可展曲面上的插值曲线研究
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摘要
在计算机辅助几何设计中,曲面上的插值曲线研究是一个十分重要的课题。本文主要讨论可展曲面上的插值曲线,经过分析,构造曲面上插值曲线的关键在于找到一个好的参数化方法,也就是如何将R~2上一条曲线对应到曲面上,使得到的曲面上曲线的扭曲尽可能小。基于这样的一种思想,结合微分几何理论,本文通过构造等距对应将可展曲面展成平面,从而将可展曲面上的曲线插值归结为通常的R~2上插值曲线的构造。分别针对已知型值点和型值点处的单位切矢以及只知道型值点而型值点处切矢未知的情况构造了C~1和G~2连续的插值曲线,并以可展曲面的三种类型:柱面、锥面、切线曲面为例构造了其上的插值曲线。文章最后还提出了一种选点修改法与能量法相结合的算法光顺可展曲面上的插值曲线。
本文主要包括如下内容:
1.对曲线曲面造型方法的相关内容作了简要介绍,简述了曲面上插值曲线的研究现状,回顾了曲线光顺技术的意义和发展现状。
2.介绍了曲线曲面的基本理论,包括曲线的切矢、曲率、挠率以及曲线论基本公式,曲面论基本公式及曲面的法曲率、测地曲率、等距对应等。
3.在已知型值点以及型值点处切矢的条件下,提出一种构造可展曲面上C~1连续的插值曲线的方法,采用积累弦长作为参数,无需反算控制顶点。
4.在已知型值点而型值点处切矢未知的情况下,采用积累弦长作为参数,利用二次三角B-样条来构造插值曲线,无需反算控制顶点,并证明了所得插值曲线是G2连续的。
5.提出一种选点修改法与能量法相结合的算法光顺可展曲面上的插值曲线,以圆柱面为例对其上的插值曲线进行了光顺处理,并推导了光顺以后控制顶点的坐标,用matlab软件绘出光顺后的曲线及曲率图,图例显示该算法具有满意的效果。
In CAGD, the research on the interpolation curves on surface is a very important topic. This text is mainly about the interpolation curves on developable surfaces. After analyzed, the key of constructing interpolation curves on surfaces is to find a good parameterization method, namely how to map a curve of R~2 onto the surface, to minimize the distortion of the obtained curve on the surface. Base on such an idea, according to the basic principles of differential geometry, it constructs a isometric correspondence, and then, the developable surface will be developed into plane. Therefore, the interpolation curve on the developable surface can be ended into the construction of interpolation curve in R~2 generally. This paper put forward methods for constructing interpolation curves of C~1 and G~2 continuous, according to two situations, getting the points on surface with specified tangent vector, or just getting the points but no tangent vector. The three types of developable surface: cylndrical surface, conical surface and tangent surface ,are used as examples to construct the interpolation curves. And it also presents a new method mixing bad points selection with energy minimization, for fairing the interpolation curves on the developable surface.
The content of this paper is mainly as following:
1. Simply introduce the relevant content of curve & surface shaping method, briefly describe the present condition of interpolation curves on surface, and review the meaning and development of curves fairing.
2. Introduce the basic theory of curve & surface, including the tangent vector、curvature、torsion of curve and the basic expressions of curve theory & surface theory, normal curvature、geodesic curvature、isometric correspondence of surface etc.
3.When the points on surface are given with specified tangent vector, a method of constructing C1 continuous interpolation curves on the developable surface is presented. Take the accumulated chord length as parameter, without calculating the control points.
4. Under the situation of just getting the points but no tangent vector, it takes the accumulated chord length as parameter,and uses the quadratic trigonometric B spline curve to construct the interpolation curve. Without calculating the control points. And it proves that the interpolation curve on the developable surface is G~2 continuous..
5. A new method mixing bad points selection with energy minimization, for fairing the interpolation curves on the developable surface is presented. It takes the cylindrical surface as an example, to fair its upper curve. Deduce the coordinates of control points after fairing. Draw the curve and its curvature after fairing with the software MATLAB, and it shows that this algorithm is effective.
