带B样条曲率线的NURBS曲面设计
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摘要
为了满足产品设计的需要,提出一种NURBS曲面设计算法用于构造插值曲率线的曲面.首先利用给定已知曲线作为公共曲率线的等参曲面束的表达式、B样条的导矢公式和2个B样条的乘积理论,给出以一条非均匀B样条曲线作为公共曲率线的等参曲面束的显式表达式;然后讨论插值曲面可用NURBS精确表示的必要条件,并给出2种表达式,得出以一条非均匀B样条平面曲线作为公共曲率线的曲面束的NURBS精确表达式,以及控制顶点的计算式.通过实例展示了曲面设计效果,表明算法是可行的.
In order to satisfy the requirements of design for the products, an algorithm of design NURBS surfaces is proposed on constructing surfaces interpolating a line of curvature. Firstly, utilizing the mathematical expression of isoparametric surface pencil with a given common line of curvature, as well as the formulae of both the derivatives and the products of B-splines, we provide an explicit expression of isoparametric surface pencil with a non-uniform B-spline common curve of curvature. Furthermore, the necessary condition to express the interpolating surface as NURBS form exactly is introduced, and two styles of the expression are given. Then, an exact NURBS expression of surface pencil with a given non-uniform B-spline plane curve of curvature and a formula for calculating the control points of the surface are derived. Finally, the effect of surface design is shown in the experimental results, which demonstrate the effectiveness of the proposed algorithm.
In order to satisfy the requirements of design for the products, an algorithm of design NURBS surfaces is proposed on constructing surfaces interpolating a line of curvature. Firstly, utilizing the mathematical expression of isoparametric surface pencil with a given common line of curvature, as well as the formulae of both the derivatives and the products of B-splines, we provide an explicit expression of isoparametric surface pencil with a non-uniform B-spline common curve of curvature. Furthermore, the necessary condition to express the interpolating surface as NURBS form exactly is introduced, and two styles of the expression are given. Then, an exact NURBS expression of surface pencil with a given non-uniform B-spline plane curve of curvature and a formula for calculating the control points of the surface are derived. Finally, the effect of surface design is shown in the experimental results, which demonstrate the effectiveness of the proposed algorithm.
引文
[1]Wang G J,Tang K,Tai C L.Parametric representations of a surface pencil with a common spatial geodesic[J].ComputerAided Design,2004,36(5):447-459
[2]Li C Y,Wang R H,Zhu C G.Parametric representation of a surface pencil with a common line of curvature[J].Computer-Aided Design,2011,43(9):1110-1117
[3]Liu Yu.Inverse design of all kinds of surfaces with constraints and curve reduction approximation[D].Hangzhou:Zhejiang University.School of Mathematical Sciences,2012(in Chinese)(刘羽.带约束的各类曲面逆向设计和曲线降阶逼近[D].杭州:浙江大学数学科学学院,2012)
[4]Li C Y,Wang R H,Zhu C G.An approach for designing a developable surface through a given line of curvature[J].Computer-Aided Design,2013,45(3):621-627
[5]Li C Y,Zhu C G,Wang R H.Spacelike developable surfaces through a common line of curvature in Minkowski 3-space[J].Journal of Advanced Mechanical Design,Systems,and Manufacturing,2015,9(4):Paper No.14-00543
[6]do Carmo M P.Differential geometry of curves and surfaces[M].Englewood Cliffs:Prentice Hall,1976:16-19
[7]Mei Xiangming,Huang Jingzhi.Differential geometry[M].Beijing:Higher Education Press,2003(in Chinese)(梅向明,黄敬之.微分几何[M].北京:高等教育出版社,2003)
[8]Wang Guojin,Wang Guozhao,Zheng Jianmin.Computer aided geometric design[M].Beijing:Higher Education Press,2001:18-55(in Chinese)(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,2001:18-55)
[9]Zhang Lilin,Wang Guojin.Inverse design of NURBS surface pencil with a geodesic[J].Journal of Computer-Aided Design&Computer Graphics,2017,29(2):295-303(in Chinese)(张礼林,王国瑾.带测地线的NURBS曲面束的逆向设计[J].计算机辅助设计与图形学学报,2017,29(2):295-303)
[10]Wang G Z,Deng C Y.On the degree elevation of B-spline curves and corner cutting[J].Computer Aided Geometric Design,2007,24(2):90-98
[11]Farouki R T,Sakkalis T.Pythagorean hodographs[J].IBM Journal of Research and Development,1990,34(5):736-752
[2]Li C Y,Wang R H,Zhu C G.Parametric representation of a surface pencil with a common line of curvature[J].Computer-Aided Design,2011,43(9):1110-1117
[3]Liu Yu.Inverse design of all kinds of surfaces with constraints and curve reduction approximation[D].Hangzhou:Zhejiang University.School of Mathematical Sciences,2012(in Chinese)(刘羽.带约束的各类曲面逆向设计和曲线降阶逼近[D].杭州:浙江大学数学科学学院,2012)
[4]Li C Y,Wang R H,Zhu C G.An approach for designing a developable surface through a given line of curvature[J].Computer-Aided Design,2013,45(3):621-627
[5]Li C Y,Zhu C G,Wang R H.Spacelike developable surfaces through a common line of curvature in Minkowski 3-space[J].Journal of Advanced Mechanical Design,Systems,and Manufacturing,2015,9(4):Paper No.14-00543
[6]do Carmo M P.Differential geometry of curves and surfaces[M].Englewood Cliffs:Prentice Hall,1976:16-19
[7]Mei Xiangming,Huang Jingzhi.Differential geometry[M].Beijing:Higher Education Press,2003(in Chinese)(梅向明,黄敬之.微分几何[M].北京:高等教育出版社,2003)
[8]Wang Guojin,Wang Guozhao,Zheng Jianmin.Computer aided geometric design[M].Beijing:Higher Education Press,2001:18-55(in Chinese)(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,2001:18-55)
[9]Zhang Lilin,Wang Guojin.Inverse design of NURBS surface pencil with a geodesic[J].Journal of Computer-Aided Design&Computer Graphics,2017,29(2):295-303(in Chinese)(张礼林,王国瑾.带测地线的NURBS曲面束的逆向设计[J].计算机辅助设计与图形学学报,2017,29(2):295-303)
[10]Wang G Z,Deng C Y.On the degree elevation of B-spline curves and corner cutting[J].Computer Aided Geometric Design,2007,24(2):90-98
[11]Farouki R T,Sakkalis T.Pythagorean hodographs[J].IBM Journal of Research and Development,1990,34(5):736-752