CAGD中可展曲面设计和球域Bézier曲线的研究
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摘要
本文的主题是对CAGD中可展Bézier曲面和可展Poisson曲面的设计以及球域Bézier曲线的边界曲面进行研究。
第一章综述了Bézier曲线曲面的理论发展过程,引申出可展曲面设计和球域Bézier曲线设计的必要性。
第二章介绍了以任意一条空间Bézier曲线为准线的一张可展Bézier曲面设计方法。该方法应用有关曲面可展充要条件的微分几何理论,Bézier曲线的升阶公式以及Bernstein基函数的线性无关性,不用解非线性特征方程,就可以直接决定可展曲面的直母线、可展锥面的顶点和切线面的脊线,完成可展曲面的设计。本章的结果对于工程外形设计具有良好的应用前景。
第三章是空间可展Poisson曲面的设计。利用Poisson基函数的线性无关性,Poisson曲线的升阶公式,以及微分几何理论中可展曲面的充要条件,设计出超越可展曲面。本章结果可广泛应用于旋转切割加工中频繁遇见的超越曲面的设计以及螺旋形管道曲面的外钣展开。
第四章是球域Bézier曲线的边界曲面的求解。利用微分几何中空间曲面族的包络算法和变量替换方法,求得球域Bézier曲线的精确边界表示;进一步利用函数逼近论中Legendre多项式的最佳一致平方逼近方法,把球域Bézier曲线的边界曲面近似地表示为一张Bézier曲面或分片Bézier曲面的组合。球域Bézier曲线是表达方式简洁、存储空间节省、运算速度较快的误差分析和误差控制工具。
In this paper, some researches have been done on design of developable Bezier surfaces and Poisson surfaces, the boundary surfaces of ball Bezier curve in Computer Aided Geometric Design (CAGD).In Chapter 1, an overview of development of Bezier curves and surfaces is presented, and hence developable surfaces and ball Bezier curve are introduced.In Chapter 2, an algorithm is presented that generates a developable Bezier surface through a Bezier curve called a directrix. The algorithm is based upon the differential geometry theory about the necessary and sufficient conditions for a surface which is developable, the degree elevation formula for parameter curves and the linear independency for Bernstein basis. No nonlinear characteristic equations have to be solved. Moreover the vertex for a cone and the edge of regression for a tangent surface can be given easily.In Chapter 3, design of developable Poisson surface is presented. Based on the necessary and sufficient conditions for a surface which is developable, the degree evaluation formula for Poisson curves and the linear independency for Poisson basis functions, an algorithm that generates developable Poisson surfaces through a Poisson curve of arbitrary degree and shape is given. These results could be applied to design of surfaces frequently met in revolution cutting, and plate expansion of spiral-like pipe surfaces.In Chapter 4, how to solve boundary surface of a ball Bezier curve is presented. With the envelope algorithm of the family of space surfaces in differential geometric and variable transformation, an accurate representation for the boundary of a ball Bezier curve was gained;and furthermore, it was approximately represented as a Bezier surface or an union of Bezier patches by using Legendre polynomial best square uniform approximation. Ball Bezier is a kind of error control and error analysis with simple expression, less storage and fast computation.
