四次Bézier曲线的光顺
|
摘要
曲线曲面的光顺是计算几何中的一个重要研究课题,它有着重要的理论和实用价值。在机械零件设计中,经常要用参数曲线曲面来插值、逼近、拟合测量得到数据点。然而,由于在测量和计算时都不可避免的存在误差,因此想得到令人满意的曲线曲面,一般都需要对由这些数据得到的曲线和曲面进行光顺处理。
对曲线的光顺既可以通过数据点的修改,也可以通过控制点的修改来进行。因此本文主要从两个方面,即通过数据点的修改和控制点的修改,来研究曲线光顺的问题。在实际中,有时可能出现一个孤立的“坏点”,有时也可能出现连续的几个“坏点”,因此我们分别讨论了一次修改一点和一次修改多点的情形。我们运用能量和带有误差控制的能量,对各种光顺算法的效果进行了分析,特别在一次修改两点的情况中,我们采用了两种光顺算法,并对这两种光顺算法的效果进行了比较。结果表明,光顺算法各有可取之处,这样我们就可以在实际应用中对一次修改一点还是多点,甚至对修改多点的不同算法做出选择,从而收到更好的光顺效果。本文以常用的四次Bézier曲线为光顺对象,研究的主要内容和结果如下:
第一部分论述了曲线曲面光顺的重要意义及它的国内外研究状况。
第二部分回顾了四次Bézier曲线的定义和性质,阐述了如何反算四次Bézier曲线的控制顶点。
第三部分和第四部分分别从控制点和数据点的修改两个方面,系统地研究了四次Bézier曲线局部修改和全局修改的问题。首先给出了一次修改一点光顺算法及其算法的步骤,并举出了算法实例。其次给出了一次修改两点的光顺算法,并对算法的特性进行分析。最后还给出了一次修改三点的光顺算法。结果表明,算法有较好的光顺作用。
第五部分在总结全文的同时,探讨了以后工作努力的方向。
本文以能量法为基础对四次Bézier曲线进行光顺,该算法既可以对四次Bézier曲线进行全局的光顺处理,也可以进行局部的光顺调节。
The fairing of curves and surfaces has been widely studied in the field of CAGD due to its great theoretical and practical value. In the mechanical parts design, parametric curves and surfaces are usually applied to interpolate, approximate and fit the giving data. Nevertheless, errors inevitably exist in measuring and calculating data, it is necessary to fair the curves and surfaces obtained from these data for a desiring curve and surface.
The faring of curves can be achieved by revising the data points or the control points. Thus the problems of fairing curves are researched from two aspects, i.e. fairing through revising the data points and revising the control points in this paper. In practice, the 'bad' point may be single or continuous, so the cases of modifying a single point or modifying more than one point in each modification step are discussed respectively. The validity of various fairing algorithms are analyzed through the energy and the energy with error-controlling complements, especially two fairing methods are given and the efficiency of the fairing methods are compared in case of modifying two points simultaneously. Experimental tests shows that the fairing algorithms of this paper have their advantages, thus we have choices between modifying one single point or modifying more than one point simultaneously, even choices between different fairing methods of modifying more than one point simultaneously in practice, so to improve the efficiency of the fairing processes. In this paper we use common fourth Bézier curves as fairing subjects. Its structure and the main results are as follow:
In the first part, the significance and the research status of the fairing are discussed.
In the second part, the concepts and properties of the fourth Bézier curves are reviewed and the inverse calculation of control points of the fourth Bézier curves is introduced.
In the third and the fourth part, the problem of global and local fairing of the fourth Bézier curves is researched systematically from two aspects, i.e. the modifying of the control points and the data point respectively. Firstly, the algorithms and the procedure of the algorithms of modifying one point in a step are given with some fairing examples. Secondly, the fairing algorithms of modifying two points simultaneously are obtained, the properties of the algorithms are also analyzed and their efficiencies are compared. Finally, the algorithm of fairing there points simultaneously is put forward. The validity of the proposed methods is demonstrated via experimental tests.
