带形状参数的Bézier曲线的能量优化
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摘要
相较于经典的Bézier曲线,带形状参数的Bézier曲线提供了独立于控制顶点的形状调整自由度,但同时又增加了设计人员选择形状参数的工作量。鉴于此,讨论了形状参数的选取方案。首先证明了已有文献中给出的Bernstein基函数的含参数扩展基为全正基,从而保证了相应的带形状参数的Bézier曲线的理论价值;然后采用能量最小化方法来确定曲线中形状参数的取值,推导了曲线的拉伸能量、弯曲能量、扭曲能量近似最小时,形状参数的计算公式,为曲线的应用提供了方便。
Compared with the classical Bézier curves,the Bézier curves with shape parameter provide the shape adjustment freedom which independent of the control points. However,the introduction of shape parameter increases the workload of designers for choosing shape parameters. In view of this,this paper mainly discussed the selection of shape parameter. Firstly,it proved that the extended basis of Bernstein basis functions with parameter presented in the existing literature was totally positive basis. It guaranteed the theory value of the corresponding Bézier curves with shape parameter. Then it used the energy minimization method to determine the shape parameter of the curves. It deduced the calculation formula of the shape parameter which made the stretch energy,strain energy and jerk energy of the curves approximate minimum. The formula was convenient for the application of the curves.
Compared with the classical Bézier curves,the Bézier curves with shape parameter provide the shape adjustment freedom which independent of the control points. However,the introduction of shape parameter increases the workload of designers for choosing shape parameters. In view of this,this paper mainly discussed the selection of shape parameter. Firstly,it proved that the extended basis of Bernstein basis functions with parameter presented in the existing literature was totally positive basis. It guaranteed the theory value of the corresponding Bézier curves with shape parameter. Then it used the energy minimization method to determine the shape parameter of the curves. It deduced the calculation formula of the shape parameter which made the stretch energy,strain energy and jerk energy of the curves approximate minimum. The formula was convenient for the application of the curves.
引文
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[16]Wang Wentao,Wang Guozhao.Bézier curves with shape parameter[J].Journal of Zhejiang University Science,2005,6A(6):497-501.
[17]吴晓勤.带形状参数的Bézier曲线[J].中国图象图形学报,2006,11(2):269-274.(Wu Xiaoqin.Bézier curve with shape parameter[J].Journal of Image and Graphics,2006,11(2):269-274.)
[18]Yan Lanlan,Liang Jiongfeng.An extension of the Bézier model[J].Applied Mathematics and Computation,2011,218(6):2863-2879.
[19]Qin Xinqiang,Hu Gang,Zhang Nianjuan,et al.A novel extension to the polynomial basis functions describing Bézier curves and surfaces of degree n with multiple shape parameters[J].Applied Mathematics and Computation,2013,223(10):1-16.
[20]严兰兰,邬国根.Bézier方法的新扩展[J].合肥工业大学学报:自然科学版,2013,36(5):625-631.(Yan Lanlan,Wu Guogen.An new extension of Bézier method[J].Journal of Hefei University of Technology:Natural Science,2013,36(5):625-631.)
[21]翟芳芳.带两个形状参数的四次Bézier曲线的扩展[J].大学数学,2016,32(1):33-37.(Zhai Fangfang.Extension of the quartic Bézier curve with two shape parameters[J].University Mathematics,2016,32(1):33-37.)
[2]Xu Gang,Wang Guozhao,Chen Wenyu.Geometric construction of energy-minimizing Bézier curves[J].Science China:Information Sciences,2011,54(7):1395-1406.
[3]Zhang Caiming,Zhang Pifu,Cheng Fuhua.Faring spline curve and surfaces by minimizing energy[J].Computer-Aided Design,2001,33(13):913-923.
[4]龙小平.局部能量最优法与曲线曲面的光顺[J].计算机辅助设计与图形学学报,2002,14(2):1109-1113.(Long Xiaoping.Fairing of curves and surfaces by local energy optimization[J].Journal of Computer-Aided Design&Computer Graphics,2002,14(2):1109-1113.)