第一章综述了Bézier曲线曲面的理论发展过程,引申出可展曲面设计和球域Bézier曲线设计的必要性。
第二章介绍了以任意一条空间Bézier曲线为准线的一张可展Bézier曲面设计方法。该方法应用有关曲面可展充要条件的微分几何理论,Bézier曲线的升阶公式以及Bernstein基函数的线性无关性,不用解非线性特征方程,就可以直接决定可展曲面的直母线、可展锥面的顶点和切线面的脊线,完成可展曲面的设计。本章的结果对于工程外形设计具有良好的应用前景。
第三章是空间可展Poisson曲面的设计。利用Poisson基函数的线性无关性,Poisson曲线的升阶公式,以及微分几何理论中可展曲面的充要条件,设计出超越可展曲面。本章结果可广泛应用于旋转切割加工中频繁遇见的超越曲面的设计以及螺旋形管道曲面的外钣展开。
第四章是球域Bézier曲线的边界曲面的求解。利用微分几何中空间曲面族的包络算法和变量替换方法,求得球域Bézier曲线的精确边界表示;进一步利用函数逼近论中Legendre多项式的最佳一致平方逼近方法,把球域Bézier曲线的边界曲面近似地表示为一张Bézier曲面或分片Bézier曲面的组合。球域Bézier曲线是表达方式简洁、存储空间节省、运算速度较快的误差分析和误差控制工具。
In this paper, some researches have been done on design of developable Bezier surfaces and Poisson surfaces, the boundary surfaces of ball Bezier curve in Computer Aided Geometric Design (CAGD).In Chapter 1, an overview of development of Bezier curves and surfaces is presented, and hence developable surfaces and ball Bezier curve are introduced.In Chapter 2, an algorithm is presented that generates a developable Bezier surface through a Bezier curve called a directrix. The algorithm is based upon the differential geometry theory about the necessary and sufficient conditions for a surface which is developable, the degree elevation formula for parameter curves and the linear independency for Bernstein basis. No nonlinear characteristic equations have to be solved. Moreover the vertex for a cone and the edge of regression for a tangent surface can be given easily.In Chapter 3, design of developable Poisson surface is presented. Based on the necessary and sufficient conditions for a surface which is developable, the degree evaluation formula for Poisson curves and the linear independency for Poisson basis functions, an algorithm that generates developable Poisson surfaces through a Poisson curve of arbitrary degree and shape is given. These results could be applied to design of surfaces frequently met in revolution cutting, and plate expansion of spiral-like pipe surfaces.In Chapter 4, how to solve boundary surface of a ball Bezier curve is presented. With the envelope algorithm of the family of space surfaces in differential geometric and variable transformation, an accurate representation for the boundary of a ball Bezier curve was gained;and furthermore, it was approximately represented as a Bezier surface or an union of Bezier patches by using Legendre polynomial best square uniform approximation. Ball Bezier is a kind of error control and error analysis with simple expression, less storage and fast computation.
引文
[Aum91] Aumarm G. Interpolation with developable Bezier patches. Computer Aided Geometric Design 1991, 8: 409-420.
[Aum03] Aumarm G. A simple algorithm for designing developable Bezier surfaces. Computer Aided Geometric Design, 2003, 20: 601-619.
[Bez72] Bezier, P., Numerical Control: Mathematics and Applications, John Wiley and Sons, New York, 1972.
[Bez74] Bezier, 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.
[Bez86] Bezier, P., The Mathematical basis of the UNISURF CAD system, Butterworths, England, 1986.
[BR93] Boddulur RMC and Ravani B. Design of developable surfaces using duality between plane and point geometries. Computer-Aided Design, 1993, 25: 621-632.
[CL00] Chen FL and Lou WP. Degree reduction of interval Bezier curves. Computer Aided Design, 2000, 32(5): 571-582.
[CM98] Chalfant JS and Maekawa T. Design for manufacturing using B-spline developable surfaces. Journal of Ship Research, 1998, 42(3): 206-214.
[CS02] Chu CH and Sequin CH. Developable Bezier patches properties and design. Computer-Aided Design 2002, 34: 511-527.
[Coo64] Coons, S. A., Surfaces for computer aided design of space Figures, MIT Project MAC-TR-255, June, 1964.
[Coo67] Coons, S. A., Surfaces for computer aided design of space forms, MIT Project MAC-TR-41, June, 1967.
[CP99] Chen HY and Pottmann H. Approximation by ruled surfaces. Journal of Computional and applied Math., 1999, 102(1): 143-156.
[doC76] do Carmo, M. P. Differential Geometry of Curves and Surfaces. New Jersey: Prentice-Hall Inc., Englewood Cliffs, 1976.
[Edg31] Edge WL. The Theory of Ruled Surfaces. Cambridge Univ. Press, Cambridge, UK, 1931.
[Far90] Farin, G., Curves and surfaces for Computer Aided Geometric Design, Second Edition, Academic Press, 1990.
[FB93] Frey WH and Bindschadler D. Computer aided design of a class of developable Bezier surfaces. GM Research Publications R&D-8057, 1993.