In the fifth part, the conclusions of this paper are summarized and the future work is put forward.
The fairing of the fourth Bézier curves is obtained by energy minimization. The methods can be applied for both global and local curves fairing.
对曲线的光顺既可以通过数据点的修改,也可以通过控制点的修改来进行。因此本文主要从两个方面,即通过数据点的修改和控制点的修改,来研究曲线光顺的问题。在实际中,有时可能出现一个孤立的“坏点”,有时也可能出现连续的几个“坏点”,因此我们分别讨论了一次修改一点和一次修改多点的情形。我们运用能量和带有误差控制的能量,对各种光顺算法的效果进行了分析,特别在一次修改两点的情况中,我们采用了两种光顺算法,并对这两种光顺算法的效果进行了比较。结果表明,光顺算法各有可取之处,这样我们就可以在实际应用中对一次修改一点还是多点,甚至对修改多点的不同算法做出选择,从而收到更好的光顺效果。本文以常用的四次Bézier曲线为光顺对象,研究的主要内容和结果如下:
第一部分论述了曲线曲面光顺的重要意义及它的国内外研究状况。
第二部分回顾了四次Bézier曲线的定义和性质,阐述了如何反算四次Bézier曲线的控制顶点。
第三部分和第四部分分别从控制点和数据点的修改两个方面,系统地研究了四次Bézier曲线局部修改和全局修改的问题。首先给出了一次修改一点光顺算法及其算法的步骤,并举出了算法实例。其次给出了一次修改两点的光顺算法,并对算法的特性进行分析。最后还给出了一次修改三点的光顺算法。结果表明,算法有较好的光顺作用。
第五部分在总结全文的同时,探讨了以后工作努力的方向。
本文以能量法为基础对四次Bézier曲线进行光顺,该算法既可以对四次Bézier曲线进行全局的光顺处理,也可以进行局部的光顺调节。
The fairing of curves and surfaces has been widely studied in the field of CAGD due to its great theoretical and practical value. In the mechanical parts design, parametric curves and surfaces are usually applied to interpolate, approximate and fit the giving data. Nevertheless, errors inevitably exist in measuring and calculating data, it is necessary to fair the curves and surfaces obtained from these data for a desiring curve and surface.
The faring of curves can be achieved by revising the data points or the control points. Thus the problems of fairing curves are researched from two aspects, i.e. fairing through revising the data points and revising the control points in this paper. In practice, the 'bad' point may be single or continuous, so the cases of modifying a single point or modifying more than one point in each modification step are discussed respectively. The validity of various fairing algorithms are analyzed through the energy and the energy with error-controlling complements, especially two fairing methods are given and the efficiency of the fairing methods are compared in case of modifying two points simultaneously. Experimental tests shows that the fairing algorithms of this paper have their advantages, thus we have choices between modifying one single point or modifying more than one point simultaneously, even choices between different fairing methods of modifying more than one point simultaneously in practice, so to improve the efficiency of the fairing processes. In this paper we use common fourth Bézier curves as fairing subjects. Its structure and the main results are as follow:
In the first part, the significance and the research status of the fairing are discussed.
In the second part, the concepts and properties of the fourth Bézier curves are reviewed and the inverse calculation of control points of the fourth Bézier curves is introduced.
In the third and the fourth part, the problem of global and local fairing of the fourth Bézier curves is researched systematically from two aspects, i.e. the modifying of the control points and the data point respectively. Firstly, the algorithms and the procedure of the algorithms of modifying one point in a step are given with some fairing examples. Secondly, the fairing algorithms of modifying two points simultaneously are obtained, the properties of the algorithms are also analyzed and their efficiencies are compared. Finally, the algorithm of fairing there points simultaneously is put forward. The validity of the proposed methods is demonstrated via experimental tests.
In the fifth part, the conclusions of this paper are summarized and the future work is put forward.
The fairing of the fourth Bézier curves is obtained by energy minimization. The methods can be applied for both global and local curves fairing.
引文
[1]Hosaka. Theory of Curves and Surface Synthesis and Their Smooth Fitting[J]. Information Processing in Japan, 1969 (9):60-68.
