广义Ball曲线、曲面的研究
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摘要
在计算机辅助几何设计中,定义在千变万化的拓扑结构上的自由型曲线曲面,存在着千变万化的形式,而广义Ball曲线则是其中一种在曲线求值及升降阶的计算速度方面明显优于Bézier曲线。本文对广义Ball曲线曲面的相关问题进行了较深入的研究。首先,通过引入多个形状参数,给出了Wang-Said型曲线的多形状参数的扩展。改变形状参数的值,可以局部或整体调控曲线的形状,同时给出了Said-Ball曲线的多形状参数的扩展。通过引入一个形状参数,生成了Wang-Ball曲线与三角域上Wang-Ball曲面的扩展。其次,给出了Wang-Ball曲线与Bézier曲线的统一表示,构造了介于二者之间的新型曲线族,并称之为Wang-Bézier曲线。同时给出了它的升阶公式、递推算法以及与Bézier曲线相互转化的公式。最后,给出Wang-Ball曲线、Said-Ball曲线与Bézier曲线三者的统一表示以及相关的升阶公式、递推算法。
Free curves/surfaces defined in different topological structure have different expressions in CAGD, and generalized Ball curves are more efficient than Bézier curves in calculation, the degree elevation and reduction。This dissertation focuses on the research about the generalized Ball curves/surfaces。Firstly, the extension of Wang-Said type generalized curves is provided by introducing some shape parameters。By changing the values of those shape parameters, the shape of these new curves can be adjusted locally or entirely, and Said-Ball curve with multiple shape parameters is provided。The extension of Wang-Ball surface is made to get a kind of new surface over triangular domain by introducing a shape parameter。Secondly, this paper presents a family of new curves which unify the representations of Bézier curve and Wang-Ball curve。The corresponding degree elevation scheme, recursive algorithm and explicit conversion formulae between this new family of curves and Bézier curve are given。Finally, a family of new basis is offered to unify the Berntein basis, Wang-Ball and Said-Ball basis。In addition, the corresponding degree elevation scheme and recursive algorithm are presented。
Free curves/surfaces defined in different topological structure have different expressions in CAGD, and generalized Ball curves are more efficient than Bézier curves in calculation, the degree elevation and reduction。This dissertation focuses on the research about the generalized Ball curves/surfaces。Firstly, the extension of Wang-Said type generalized curves is provided by introducing some shape parameters。By changing the values of those shape parameters, the shape of these new curves can be adjusted locally or entirely, and Said-Ball curve with multiple shape parameters is provided。The extension of Wang-Ball surface is made to get a kind of new surface over triangular domain by introducing a shape parameter。Secondly, this paper presents a family of new curves which unify the representations of Bézier curve and Wang-Ball curve。The corresponding degree elevation scheme, recursive algorithm and explicit conversion formulae between this new family of curves and Bézier curve are given。Finally, a family of new basis is offered to unify the Berntein basis, Wang-Ball and Said-Ball basis。In addition, the corresponding degree elevation scheme and recursive algorithm are presented。
引文
[1]丁友东,李敏.广义Ball曲线的性质及其应用[J],应用数学学报, 2000, 23 (4): 580-595.
[2]韩旭里,刘圣军.二次曲线Bézier的扩展[J].中南工业大学学报(自然科学版),2003,34(2):214-217.
[3]韩旭里,刘圣军.三次均匀B样条曲线的扩展[J].计算机辅助设计与图形学学报, 2003,15(5):576-582).
[4] CAD系统数据通讯中若干问题的研究[D],杭州,浙江大学,1996.
[5]刘松涛,刘根洪.广义Ball样条曲线及三角域上曲面的升阶公式和转换算法[J],应用数学学报, 1996, 19(2): 243-253.
[6]刘植. Bézier曲线的扩展[J].合肥工业大学学报,2004,27(8):976-979.
[7]沈莞蔷,汪国昭. Ball基的推广[J].软件学报,2005,16(11):1992-1999.
[8]王国瑾.高次Ball曲线及其应用[J].高校应用数学学报,1987,2(1):126-140.
[9]王国瑾,汪国昭,郑建民.计算机辅助几何设计[M].北京:高等教育出版社,施普林格出版社,2001.
[10]邬弘毅.两类新的广义Ball曲线[J].应用数学学报, 2000, 23(2): 196-205.
[11]邬弘毅,夏成林.带多个形状参数的Bézier曲线与曲面的扩展[J].计算机辅助设计与图形学学报,2005, 17(12): 2607-2612.