本文主要包括如下内容:
1.对曲线曲面造型方法的相关内容作了简要介绍,简述了曲面上插值曲线的研究现状,回顾了曲线光顺技术的意义和发展现状。
2.介绍了曲线曲面的基本理论,包括曲线的切矢、曲率、挠率以及曲线论基本公式,曲面论基本公式及曲面的法曲率、测地曲率、等距对应等。
3.在已知型值点以及型值点处切矢的条件下,提出一种构造可展曲面上C~1连续的插值曲线的方法,采用积累弦长作为参数,无需反算控制顶点。
4.在已知型值点而型值点处切矢未知的情况下,采用积累弦长作为参数,利用二次三角B-样条来构造插值曲线,无需反算控制顶点,并证明了所得插值曲线是G2连续的。
5.提出一种选点修改法与能量法相结合的算法光顺可展曲面上的插值曲线,以圆柱面为例对其上的插值曲线进行了光顺处理,并推导了光顺以后控制顶点的坐标,用matlab软件绘出光顺后的曲线及曲率图,图例显示该算法具有满意的效果。
In CAGD, the research on the interpolation curves on surface is a very important topic. This text is mainly about the interpolation curves on developable surfaces. After analyzed, the key of constructing interpolation curves on surfaces is to find a good parameterization method, namely how to map a curve of R~2 onto the surface, to minimize the distortion of the obtained curve on the surface. Base on such an idea, according to the basic principles of differential geometry, it constructs a isometric correspondence, and then, the developable surface will be developed into plane. Therefore, the interpolation curve on the developable surface can be ended into the construction of interpolation curve in R~2 generally. This paper put forward methods for constructing interpolation curves of C~1 and G~2 continuous, according to two situations, getting the points on surface with specified tangent vector, or just getting the points but no tangent vector. The three types of developable surface: cylndrical surface, conical surface and tangent surface ,are used as examples to construct the interpolation curves. And it also presents a new method mixing bad points selection with energy minimization, for fairing the interpolation curves on the developable surface.
The content of this paper is mainly as following:
1. Simply introduce the relevant content of curve & surface shaping method, briefly describe the present condition of interpolation curves on surface, and review the meaning and development of curves fairing.
2. Introduce the basic theory of curve & surface, including the tangent vector、curvature、torsion of curve and the basic expressions of curve theory & surface theory, normal curvature、geodesic curvature、isometric correspondence of surface etc.
3.When the points on surface are given with specified tangent vector, a method of constructing C1 continuous interpolation curves on the developable surface is presented. Take the accumulated chord length as parameter, without calculating the control points.
4. Under the situation of just getting the points but no tangent vector, it takes the accumulated chord length as parameter,and uses the quadratic trigonometric B spline curve to construct the interpolation curve. Without calculating the control points. And it proves that the interpolation curve on the developable surface is G~2 continuous..
5. A new method mixing bad points selection with energy minimization, for fairing the interpolation curves on the developable surface is presented. It takes the cylindrical surface as an example, to fair its upper curve. Deduce the coordinates of control points after fairing. Draw the curve and its curvature after fairing with the software MATLAB, and it shows that this algorithm is effective.
引文
[1]莫蓉,吴英,常智勇.计算机辅助几何造型技术[M].北京:科学出版社,2004:56~77
[2]朱心雄等.自由曲线曲面造型技术[M].北京:科学出版社,2000:66~87,348~365
[3]施法中.计算机辅助几何设计与非均匀有理B样条(CAGD&NURBS).北京航空航天大学出版社,1994,448~450
[4]陈省身,陈维桓.微分几何讲义[M].(第二版).北京:北京大学出版社,2001:322~352
[5]苏步青,刘鼎元.初等微分几何[M].上海:上海科学技术出版社,1985:177~220
[6] Bezier P E.Numerical control-Mathematics and applications[M].Forrest trans.Wiley,London,1972.
[7] de Boor, C.On calculating with B-splines[J].Journal of Approximation Theory, 1972, 6:50~62
[8] Piegl, L.Curve fitting algorithm for rough cutting[J].Computer Aided Design, 1986, 18:79~82
[9] Tiller,W.Rational B-splines for curve and surface representation [J]. IEEE Computer Graphics and its Application, 1983, 3:61~69
[10] Piegl, L.and Tiller,W..Curve and Surface construction using B-splines [J].Computer Aided Design, 1987, 19:487-498
[11] Ken S . Animation rotation with quaternion curves[J] . ACM SIGGRAPH ,1985,(19):245~254
[12]张怀.球面Bézier曲线的生成及其性质[D]:[博士学位论文].上海:复旦大学,1994.
[13]张怀.限制在光滑曲面上的C1曲线插值方法[J].计算机辅助设计与图形学学报,1997,9(5):385~390
[14]王小平,周儒荣,余湛悦等.参数曲面上的插值与混合[J].软件学报,2004,15(3):451~460
[15]蔺宏伟,王国瑾.光滑曲面上的G1插值曲线[J].计算机辅助设计与图形学学报,2003,15(5):541~546
[16]王小平,周儒荣,王志国等.利用Coons曲面构造曲面上的G1插值曲线[J].东南大学学报(英文版),2004,20(2):187~190
[17] Hartmann E.G2interpolation and blending on surfaces [J].The Visual Computer,1996,12(4): 181~192.