[2]苏步青,刘鼎元.计算几何[M].上海:上海科学技术出版社,1981.
[3]复旦大学数学系,江南造船厂.单根曲线光顺的基样条法,江南造船厂技术资料,1974年7月.
[4]齐东旭等.曲线拟合的数值磨光方法[J].数学学报,1975,vol.18:173-184.
[5]董光吕.船体数学放样-回弹法[M].北京:科学出版社,1978.
[6]Kjellander,J.A.P.Smoothing of Cubic Parametric Spline[J].CAD,1983,Vol.15,No.3:175- 179.
[7]Kjellander,J.A.P.Smoothing of Cubic Parametric Surface[J].CAD,1983,Vol.15,No.5:288- 293.
[8]Farin G,Rein G,Spidis N,et al.Fairing Cubic B-Spline Curves[J].CAGD,1987,Vol.4: 91-103.
[9]D.Terzopoulos,J.PlattandA.Barr.Elastically Deformable Models[J].Computer Graphics,1987,Vol.24,No.4,205-214.
[10]D.Terzopoulos and H.Qin,Dynamic NURBS with Geometric Constraints for Interactive Scu1pting[J],ACM,Transactionson on Graphics,1994,Vol.13,No.2:103-136.
[11]Lott N.J.and Pullin D.I.Method for Fairing B-Spline Surfaces[J],CAD,1988,Vol.20,No,10:597-604.
[12]G. Celniker and D.Gossard.Continuous Deformable Curves and Their Application to Fairing 3D Geometry,Proc.IFIP TC5/WG5.2 Working Conf Geometric Modelling,Rensselaeville, NY,USA,1991,5-29.
[13]G.Celniker and D.Gossard.Deformable Curves and Surface Finite-Elements for Free- Form Shape Design[J].ACM Computer Graphics,1991,Vol.25,No.4,257-266.
[14]G.Celniker and W.Welch.Linear Constraints for Deformable B-Spline Surfaces[M].In Proceedings of the Symposium on interactive 3D Graphics, ACM,New York,1992,61-68.
[15]W.Welch and A.Witckin.Variational Surface Modelling[J].ACM Computer Graphics,Vol.26,1992,No.2,157-166.
[16]H.Hagen and G.Schulze.Automatic Smoothing with Geometric Surface Patches[J].CAGD,1987,Vol.4,231-236.
[17]H.Hagen,Kaiserslautern and G.P.Bonneau.Variational. Design of Smooth Rational Bezier-Surfaces[M],in Geometric Modelling,G.Farin,H.Hagen and H.Noltemeier,Eds,Springer-Verlag,1993,133-138.
[18]J.C.Leon and P.Trompette.A New Approach towards Free-Form Surfaces Control[J].CAGD,1995,Vol.12,395-416.
[19]Adam Finkelstein , David H Salesin.Multiresolution curves.In: Computer Graphics Proeeedings,Annual Conference Series,ACM SIGGRAPH,Orlando,Florida,1994:261-268.
[20]Giancarlo Amati.A multi-level filtering approach for fairing planner cubic B-spline curves[J].CAGD,2007,Vol.24,53-66.
[21]石岩,彭国华等.基于小波分析和三次有理Bezier样条曲线算法[J].科学技术与工程,2006,Vol.6,3291-3294.
[22]张力宁,张定华等.基于曲率图小波分解的平面曲线光顺方法[J].计算机应用研究,2005,Vol.11,250-252.
[23]穆国旺,臧婷.基于小波的B样条曲线局部光顺算法[J].工程图学学报,2006,Vol.2,84-89.
[24]柯映林,李奇敏.基于非均匀B样条小波分解的NURBS曲线光顺[J].浙江大学学报,2005,Vol.7,953-956.
[25]纪小刚,龚光容.半正交B样条小波及其在曲线曲面光顺中的应用[J].机械工程学报,2006,Vol.42:54-60.
[26]张荣鑫,林焰.基于小波变换的船体NURBS曲面光顺方法研究[J].哈尔滨工程大学学报,2007,Vol,28:955-959.
