C-曲线光顺算法的研究与实现
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摘要
自由曲线和曲面在飞机、汽车、船舶、家电外形设计和反求工程中有着广泛的应用。在CAGD中,经常用参数曲线曲面来插值、逼近、拟合测量得到的数据点。然而,由于计算和测量数据都不可避免的存在误差,由这些数据设计得到的曲线曲面一般都需要进行光顺处理。因此曲线曲面的光顺处理成为CAGD中一个非常重要的研究课题,受到人们的普遍重视。但是由于光顺处理的复杂性,直到现在,此问题还没有得到彻底的解决,对它的研究仍在进行之中。C-曲线作为一种新颖的曲线造型方法,它不仅能够处理自由形式的曲线,而且能够精确地表示一些常规二次曲线。因此研究C-曲线有着很重要的理论和应用价值。本文针对目前C-曲线光顺算法存在的问题,提出了一种光顺C-曲线的新算法,该算法的基本思想是：通过调整控制参数α和控制顶点使得曲线的能量最小,得到最优的光顺逼近曲线。通过最小二乘法和非线性泛函极小值的优化计算,对平面数据点进行光顺逼近,达到光顺的目的,并给出了数值算例来说明本文算法的有效性。
本文的主要研究内容如下：
1.介绍了曲线曲面光顺技术和C-曲线曲面理论的研究意义和发展概况；
2.介绍了C-曲线的定义和性质以及光顺的概念、光顺准则和一些经典光顺法；
3.提出了一种新的曲线光顺准则,将其分别应用于C-Bezier曲线和C-B样条曲线,构造了光顺算法；
4.给出了由数据拟合的C-Bezier曲线和C-B样条曲线光顺的实例,通过对数值算例的计算,显示了本文提出的算法的可行性和有效性。
Free-form curves and surfaces are widely used in airplane, automobile, shipping and electrical appliance shape design and inverse design in engineering. In CAGD, parametric curves and surfaces are usually applied to interpolate, approximate and fit the giving data. Nevertheless, as error exists due to measuring and calculating data, it is necessary to do fairing operations on the curves and surfaces obtained by these data, thus to get the aesthetic curves and surfaces. So the fairing of curves and surfaces is becoming one of the most important tasks in CAGD, and great importance is attached to it. But fairing process is so complicated that until now it is not completely resolved and the research for it is going on. Serving as a new method of curve/surface modeling, C-curves not only can deal with free form curves and surfaces, but also can present conic exactly. So the research of C-curves has the important value of theory and application. A new method of fairing C-curves is given in this paper in view of some disadvantages of fairing of C-curves at present.The mean idea is:fairing of C-curves is fulfilled by adjusting the value of parameter a and control points to reduce the implied energy. By using the technique of least square approximation and non-linear functional minimization, the data points of plane can be faired approximately. The examples of C-curves fairing show the efficiency of our method.
The main results in this paper are as following:
1. The meaning and general situation in fairing of curves and surfaces and the theories of C-curves and C-surfaces are introduced;
2. The definition and property of C-curves, and the concept of fairing, fairness indicators and some classical fairing methods are introduced;
3. A new fairness indicator of curves is put forward, and it is applied to C-Bezier curve and C-B spline curve, the fairing algorithm are constructed;
4. The examples of C-Bezier curve and C-B spline curve fairing are given. The validity and efficiency of the algorithm in this paper are shown by calculating these examples.