[12] A.A. Ball. CONSURF Part 1: Introduction of conic lofting title[J]. Computer Aided Design, 1974,6(4): 243-249.
[13] A.A. Ball. CONSURF Part 2: Description of the algorithms[J]. Computer Aided Design, 1975, 7(4): 237-242.
[14] A. A. Ball. CONSURF Part 3: How the program is used[J], Computer Aided Design, 1977, 9: 9-12.
[15] P. E. Bézier. Example of an existing system in motor industry: the UNISURF system[J], Proc. Roy. Soc. of London, 1971, Vol. A321: 207-218.
[16] P. Bézier. Numerical Control: Mathematics and Applications[M], John Wiley and Sons, New York, 1972.
[17] P. Bézier. Mathematical and practical possibilities of UNISURF[J]. Computer Aided Geometric Design, in: Barnhill, R.E. and Riesenfield, R.E., eds.,Academic Press, New York, 1974, 127-152.
[18] P. Bézier. The Mathematical basis of the UNISURF CAD system[M]. Butterworths, England, 1986.
[19] C. de Boor. On calculating with B-spline[J]. J. Approx. Theory, 1972, Vol. 6: 50-62.
[20] J. Clark. Designing surfaces in 3D[J]. Comm. ACM, 1976, 19(8): 454-460.
[21] S.A. Coons. Surfaces for computer aided design of space Figures. MIT Project MAC-TR-255, June, 1964.
[22] S.A. Coons. Surfaces for computer aided design of space forms. MIT Project MAC-TR-41, June, 1967.
[23] M.G. Cox. The numerical evaluation of B-spline. Report No. NPL-DNACS-4, National Laboratory, 1971.
[24] C. de Boor. On Calculation with B-Spline[J]. Journal of Approximation Theory, 1972, 6:50-62.
[25] G. Farin. Curves and Surfaces for Computer Aided Geometric Design[M]. Second Edition, Academic Press, 1990.
[26] J.C. Ferguson. Multivariable curve interpolation. Journal[A]. ACM, 1964, 11 (2), 221-228.
[27] Goodman T N T, Said H B. Shape-preserving Properties of the generalized Ball Basis [J]. Computer Aided Geometric Design, 1991,8:115-121.
[28] Goodman T N T, Said H B. Properties of the generalized Ball Curves and Surfaces [J].Computer-Aided Design, 1991, 23: 554-560.
[29] W.J. Gordon, and R.F. Riesenfeld. Bernstein Bézier methods for the Computer Aided Geometric Design of free-form curves and surfaces[A]. Journal of ACM, 1974, 21:293-310.
[30] W.J. Gordon, and R.F. Riesenfeld. B-spline curves and surfaces[J]. in: Barnhill, R.E. and Riesenfeld, R.F. eds., Computer Aided Geometric Design, Academic Press, New York, 1974, 95-126.
[31] T.N.T. Goodman, and H.B. Said. Further properties of the generalized Ball basis[M]. Univ. of Dundee, Report AA/897, UK, 1989.
[32]T.N.T. Goodman, and H.B. Said. Shape-preserving properties of thegeneralized Ball basis[J]. Computer Aided Geometric Design, 1991, 8(2): 115-121.
[33] T.N.T. Goodman, and H.B. Said. Properties of the generalized Ball basis and surfaces[J]. Computer Aided Design, 1991, 23(8): 554-560.
[34] Han Xuli. Piecewise quartic polynomial curves with a local shape parameter [J]. Computer Aided Geometric Design, 195(2006) 34-45.
[35] S.M. Hu, and T.G. Jin, Degree reductive approximation of Bézier curves[J]。Proc。Symposium on Computational Geometry Hangzhou, China(1992), 110-126。
[36] S。M。Hu, G。J。Wang, and T。G。Jin。Degree reduction of Bézier surface patches[J]。Proc。CAD/Graphics’93 Beijing, China, 285-289。
[37] S。M。Hu, G。Z。Wang and T。G。Jin。Properties of two types of generalized Ball curves[J]. Computer Aided Design, 1996,28(2): 125-133.
[38] S.M. Hu, G. J. Wang and J.G. Sun. A type of triangular Ball surface and its Properties[J]. J. Comput. Sci. Technol, 1998, 13: 63-72.
[39] P. Jiang, H.Y Wu, J.Q Tan. The dual functionals for the generalized Ball basis of Wang-Said type and basis transformation formulas[J]. Numerical Mathematics A Journal of Chinese Universities(English Series), 2006 ,15(1):248-256.