[5]王远军,曹沅.非均匀三次参数样条曲线的能量最优光顺算法[J].计算机辅助设计与图形学学报,2005,17(9):1969-1975.(Wang Yuanjun,Cao Yuan.Energy optimization fairing algorithm of cubic parametric splines[J].Journal of Computer-Aided Design&Computer Graphics,2005,17(9):1969-1975.)
[6]孙义环.曲率变化最小的五次G2插值光顺曲线[D].杭州:浙江工商大学,2015.(Sun Yihuan.Quintic G2 interpolation fair curves via curvature variation minimization[D].Hangzhou:Zhejiang Gongshang University,2015.)
[7]Yong J H,Cheng F.Geometric hermite curves with minimum strain energy[J].Computer Aided Geometric Design,2004,21(3):913-923.
[8]Zhang Aiwu,Zhang Caiming.Shape interpolating geometric hermite curves with minimum strain energy[J].Journal of Information&Computational Science,2006,3(4):1025-1033.
[9]韩旭里,李建军,刘子奇.一类三次几何Hermite插值曲线及其优化[J].数学理论与应用,2008,28(1):20-24.(Han Xuli,Li Jianjun,Liu Ziqi.A class of cubic geometric Hermite interpolation curves with minimum strain energy[J].Mathematical Theory and Applications,2008,28(1):20-24.)
[10]Jakli2ab G,agara E.Planar cubic G1 interpolatory spline with small strain energy[J].Journal of Computational and Applied Mathematics,2011,235(8):2758-2765.
[11]Ling C C,Abbas M,Ali J M.Minimum energy curve in polynomial interpolation[J].Mathematika,2011,27(2):159-169.
[12]严兰兰,李水平.形状可调插值曲线曲面的参数选择[J].中国图象图形学报,2016,21(12):1685-1695.(Yan Lanlan,Li Shuiping.Parameter selection of shape-adjustable interpolation[J].Journal of Image and Graphics,2016,21(12):1685-1695.)
[13]李军成,严兰兰,刘成志.形状可调的5次组合样条及其参数选择[J].中国图象图形学报,2017,22(2):197-204.(Li Juncheng,Yan Lanlan,Liu Chengzhi.Quintic composite spline with adjustable shape and parameter selection[J].Journal of Image and Graphics,2017,22(2):197-204.)
[14]Li Xuemei,Zhang Yongxia,Ma Long,et al.Discussion on relationship between minimal energy and curve shapes[J].Applied Mathematics:A Journal of Chinese Universities(Series B),2014,29(4):379-390.
[15]韩旭里,刘圣军.二次Bézier曲线的扩展[J].中南工业大学学报:自然科学版,2003,34(2):214-217.(Han Xuli,Liu Shengjun.Extension of quadratic Bézier curve[J].Journal of Central South University of Technology:Natural Science,2003,34(2):214-217.)
[16]Wang Wentao,Wang Guozhao.Bézier curves with shape parameter[J].Journal of Zhejiang University Science,2005,6A(6):497-501.
[17]吴晓勤.带形状参数的Bézier曲线[J].中国图象图形学报,2006,11(2):269-274.(Wu Xiaoqin.Bézier curve with shape parameter[J].Journal of Image and Graphics,2006,11(2):269-274.)
[18]Yan Lanlan,Liang Jiongfeng.An extension of the Bézier model[J].Applied Mathematics and Computation,2011,218(6):2863-2879.
[19]Qin Xinqiang,Hu Gang,Zhang Nianjuan,et al.A novel extension to the polynomial basis functions describing Bézier curves and surfaces of degree n with multiple shape parameters[J].Applied Mathematics and Computation,2013,223(10):1-16.
[20]严兰兰,邬国根.Bézier方法的新扩展[J].合肥工业大学学报:自然科学版,2013,36(5):625-631.(Yan Lanlan,Wu Guogen.An new extension of Bézier method[J].Journal of Hefei University of Technology:Natural Science,2013,36(5):625-631.)
[21]翟芳芳.带两个形状参数的四次Bézier曲线的扩展[J].大学数学,2016,32(1):33-37.(Zhai Fangfang.Extension of the quartic Bézier curve with two shape parameters[J].University Mathematics,2016,32(1):33-37.)