本文的主要研究内容如下：
1.介绍了曲线曲面光顺技术和C-曲线曲面理论的研究意义和发展概况；
2.介绍了C-曲线的定义和性质以及光顺的概念、光顺准则和一些经典光顺法；
3.提出了一种新的曲线光顺准则,将其分别应用于C-Bezier曲线和C-B样条曲线,构造了光顺算法；
4.给出了由数据拟合的C-Bezier曲线和C-B样条曲线光顺的实例,通过对数值算例的计算,显示了本文提出的算法的可行性和有效性。
Free-form curves and surfaces are widely used in airplane, automobile, shipping and electrical appliance shape design and inverse design in engineering. In CAGD, parametric curves and surfaces are usually applied to interpolate, approximate and fit the giving data. Nevertheless, as error exists due to measuring and calculating data, it is necessary to do fairing operations on the curves and surfaces obtained by these data, thus to get the aesthetic curves and surfaces. So the fairing of curves and surfaces is becoming one of the most important tasks in CAGD, and great importance is attached to it. But fairing process is so complicated that until now it is not completely resolved and the research for it is going on. Serving as a new method of curve/surface modeling, C-curves not only can deal with free form curves and surfaces, but also can present conic exactly. So the research of C-curves has the important value of theory and application. A new method of fairing C-curves is given in this paper in view of some disadvantages of fairing of C-curves at present.The mean idea is:fairing of C-curves is fulfilled by adjusting the value of parameter a and control points to reduce the implied energy. By using the technique of least square approximation and non-linear functional minimization, the data points of plane can be faired approximately. The examples of C-curves fairing show the efficiency of our method.
The main results in this paper are as following:
1. The meaning and general situation in fairing of curves and surfaces and the theories of C-curves and C-surfaces are introduced;
2. The definition and property of C-curves, and the concept of fairing, fairness indicators and some classical fairing methods are introduced;
3. A new fairness indicator of curves is put forward, and it is applied to C-Bezier curve and C-B spline curve, the fairing algorithm are constructed;
4. The examples of C-Bezier curve and C-B spline curve fairing are given. The validity and efficiency of the algorithm in this paper are shown by calculating these examples.
引文
[1]Barnhill, R.E.and Risenfeld, R.F., Computer Aided Geometric Design, Academic Press,1974.
[2]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,1994.
[3]朱心雄等.自由曲线曲面造型技术[M].北京：科学出版社,2000.
[4]Jiwen Zhang.C-curves:An entension of cubic curves[J].CAGD,1996,13:199-217.
[5]Jiwen Zhang.Two different forms of C-B-splines[J].CAGD,1997,14:31-41.
[6]Jiwen Zhang.C-Bezier curves and surfaces[J].Graphical Models and Image Processing,1999,61:2-15.
[7]Mainar E,Pena J.M,Sanchez-Reyes J.Shape preserving alternatives to the rational Bezier model[J]. Computer Aided Geometric Design,2001,18,37-60.
[8]闵春艳,汪国昭.C-B样条基是B基[J].浙江大学学报,2004,31(2),148-150.
[9]樊建华,张纪文,邬义杰.C-Bezier曲线的形状修改[J].软件学报,2002,13(11)：2194-2199.
[10]樊建华,邬义杰,林兴.C-Bezier曲线分割算法及G’拼接条件[J].计算机辅助设计与图形学报,2002,14(5)：421-424.
[11]陈秦玉,杨勋年.四次C-曲线的性质及其应用[J].高等应用数学学报A辑,2003,18(1)：45-50.
[12]刘华勇.C2连续的C-B样条的插值和拟合方法[J].湖南工程学院学报,2005,15(4)：68-71.
[13]李鹏,李原,刘平,张开富.C-B样条曲线及曲面的光滑拼接与应用[J].西北工业大学报,2007,25(6)：890～895.
[14]Hosaka.Theory of Curve and Surface Synthesis and their Smoothing Fitting[J].Information Processing in Japan,1969,9:60-68.
[15]Kjellander.Smoothing of Cubic Parametric Spline[J].Computer Aided Design,1983,15(3):175-179.
[16]Kjellander.Smoothing of Cubic Parametric Surface[J].Computer Aided Design,1983,15(5):288-293.
[17]Farin G,Rein G,Sapidis N and Worsey A.Fairing Cubic B-Spline Curves[J]. CAGD,1987,4:91-103.
[18]Sapidis N and Farin G.Automatic Fairing Algorithm for B-Spline Curves[J]. CAD,1990,22(2): 121-129.
[19]Pigounakis K.G.and Kablis P.D.Convexity-Preserving fairing[J].CAD,1996,28(12):981-994.