[27]刘华勇.用遗传算法光顺基于点表示的曲线曲面[J].机械科学与技术,2006,Vol.25,826-830.
[28]穆国旺,臧婷.用改进遗传算法确B样条曲线的节点矢量[J].计算机工程与应用,2006,Vol.11,88-90.
[29]臧婷,穆国旺.基于遗传算法的B样条曲线自动光顺算法[J].计算机工程与应用,2006,Vol.12,68-70.
[30]Zheng Xing-gao,Tan Jie-qing.Fairing of cubic Bezier curves[J].Journal of Hefei University ofTechnology,2005 (3):109-112.
[31]Caiming Zhang,Pifu zhang,Fuhua (Frank) Cheng.Fairing spline curves and surfaces by minimizing energy[J].CAD,2001,33:913-923.
[32]Xuefu Wang.Energy and B-spline interproximation[J].CAGD,1997,29(7):485-496.
[33]龙小平.局部能量最优化法与曲线曲面的光顺穆国旺[J].计算机辅助几何设计与图形学学报,2002,14 (12):1109-1103.
[34]穆国旺,宋秀琴,臧婷.一种选点法和能量法相结合的曲线光顺方法[J].工程图学学报,2005,Vol.6,118-121.
[35]何时剑.B样条曲线的自动光顺算法—最小二乘能量法[J].机电产品开发与创新,2005,Vol.18,No.1:93-95.
[36]郑康平,姜虹,吴坚等.一种光顺B-Spline曲线的实用方法[J].计算机工程与应用,2003:92-94.
[37]谢俊.贝赛尔曲线设计离心泵叶轮的叶片型线[J].排灌机械,2000,Vol.18,1-3.
[38]汪建华,童新华.Bézier曲线在离心泵叶轮轴面流到设计中的应用[J].长江大学学报,2007,Vol.4 (3):93-97.
[39]施法中.计算机辅助几何设计与非均匀有理B样条[M].北京:北京航空航天大学出版社,2001,114-120.
[2]苏步青,刘鼎元.计算几何[M].上海:上海科学技术出版社,1981.
[3]复旦大学数学系,江南造船厂.单根曲线光顺的基样条法,江南造船厂技术资料,1974年7月.
[4]齐东旭等.曲线拟合的数值磨光方法[J].数学学报,1975,vol.18:173-184.
[5]董光吕.船体数学放样-回弹法[M].北京:科学出版社,1978.
[6]Kjellander,J.A.P.Smoothing of Cubic Parametric Spline[J].CAD,1983,Vol.15,No.3:175- 179.
[7]Kjellander,J.A.P.Smoothing of Cubic Parametric Surface[J].CAD,1983,Vol.15,No.5:288- 293.
[8]Farin G,Rein G,Spidis N,et al.Fairing Cubic B-Spline Curves[J].CAGD,1987,Vol.4: 91-103.
[9]D.Terzopoulos,J.PlattandA.Barr.Elastically Deformable Models[J].Computer Graphics,1987,Vol.24,No.4,205-214.
[10]D.Terzopoulos and H.Qin,Dynamic NURBS with Geometric Constraints for Interactive Scu1pting[J],ACM,Transactionson on Graphics,1994,Vol.13,No.2:103-136.
[11]Lott N.J.and Pullin D.I.Method for Fairing B-Spline Surfaces[J],CAD,1988,Vol.20,No,10:597-604.
[12]G. Celniker and D.Gossard.Continuous Deformable Curves and Their Application to Fairing 3D Geometry,Proc.IFIP TC5/WG5.2 Working Conf Geometric Modelling,Rensselaeville, NY,USA,1991,5-29.
[13]G.Celniker and D.Gossard.Deformable Curves and Surface Finite-Elements for Free- Form Shape Design[J].ACM Computer Graphics,1991,Vol.25,No.4,257-266.
[14]G.Celniker and W.Welch.Linear Constraints for Deformable B-Spline Surfaces[M].In Proceedings of the Symposium on interactive 3D Graphics, ACM,New York,1992,61-68.