[Fer63] Ferguson, J. C., Multivariable curve interpolation, Report No. D2-22504, The Boeing Co., Seattle, Washington, 1963.
[Fer64] Ferguson JC., Multivariable curve interpolation, Journal ACM, 11(1964), 221-228.
[For72] Forrest AR. Interactive interpolation and approximation by Bezier curve, The Computer Journal, 15(1972), 71-79.
[For98] Forrest AR. Computational geometry and software engineering: towards a geometric computing environment, In: Rodgers D. and Earnshaw R. eds., Techniques for Computer Graphics. Springer, New Work, 1998.
[Ge96] Ge QJ. Kinematics-driven geometric modeling: a framework for simultaneous NC tool-path generation and sculptured surface design. Proc. IEEE Intemat. Conf. On Robotics and Automation, Minneapolis, MN, 1996, 1819-1824.
[GR74] Gordon, W. J. and Riesenfeld, R. F., Bemstein Bezier methods for the Computer Aided Geometric Design of free-form curves and surfaces, Journal of ACM, 21(1974), 293-310.
[Her02] Hermann T. Degree elevation for generalized Poisson functions. Computer Aided Geometric Design, 2002, 19: 65-70.
[HMP97] Hu CY, Maekawa EC, Patrikalakis NM and Ye XZ. Robust interval algorithm for surface intersections. Computer-Aided Design, 1997, 29(9): 617-627.
[Hos71] Hoschek J. Liniengeometrie, Bibliograph. Institut, Zurich, 1971.
[蒋91] 蒋骏,理有亲,张中元等.钛合金零件制造技术(飞机制造技术丛书).1991年4月.
[Kim03] Kim J. Minimum distance between a canal surface and a simple surface. Computer Aided Design, 2003, 35(10): 871-879.
[Kli78] Klingenberg W. A course in Differential Geometry. Translated by Hoffman D. New York: Springer-Verlag, 1978.
[LLW02] Lin HW, Liu LG and Wang GJ. Boundary evaluation for interval Bezier curve. Computer-Aided Design, 2002, 34(9): 637-646.
[LR92] Lang J and Roschel O. Developable (1,n)- Bezier Surfaces. Computer Aided Geometric Design 1992, 9: 291-298.
[LR98] Lin Q and Rokne JG.. Disk Bezier curves. Computer Aided Geometric Design, 1998, 15(7): 721-737.
[刘00] 刘利刚,王国瑾,寿华好.区间B6zier曲面逼近.计算机辅助设计与图形学学报,2000,12(9):645-650.
[Mac98] Maekawa T. Design and tessellation of B-spline developable surfaces. ASME Trans. Journal of Mechnanical Design, 1998, 120(3): 453-461.
[MF92] Mancewicz MJ and Frey WH. Developable surfaces properties, representations and methods of design. GM Research Publications GMR-7637, 1992.
[MG00] Morin G and Goldman RN. A subdivision scheme for Poisson curves and surfaces. Computer Aided Geometric Design, 2000, 17: 813-833.
[Moo66] Moore R. E. Interval Analysis. Prentice-Hall. Eaglewood Cliffs, New Jersey, 1966.
[PF95] Pottmann H and Farin G. Developable rational Bezier and B-spline surfaces. Computer Aided Geometric Design, 1995, 12: 513-531.
[PPR98] Pottmann H, Petemell M and Ravani B. Approximation in line space: applications in robot kinematics and surface reconstruction. In: Lenarcis J andHusty Meds. Advances in Robot Kinematics: Analysis and Control, Kluwer, 1998,403-412.
[PPR99] Peternell M, Pottmann H and Ravani B. On the computational geometry of ruled surfaces. CAD, 1999, 31(1): 17-31.
[PW99] Pottrnann H and Wallner J. Approximation algorithms for developable surfaces, CAGD, 1999;16(6): 538-556.
[PW01] Pottmann H and Wallner J. Computational Line Geometry. Springer, Berlin, 2001.
[RC86] Ravani B and Chen YJ. Computer-aided design and machining of composite ruled surfaces. J. Mech. Trans. Automat. Design 1986, 108: 217-223.