[20]Lott N.J.and Pullin D.I.Method for Fairing B-Spline Surfaces[J].CAD,1988,20(10):597-604.
[21]苏步青,刘鼎元.计算几何[M].上海：上海科学技术出版社,1981.
[22]复旦大学数学系,江南造船厂.单根曲线光顺的基样条法[J].江南造船厂技术资料,1974年7月.
[23]齐东旭.曲线拟合的数值方法[J].数学学报,1975,18：173-184.
[24]董光昌.船体数学放样—回弹法[M].北京：科学出版社,1978.
[25]蔡中义.基于能量极小准则的曲面再现与光顺[J].计算机辅助设计与图形学学报,2002,14(8)：758-762.
[26]蔡中义,李明哲.散乱分布数据曲面重构的光顺—有限元方法[J].软件学报,2003,14(4)：838-844.
[27]蔡中义.C1连续曲面重构与光顺的有限元算法[J].工程图学学报,2002,4：79-86.
[28]刘华勇,钱江.曲线曲面的有限元光顺方法[J].黑龙江水专学报,2005,32(4)：68-72.
[29]刘华勇,钱江.基于点的曲线曲面有限元光顺方法[J].组合机床与自动化加工技术,2005,11：12-16.
[30]Yankui Sun and xinxiong Zhu. Wavelet-Based Fairing of B-Spline Surface[J].Chinese Journal of Aeronautics,1999,12(3):176-182.
[31]孙延奎,朱心雄.B样条曲线的小波光顺法[J].工程图学学报,1998,4：51-58.
[32]石岩,彭国华,张力宁,周敏.基于小波分析和三次有理Bezier样条的曲线光顺算法[J].科学技术与工程,2006,6(20)：3291-3294.
[33]张力宁,张定华,刘元朋,赵西峰.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,2005,11：250-252.
[34]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,2005,39(7)：953-956.
[35]穆国旺,臧婷,赵罡.基于小波的B样条曲线局部光顺算法[J].工程图学学报,2006,2：84-89.
[36]彭程,彭芳瑜,陈吉红.基于小波分析的B样条曲线的光顺[J].机床与液压,2003,5：184-186.
[37]张荣鑫,林焰,纪卓尚.基于小波变换的船体NURBS曲线光顺方法[J].中国造船,2007,48(4)：58-63.
[38]Adam Finkelstein,David H Salesin.Multiresolution curves.In:Computer Graphics Proceedings,Annual Conference Series,ACM SIGGRAPH,Orlando,Florida,1994:261-268.
[39]孙延奎,朱心雄.任意B样条曲面的多分辨表示及光顺[J].工程图学学报,1998(3)：49-54.
[40]秦开怀,唐泽圣.B样条曲线小波分解的快速算法[J].清华大学学报(自然科学版),1999,39(5)：85-88.
[41]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,2006,12：68-70.
[42]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,2006,25(7)：826-830.
[43]甘屹,齐从谦,陈亚洲.基于遗传算法的曲线曲面光顺[J].同济大学学报,2002,30(3)：322-325.
[44]林忠钦,董洪智,李玉璇等.基于遗传优化算法的曲线曲面光顺[J].中国机械工程,2002,13(3)：217-220.
[45]Qinyu Chen,Guozhao Wang.A class of Bezier-like curves[J].CAGD,2003,20:29-39.
[46]马利庄,石教英.曲线曲面的几何光顺算法[J].计算机学报,1996,19：210-216.
[47]刘华勇.C-Bezier曲线的光顺[J].工程图学学报,2006,02：103-107.
[48]刘华勇.基于光顺要求的样C-B样条拟合方法[J].宁波职业技术学院学报,2005,9(5)：13-15.
[49]丁友东,华宣积.基于光顺优化的NURBS曲线权因子估计方法[J].计算机辅助设计与图形学学报,2000,12(5):325-329.
[50]Hagen H,Bonneau G.Variational design of smooth rational Bezier curves[J]. CAGD,1991,8:393-399.