[40] W.A.M. Othman, and R. N. Goldman. The dual basis functions for the generalized Ball basis of odd degree[J]. Computer Aided Geometric Design, 1997, 14: 571-582.
[41] H. N. Phien and N. Dejdumrong. Efficient algorithms for Bézier curves[J]. Computer Aided Geometric Design, 2000, 17(3): 247-250 .
[42] L. Piegl. Representation of quadric primitives by rational polynomials[J]. Computer Aided Geometric Design, 1985, 2(1-3):151-155.
[43] L. Piegl. Modifying the shape of rational B-splines[J]. Part 2: Computer Aided Design, 1989, 21(9): 538-546.
[44] L. Piegl, W. Tiller. The NURBS Book[M]. Springer, New York, 1995.
[45] H.B. Said. Generalized Ball curve and its recursive algorithm[A]. ACM Trans. Graph, 1989, 8(4): 360-371.
[46] I.J. Schoenberg. Contributions to the problem of approximation of equidistant data by analytic function[J]. Quart, Applied Mathematics, 1946, 4:45-99,112-141.
[47] W. Tiller, Rational B-splines for curves and surfaces representation[J]. IEEE CG&A, 1983, September: 61-69 .
[48] G. J. Wang, and M. Cheng, New algorithms for evaluating parametric surface[J]. Progress in Natural Science, 2001, 11(2): 142-148.
[49] H.Y. Wu. Unifying representation of Bézier curve and generalized Ball curves [J]. Appl. Math. J. Chinese Univ. Ser. B, 2000,15 (1): 109-121.
[2]韩旭里,刘圣军.二次曲线Bézier的扩展[J].中南工业大学学报(自然科学版),2003,34(2):214-217.
[3]韩旭里,刘圣军.三次均匀B样条曲线的扩展[J].计算机辅助设计与图形学学报, 2003,15(5):576-582).
[4] CAD系统数据通讯中若干问题的研究[D],杭州,浙江大学,1996.
[5]刘松涛,刘根洪.广义Ball样条曲线及三角域上曲面的升阶公式和转换算法[J],应用数学学报, 1996, 19(2): 243-253.
[6]刘植. Bézier曲线的扩展[J].合肥工业大学学报,2004,27(8):976-979.
[7]沈莞蔷,汪国昭. Ball基的推广[J].软件学报,2005,16(11):1992-1999.
[8]王国瑾.高次Ball曲线及其应用[J].高校应用数学学报,1987,2(1):126-140.
[9]王国瑾,汪国昭,郑建民.计算机辅助几何设计[M].北京:高等教育出版社,施普林格出版社,2001.
[10]邬弘毅.两类新的广义Ball曲线[J].应用数学学报, 2000, 23(2): 196-205.
[11]邬弘毅,夏成林.带多个形状参数的Bézier曲线与曲面的扩展[J].计算机辅助设计与图形学学报,2005, 17(12): 2607-2612.
[12] A.A. Ball. CONSURF Part 1: Introduction of conic lofting title[J]. Computer Aided Design, 1974,6(4): 243-249.
[13] A.A. Ball. CONSURF Part 2: Description of the algorithms[J]. Computer Aided Design, 1975, 7(4): 237-242.
[14] A. A. Ball. CONSURF Part 3: How the program is used[J], Computer Aided Design, 1977, 9: 9-12.
[15] P. E. Bézier. Example of an existing system in motor industry: the UNISURF system[J], Proc. Roy. Soc. of London, 1971, Vol. A321: 207-218.
[16] P. Bézier. Numerical Control: Mathematics and Applications[M], John Wiley and Sons, New York, 1972.
[17] P. Bézier. Mathematical and practical possibilities of UNISURF[J]. Computer Aided Geometric Design, in: Barnhill, R.E. and Riesenfield, R.E., eds.,Academic Press, New York, 1974, 127-152.
[18] P. Bézier. The Mathematical basis of the UNISURF CAD system[M]. Butterworths, England, 1986.
[19] C. de Boor. On calculating with B-spline[J]. J. Approx. Theory, 1972, Vol. 6: 50-62.
[20] J. Clark. Designing surfaces in 3D[J]. Comm. ACM, 1976, 19(8): 454-460.
[21] S.A. Coons. Surfaces for computer aided design of space Figures. MIT Project MAC-TR-255, June, 1964.
[22] S.A. Coons. Surfaces for computer aided design of space forms. MIT Project MAC-TR-41, June, 1967.
