): 421-424.
[53]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.
[54]樊建华,罗国明,王卫民, C- Bézier曲面分割、拼接及其应用.工程图学学报,2002,3: 133-138.
[55]樊建华,罗国明,林兴,王卫民, C- Bézier构造工程曲面.机械工程学报,2003,39(1): 89-92.
[56]樊建华, C- Bézier曲线曲面造型理论及算法研究.浙江大学硕士学位论文,2001.
[57]陈秦玉,汪国昭,圆弧的C- Bézier曲线表示.软件学报,2002,13(11): 2155-2161.
[58]陈秦玉,杨勋年,汪国昭,四次C-曲线的性质及其应用.高等应用数学学报A辑,2003,18(1): 45-50.
[59]金义明,五次C-曲线的生成及其性质.浙江大学硕士学位论文,2001.
[60]金义明,汪国昭,五次C-Bézier曲线的生成及其性质.高校应用数学学报A辑,2005,20(1): 91-96.
[61]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,2004.
[62] Yang Qinmin, Wang Guozhao, Inflection points and singularities on C-curve. CAGD, 2004, 21: 207-213.
[63]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.
[64]汪仁泰,陈建兰,空间曲线的C-Bézier插值.浙江工业大学学报2004,32(2): 225-233.
[65]王刘强,刘旭敏, C-曲线间G 1拼接条件及在曲面造型中的应用.计算机工程与应用,2004,43(18): 45-46.
[66]林上华,汪国昭, C-曲线定义区间的扩展.计算机辅助设计与图形学学报,2005,17(10): 2281-2285.
[67]叶正麟,吴荣军,平面C-Bézier曲线的奇拐点分析.计算数学,2005,27(1): 63-70.
[68]戴中寅,杨松林, C- Bézier曲线与NURBS曲线的光滑拼接.江苏大学学报,2005,26(6A): 95-97.
[69]赵玉林,高阶C- Bézier曲线曲面性质研究及其应用.浙江大学硕士学位论文,2005.
[70]吴荣军,叶正麟,罗卫民,有理C-Bézier曲线的形状分析.计算机学报,2007,30(11): 2055-2059.
[71]周贵宾, C曲线和H曲线的Path.浙江大学硕士学位论文,2007.
[72]林新辉, C-Bézier曲线降阶逼近.浙江大学硕士学位论文,2007.
[73] Lü, Y G, Uniform hyperbolic polynomial B-spline curves. CAGD, 2002, 19: 379-393.
[74] Wang Guozhao, Yang Qinmin, Planar cubic hybrid hyperbolic polynomial curve and its shape classification. Progress in Natural Science, 2004, 14(1): 41-46.
[75]杨勤民,三次混合多项式曲线和区间曲面的研究.浙江大学硕士学位论文,2002.
[76]王媛,康宝生,代数双曲混合H-Bézier函数及其性质.西北大学学报,2006, 36(5): 693-697.
[77]王媛, H-Bézier曲线的理论及应用研究.西北大学硕士学位论文,2006.
[78]胡晴峰,双曲混合多项式曲线及其性质.浙江大学硕士学位论文,2003.
[79]钱江,唐月红, H Bézier-Like曲线在工程中的应用.数值计算与计算机应用,2007,28(3): 167-178.
[80]吴荣军,平面三次H-Bézier曲线的形状分析.应用数学学报, 2007, 30(5) : 816-821.
[81] Hering L., Closed( C 2- and C 3-Continuous) Bézier and B-spline curve with given tangent polygons. CAD, 1983, 15(1): 3-6.
[82]方逵,与给定的多边形相切的闭( G 2-连续)Bézier曲线.计算数学,1991,13(2): 34-37.
[83]方逵,蔡放,谭建荣,带有给定切线多边形的C 2和C 3连续Bézier闭样条曲线.计算机辅助设计与图形学学报,2000,12(5): 330-332.
[84]施法中,计算机辅助几何设计与非均匀有理B样条.北京:高等教育出版社,2001: 92-100.
[85]姜卫,服装CAD.上海:中国纺织大学出版社,1999.
[86] Pottmann H., The geometry of Tchebycheffian spline. Computer Aided Geometric Design, 1993, 10(2): 181-210.
[87] Mazure M.L., Chebyshev-Bernstein bases. Computer Aided Geometric Design, 1999, 16(7): 649-669.