[51]Wang X F,Cheng F H,Barsky B A.Energy and B-spline interproximation[J]. CAD,1997,29(7):485-496.
[52]Li An-Ping, Jiang Da-Wei. The weights optimization fairing algorithm for the cubic uniform rational B-spline curves [J].Journal of Computer-Aided Design & Computer Graphics,1997,9(6):562-567.
[53]Tzvetom ir I V. Fair interpolation and approximation of B-spline by energy minimization and points insertion[J]. CAD,1996,28(9):753-760.
[54]曾庭俊,王卫民,张纪文.C-B样条旋转曲面造型研究[J].工程图学学报,2004(2)：104-108.
[55]Mainar E,Pea J M.A Basis of C-Bezier Splines with Optimal Properties[J]. Computer Aided Geometric Design,2002,19:291-295.
[2]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,1994.
[3]朱心雄等.自由曲线曲面造型技术[M].北京：科学出版社,2000.
[4]Jiwen Zhang.C-curves:An entension of cubic curves[J].CAGD,1996,13:199-217.
[5]Jiwen Zhang.Two different forms of C-B-splines[J].CAGD,1997,14:31-41.
[6]Jiwen Zhang.C-Bezier curves and surfaces[J].Graphical Models and Image Processing,1999,61:2-15.
[7]Mainar E,Pena J.M,Sanchez-Reyes J.Shape preserving alternatives to the rational Bezier model[J]. Computer Aided Geometric Design,2001,18,37-60.
[8]闵春艳,汪国昭.C-B样条基是B基[J].浙江大学学报,2004,31(2),148-150.
[9]樊建华,张纪文,邬义杰.C-Bezier曲线的形状修改[J].软件学报,2002,13(11)：2194-2199.
[10]樊建华,邬义杰,林兴.C-Bezier曲线分割算法及G’拼接条件[J].计算机辅助设计与图形学报,2002,14(5)：421-424.
[11]陈秦玉,杨勋年.四次C-曲线的性质及其应用[J].高等应用数学学报A辑,2003,18(1)：45-50.
[12]刘华勇.C2连续的C-B样条的插值和拟合方法[J].湖南工程学院学报,2005,15(4)：68-71.
[13]李鹏,李原,刘平,张开富.C-B样条曲线及曲面的光滑拼接与应用[J].西北工业大学报,2007,25(6)：890～895.
[14]Hosaka.Theory of Curve and Surface Synthesis and their Smoothing Fitting[J].Information Processing in Japan,1969,9:60-68.
[15]Kjellander.Smoothing of Cubic Parametric Spline[J].Computer Aided Design,1983,15(3):175-179.
[16]Kjellander.Smoothing of Cubic Parametric Surface[J].Computer Aided Design,1983,15(5):288-293.
[17]Farin G,Rein G,Sapidis N and Worsey A.Fairing Cubic B-Spline Curves[J]. CAGD,1987,4:91-103.
[18]Sapidis N and Farin G.Automatic Fairing Algorithm for B-Spline Curves[J]. CAD,1990,22(2): 121-129.
[19]Pigounakis K.G.and Kablis P.D.Convexity-Preserving fairing[J].CAD,1996,28(12):981-994.
[20]Lott N.J.and Pullin D.I.Method for Fairing B-Spline Surfaces[J].CAD,1988,20(10):597-604.
[21]苏步青,刘鼎元.计算几何[M].上海：上海科学技术出版社,1981.
[22]复旦大学数学系,江南造船厂.单根曲线光顺的基样条法[J].江南造船厂技术资料,1974年7月.
[23]齐东旭.曲线拟合的数值方法[J].数学学报,1975,18：173-184.
[24]董光昌.船体数学放样—回弹法[M].北京：科学出版社,1978.
[25]蔡中义.基于能量极小准则的曲面再现与光顺[J].计算机辅助设计与图形学学报,2002,14(8)：758-762.
[26]蔡中义,李明哲.散乱分布数据曲面重构的光顺—有限元方法[J].软件学报,2003,14(4)：838-844.