[23] M.G. Cox. The numerical evaluation of B-spline. Report No. NPL-DNACS-4, National Laboratory, 1971.
[24] C. de Boor. On Calculation with B-Spline[J]. Journal of Approximation Theory, 1972, 6:50-62.
[25] G. Farin. Curves and Surfaces for Computer Aided Geometric Design[M]. Second Edition, Academic Press, 1990.
[26] J.C. Ferguson. Multivariable curve interpolation. Journal[A]. ACM, 1964, 11 (2), 221-228.
[27] Goodman T N T, Said H B. Shape-preserving Properties of the generalized Ball Basis [J]. Computer Aided Geometric Design, 1991,8:115-121.
[28] Goodman T N T, Said H B. Properties of the generalized Ball Curves and Surfaces [J].Computer-Aided Design, 1991, 23: 554-560.
[29] W.J. Gordon, and R.F. Riesenfeld. Bernstein Bézier methods for the Computer Aided Geometric Design of free-form curves and surfaces[A]. Journal of ACM, 1974, 21:293-310.
[30] W.J. Gordon, and R.F. Riesenfeld. B-spline curves and surfaces[J]. in: Barnhill, R.E. and Riesenfeld, R.F. eds., Computer Aided Geometric Design, Academic Press, New York, 1974, 95-126.
[31] T.N.T. Goodman, and H.B. Said. Further properties of the generalized Ball basis[M]. Univ. of Dundee, Report AA/897, UK, 1989.
[32]T.N.T. Goodman, and H.B. Said. Shape-preserving properties of thegeneralized Ball basis[J]. Computer Aided Geometric Design, 1991, 8(2): 115-121.
[33] T.N.T. Goodman, and H.B. Said. Properties of the generalized Ball basis and surfaces[J]. Computer Aided Design, 1991, 23(8): 554-560.
[34] Han Xuli. Piecewise quartic polynomial curves with a local shape parameter [J]. Computer Aided Geometric Design, 195(2006) 34-45.
[35] S.M. Hu, and T.G. Jin, Degree reductive approximation of Bézier curves[J]。Proc。Symposium on Computational Geometry Hangzhou, China(1992), 110-126。
[36] S。M。Hu, G。J。Wang, and T。G。Jin。Degree reduction of Bézier surface patches[J]。Proc。CAD/Graphics’93 Beijing, China, 285-289。
[37] S。M。Hu, G。Z。Wang and T。G。Jin。Properties of two types of generalized Ball curves[J]. Computer Aided Design, 1996,28(2): 125-133.
[38] S.M. Hu, G. J. Wang and J.G. Sun. A type of triangular Ball surface and its Properties[J]. J. Comput. Sci. Technol, 1998, 13: 63-72.
[39] P. Jiang, H.Y Wu, J.Q Tan. The dual functionals for the generalized Ball basis of Wang-Said type and basis transformation formulas[J]. Numerical Mathematics A Journal of Chinese Universities(English Series), 2006 ,15(1):248-256.
[40] W.A.M. Othman, and R. N. Goldman. The dual basis functions for the generalized Ball basis of odd degree[J]. Computer Aided Geometric Design, 1997, 14: 571-582.
[41] H. N. Phien and N. Dejdumrong. Efficient algorithms for Bézier curves[J]. Computer Aided Geometric Design, 2000, 17(3): 247-250 .
[42] L. Piegl. Representation of quadric primitives by rational polynomials[J]. Computer Aided Geometric Design, 1985, 2(1-3):151-155.
[43] L. Piegl. Modifying the shape of rational B-splines[J]. Part 2: Computer Aided Design, 1989, 21(9): 538-546.
[44] L. Piegl, W. Tiller. The NURBS Book[M]. Springer, New York, 1995.
[45] H.B. Said. Generalized Ball curve and its recursive algorithm[A]. ACM Trans. Graph, 1989, 8(4): 360-371.
[46] I.J. Schoenberg. Contributions to the problem of approximation of equidistant data by analytic function[J]. Quart, Applied Mathematics, 1946, 4:45-99,112-141.
[47] W. Tiller, Rational B-splines for curves and surfaces representation[J]. IEEE CG&A, 1983, September: 61-69 .
[48] G. J. Wang, and M. Cheng, New algorithms for evaluating parametric surface[J]. Progress in Natural Science, 2001, 11(2): 142-148.
[49] H.Y. Wu. Unifying representation of Bézier curve and generalized Ball curves [J]. Appl. Math. J. Chinese Univ. Ser. B, 2000,15 (1): 109-121.