[RW91] Ravani B and Wang JW. Computer. aided design of line constructs. ASME J. Mechanical Design, 1991, 113: 363-371.
[SF92] Sederberg TW and Farouki RT.. Approximated by interval Bezier curves. IEEE Computer Graphics and its Application, 1992, 15(2): 87-95.
[寿98a] 寿华好,王国瑾.区间B6zier曲线的边界.高校应用数学学报,13A(增刊),1998:37-44.
[寿98b] 寿华好,王国瑾.区间曲线曲面与Offset曲线曲面的关系.工程图学学报,1998,19(3):55-59.
[苏79] 苏步青,胡和生等.微分几何.北京:高等教育出版社,1979.
[吴81] 吴大任.微分几何讲义(第四版).北京:人民教育出版社,1981.
[杨03] 杨武.几何计算中误差控制与分析的几个问题.博士论文,中国科学技术大学,合肥,2003.
[张06] 张兴旺,王国瑾.球域B6zier曲线的边界.浙江大学学报(工学版),2006,40(2):197-201
[Aum03] Aumarm G. A simple algorithm for designing developable Bezier surfaces. Computer Aided Geometric Design, 2003, 20: 601-619.
[Bez72] Bezier, P., Numerical Control: Mathematics and Applications, John Wiley and Sons, New York, 1972.
[Bez74] Bezier, 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.
[Bez86] Bezier, P., The Mathematical basis of the UNISURF CAD system, Butterworths, England, 1986.
[BR93] Boddulur RMC and Ravani B. Design of developable surfaces using duality between plane and point geometries. Computer-Aided Design, 1993, 25: 621-632.
[CL00] Chen FL and Lou WP. Degree reduction of interval Bezier curves. Computer Aided Design, 2000, 32(5): 571-582.
[CM98] Chalfant JS and Maekawa T. Design for manufacturing using B-spline developable surfaces. Journal of Ship Research, 1998, 42(3): 206-214.
[CS02] Chu CH and Sequin CH. Developable Bezier patches properties and design. Computer-Aided Design 2002, 34: 511-527.
[Coo64] Coons, S. A., Surfaces for computer aided design of space Figures, MIT Project MAC-TR-255, June, 1964.
[Coo67] Coons, S. A., Surfaces for computer aided design of space forms, MIT Project MAC-TR-41, June, 1967.
[CP99] Chen HY and Pottmann H. Approximation by ruled surfaces. Journal of Computional and applied Math., 1999, 102(1): 143-156.
[doC76] do Carmo, M. P. Differential Geometry of Curves and Surfaces. New Jersey: Prentice-Hall Inc., Englewood Cliffs, 1976.
[Edg31] Edge WL. The Theory of Ruled Surfaces. Cambridge Univ. Press, Cambridge, UK, 1931.
[Far90] Farin, G., Curves and surfaces for Computer Aided Geometric Design, Second Edition, Academic Press, 1990.
[FB93] Frey WH and Bindschadler D. Computer aided design of a class of developable Bezier surfaces. GM Research Publications R&D-8057, 1993.
[Fer63] Ferguson, J. C., Multivariable curve interpolation, Report No. D2-22504, The Boeing Co., Seattle, Washington, 1963.
[Fer64] Ferguson JC., Multivariable curve interpolation, Journal ACM, 11(1964), 221-228.
[For72] Forrest AR. Interactive interpolation and approximation by Bezier curve, The Computer Journal, 15(1972), 71-79.
[For98] Forrest AR. Computational geometry and software engineering: towards a geometric computing environment, In: Rodgers D. and Earnshaw R. eds., Techniques for Computer Graphics. Springer, New Work, 1998.
[Ge96] Ge QJ. Kinematics-driven geometric modeling: a framework for simultaneous NC tool-path generation and sculptured surface design. Proc. IEEE Intemat. Conf. On Robotics and Automation, Minneapolis, MN, 1996, 1819-1824.
[GR74] Gordon, W. J. and Riesenfeld, R. F., Bemstein Bezier methods for the Computer Aided Geometric Design of free-form curves and surfaces, Journal of ACM, 21(1974), 293-310.