[27]蔡中义.C1连续曲面重构与光顺的有限元算法[J].工程图学学报,2002,4：79-86.
[28]刘华勇,钱江.曲线曲面的有限元光顺方法[J].黑龙江水专学报,2005,32(4)：68-72.
[29]刘华勇,钱江.基于点的曲线曲面有限元光顺方法[J].组合机床与自动化加工技术,2005,11：12-16.
[30]Yankui Sun and xinxiong Zhu. Wavelet-Based Fairing of B-Spline Surface[J].Chinese Journal of Aeronautics,1999,12(3):176-182.
[31]孙延奎,朱心雄.B样条曲线的小波光顺法[J].工程图学学报,1998,4：51-58.
[32]石岩,彭国华,张力宁,周敏.基于小波分析和三次有理Bezier样条的曲线光顺算法[J].科学技术与工程,2006,6(20)：3291-3294.
[33]张力宁,张定华,刘元朋,赵西峰.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,2005,11：250-252.
[34]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,2005,39(7)：953-956.
[35]穆国旺,臧婷,赵罡.基于小波的B样条曲线局部光顺算法[J].工程图学学报,2006,2：84-89.
[36]彭程,彭芳瑜,陈吉红.基于小波分析的B样条曲线的光顺[J].机床与液压,2003,5：184-186.
[37]张荣鑫,林焰,纪卓尚.基于小波变换的船体NURBS曲线光顺方法[J].中国造船,2007,48(4)：58-63.
[38]Adam Finkelstein,David H Salesin.Multiresolution curves.In:Computer Graphics Proceedings,Annual Conference Series,ACM SIGGRAPH,Orlando,Florida,1994:261-268.
[39]孙延奎,朱心雄.任意B样条曲面的多分辨表示及光顺[J].工程图学学报,1998(3)：49-54.
[40]秦开怀,唐泽圣.B样条曲线小波分解的快速算法[J].清华大学学报(自然科学版),1999,39(5)：85-88.
[41]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,2006,12：68-70.
[42]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,2006,25(7)：826-830.
[43]甘屹,齐从谦,陈亚洲.基于遗传算法的曲线曲面光顺[J].同济大学学报,2002,30(3)：322-325.
[44]林忠钦,董洪智,李玉璇等.基于遗传优化算法的曲线曲面光顺[J].中国机械工程,2002,13(3)：217-220.
[45]Qinyu Chen,Guozhao Wang.A class of Bezier-like curves[J].CAGD,2003,20:29-39.
[46]马利庄,石教英.曲线曲面的几何光顺算法[J].计算机学报,1996,19：210-216.
[47]刘华勇.C-Bezier曲线的光顺[J].工程图学学报,2006,02：103-107.
[48]刘华勇.基于光顺要求的样C-B样条拟合方法[J].宁波职业技术学院学报,2005,9(5)：13-15.
[49]丁友东,华宣积.基于光顺优化的NURBS曲线权因子估计方法[J].计算机辅助设计与图形学学报,2000,12(5):325-329.
[50]Hagen H,Bonneau G.Variational design of smooth rational Bezier curves[J]. CAGD,1991,8:393-399.
[51]Wang X F,Cheng F H,Barsky B A.Energy and B-spline interproximation[J]. CAD,1997,29(7):485-496.
[52]Li An-Ping, Jiang Da-Wei. The weights optimization fairing algorithm for the cubic uniform rational B-spline curves [J].Journal of Computer-Aided Design & Computer Graphics,1997,9(6):562-567.
[53]Tzvetom ir I V. Fair interpolation and approximation of B-spline by energy minimization and points insertion[J]. CAD,1996,28(9):753-760.
[54]曾庭俊,王卫民,张纪文.C-B样条旋转曲面造型研究[J].工程图学学报,2004(2)：104-108.
[55]Mainar E,Pea J M.A Basis of C-Bezier Splines with Optimal Properties[J]. Computer Aided Geometric Design,2002,19:291-295.