[15]W.Welch and A.Witckin.Variational Surface Modelling[J].ACM Computer Graphics,Vol.26,1992,No.2,157-166.
[16]H.Hagen and G.Schulze.Automatic Smoothing with Geometric Surface Patches[J].CAGD,1987,Vol.4,231-236.
[17]H.Hagen,Kaiserslautern and G.P.Bonneau.Variational. Design of Smooth Rational Bezier-Surfaces[M],in Geometric Modelling,G.Farin,H.Hagen and H.Noltemeier,Eds,Springer-Verlag,1993,133-138.
[18]J.C.Leon and P.Trompette.A New Approach towards Free-Form Surfaces Control[J].CAGD,1995,Vol.12,395-416.
[19]Adam Finkelstein , David H Salesin.Multiresolution curves.In: Computer Graphics Proeeedings,Annual Conference Series,ACM SIGGRAPH,Orlando,Florida,1994:261-268.
[20]Giancarlo Amati.A multi-level filtering approach for fairing planner cubic B-spline curves[J].CAGD,2007,Vol.24,53-66.
[21]石岩,彭国华等.基于小波分析和三次有理Bezier样条曲线算法[J].科学技术与工程,2006,Vol.6,3291-3294.
[22]张力宁,张定华等.基于曲率图小波分解的平面曲线光顺方法[J].计算机应用研究,2005,Vol.11,250-252.
[23]穆国旺,臧婷.基于小波的B样条曲线局部光顺算法[J].工程图学学报,2006,Vol.2,84-89.
[24]柯映林,李奇敏.基于非均匀B样条小波分解的NURBS曲线光顺[J].浙江大学学报,2005,Vol.7,953-956.
[25]纪小刚,龚光容.半正交B样条小波及其在曲线曲面光顺中的应用[J].机械工程学报,2006,Vol.42:54-60.
[26]张荣鑫,林焰.基于小波变换的船体NURBS曲面光顺方法研究[J].哈尔滨工程大学学报,2007,Vol,28:955-959.
[27]刘华勇.用遗传算法光顺基于点表示的曲线曲面[J].机械科学与技术,2006,Vol.25,826-830.
[28]穆国旺,臧婷.用改进遗传算法确B样条曲线的节点矢量[J].计算机工程与应用,2006,Vol.11,88-90.
[29]臧婷,穆国旺.基于遗传算法的B样条曲线自动光顺算法[J].计算机工程与应用,2006,Vol.12,68-70.
[30]Zheng Xing-gao,Tan Jie-qing.Fairing of cubic Bezier curves[J].Journal of Hefei University ofTechnology,2005 (3):109-112.
[31]Caiming Zhang,Pifu zhang,Fuhua (Frank) Cheng.Fairing spline curves and surfaces by minimizing energy[J].CAD,2001,33:913-923.
[32]Xuefu Wang.Energy and B-spline interproximation[J].CAGD,1997,29(7):485-496.
[33]龙小平.局部能量最优化法与曲线曲面的光顺穆国旺[J].计算机辅助几何设计与图形学学报,2002,14 (12):1109-1103.
[34]穆国旺,宋秀琴,臧婷.一种选点法和能量法相结合的曲线光顺方法[J].工程图学学报,2005,Vol.6,118-121.
[35]何时剑.B样条曲线的自动光顺算法—最小二乘能量法[J].机电产品开发与创新,2005,Vol.18,No.1:93-95.
[36]郑康平,姜虹,吴坚等.一种光顺B-Spline曲线的实用方法[J].计算机工程与应用,2003:92-94.
[37]谢俊.贝赛尔曲线设计离心泵叶轮的叶片型线[J].排灌机械,2000,Vol.18,1-3.
[38]汪建华,童新华.Bézier曲线在离心泵叶轮轴面流到设计中的应用[J].长江大学学报,2007,Vol.4 (3):93-97.
[39]施法中.计算机辅助几何设计与非均匀有理B样条[M].北京:北京航空航天大学出版社,2001,114-120.