[Her02] Hermann T. Degree elevation for generalized Poisson functions. Computer Aided Geometric Design, 2002, 19: 65-70.
[HMP97] Hu CY, Maekawa EC, Patrikalakis NM and Ye XZ. Robust interval algorithm for surface intersections. Computer-Aided Design, 1997, 29(9): 617-627.
[Hos71] Hoschek J. Liniengeometrie, Bibliograph. Institut, Zurich, 1971.
[蒋91] 蒋骏,理有亲,张中元等.钛合金零件制造技术(飞机制造技术丛书).1991年4月.
[Kim03] Kim J. Minimum distance between a canal surface and a simple surface. Computer Aided Design, 2003, 35(10): 871-879.
[Kli78] Klingenberg W. A course in Differential Geometry. Translated by Hoffman D. New York: Springer-Verlag, 1978.
[LLW02] Lin HW, Liu LG and Wang GJ. Boundary evaluation for interval Bezier curve. Computer-Aided Design, 2002, 34(9): 637-646.
[LR92] Lang J and Roschel O. Developable (1,n)- Bezier Surfaces. Computer Aided Geometric Design 1992, 9: 291-298.
[LR98] Lin Q and Rokne JG.. Disk Bezier curves. Computer Aided Geometric Design, 1998, 15(7): 721-737.
[刘00] 刘利刚,王国瑾,寿华好.区间B6zier曲面逼近.计算机辅助设计与图形学学报,2000,12(9):645-650.
[Mac98] Maekawa T. Design and tessellation of B-spline developable surfaces. ASME Trans. Journal of Mechnanical Design, 1998, 120(3): 453-461.
[MF92] Mancewicz MJ and Frey WH. Developable surfaces properties, representations and methods of design. GM Research Publications GMR-7637, 1992.
[MG00] Morin G and Goldman RN. A subdivision scheme for Poisson curves and surfaces. Computer Aided Geometric Design, 2000, 17: 813-833.
[Moo66] Moore R. E. Interval Analysis. Prentice-Hall. Eaglewood Cliffs, New Jersey, 1966.
[PF95] Pottmann H and Farin G. Developable rational Bezier and B-spline surfaces. Computer Aided Geometric Design, 1995, 12: 513-531.
[PPR98] Pottmann H, Petemell M and Ravani B. Approximation in line space: applications in robot kinematics and surface reconstruction. In: Lenarcis J andHusty Meds. Advances in Robot Kinematics: Analysis and Control, Kluwer, 1998,403-412.
[PPR99] Peternell M, Pottmann H and Ravani B. On the computational geometry of ruled surfaces. CAD, 1999, 31(1): 17-31.
[PW99] Pottrnann H and Wallner J. Approximation algorithms for developable surfaces, CAGD, 1999;16(6): 538-556.
[PW01] Pottmann H and Wallner J. Computational Line Geometry. Springer, Berlin, 2001.
[RC86] Ravani B and Chen YJ. Computer-aided design and machining of composite ruled surfaces. J. Mech. Trans. Automat. Design 1986, 108: 217-223.
[RW91] Ravani B and Wang JW. Computer. aided design of line constructs. ASME J. Mechanical Design, 1991, 113: 363-371.
[SF92] Sederberg TW and Farouki RT.. Approximated by interval Bezier curves. IEEE Computer Graphics and its Application, 1992, 15(2): 87-95.
[寿98a] 寿华好,王国瑾.区间B6zier曲线的边界.高校应用数学学报,13A(增刊),1998:37-44.
[寿98b] 寿华好,王国瑾.区间曲线曲面与Offset曲线曲面的关系.工程图学学报,1998,19(3):55-59.
[苏79] 苏步青,胡和生等.微分几何.北京:高等教育出版社,1979.
[吴81] 吴大任.微分几何讲义(第四版).北京:人民教育出版社,1981.
[杨03] 杨武.几何计算中误差控制与分析的几个问题.博士论文,中国科学技术大学,合肥,2003.
[张06] 张兴旺,王国瑾.球域B6zier曲线的边界.浙江大学学报(工学版),2006,40(2):197-201