[18]胡淼、杨勋年.保测地曲率的曲面曲线设计[J].计算机辅助设计与图形学学报,2005, 17(5):981~985
[19 ] Li J,Hoschek J ,Hartmann E.Gn-1 functional splines for interpolation and approximation of Curves,surfaces and solids [J].Computer Aided Geometric Design ,1990,7(2) : 209~220.
[20]唐月红,李卫国,孙本华.基于曲面上光滑插值方法的飞机表面Cp值重建[J].航空学报,2001,22(3):250~252
[21]徐国良.曲面上数据的C1有理插值[J].计算数学, 1997,19 (4) : 431~437.
[22] Pottmann H.Interpolation on surfaces using minimum norm networks[J].CAGD, 1992, 9 (1) :51~67
[23]李晓武,雷毅.平面曲线向自由曲面映射的探讨[A].见:第1届全国几何设计与计算学术会议论文集[C].青岛:石油大学出版社,2002.252-256
[24]宋琨,张渭源.用贝塞尔曲线拟合服装结构曲线的处理方法[J].国际纺织导报,2001,4:76~79
[25]徐良宏,孟勇,陈铁.给定两端点及端点处切方向和曲率的空间Bézier曲线的插值问题[J].数值计算与计算机应用,2001,2:81~86
[26] Ravi Kumar G V V , Prabha S , et al.Geodesic curve computations onsurfaces[J].Computer- Aided Geometric Design,2003 , 20 (2):119～133
[27] Azariadis P N , Aspragathos N A.Geodesic curvature preservation in surface flattening through constrained global optimization[J].Computer-Aided Design,2001,33 (8):581～591
[28] Do Carmo M P.Differential Geometry of Curves and Surfaces[M].Prentice-Hall, Englewood Cliffs,1976
[29] Hotz I,Hagen H,Visualizing geodesics[A].In: Proceedings IEEE Visualization.Salt Lake City,2000.311～318
[30] H Tam , H Xu , Z-Zhou.Iso-planar interpolation for the machining of implicit surfaces [J].Computer-Aided Design, 2002, 34 (2) : 125～136
[31]王仲才,欧阳崇珍.可展曲面上的曲线展平线的一般计算方法[J].数学的实践与认识,1976,1:21~25
[32]王仲才,欧阳崇珍,裘伯铭.平面曲线的弯曲线的一般计算方法[J].南昌大学学报,1979,2:20~29
[33]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
[34]满家巨,胡事民,雍俊海等.B样条曲线的节点去除与光顺[J].软件学报,2001,12(1): 143~147
[35] Kjellander J A P.Smoothing of cubic parametric splines[J].CAD, 1983,15(3): 175~179.
[36] Kjellander J A P.Smoothing of cubic parametric surface [J].CAD, 1983, 15(5): 288~293.
[37]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
[38]忻云龙,双三次样条函数及其在曲面光顺中的应用[J],复旦学报,自然科学版,1977, 63-68
[39]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
[40]刘舸,李柏林.一种新的曲线光顺优化模型[J].西南交通大学学报,2002,37(5):584~587
[41]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
[42]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,2003,32:92~94
[43]穆国旺,宋秀琴.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,2005,6:118~121
[44]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(12):1109~1113
[45]戴霞.计算机辅助几何设计中的曲线光顺算法研究[D]:[硕士学位论文].陕西:西北工业大学,2004
[36]刘定焜.曲线曲面光顺的理论研究与实现[D]:[硕士学位论文].陕西:西北工业大学,2001
[47]袁琦睦,林意,刘飞.一种新的自由曲线的造型方法[J].东华大学学报,2005,31(4):121~124
[48]吴晓勤,唐运梅.曲率连续的三角B样条曲线与曲面[J].计算机应用与软件,2005,22(1):118~120
[49]孙倩.一类含参数均匀三角样条及其插值曲线[D]:[硕士学位论文].安徽:合肥工业大学,2006
[2]朱心雄等.自由曲线曲面造型技术[M].北京:科学出版社,2000:66~87,348~365
[3]施法中.计算机辅助几何设计与非均匀有理B样条(CAGD&NURBS).北京航空航天大学出版社,1994,448~450
[4]陈省身,陈维桓.微分几何讲义[M].(第二版).北京:北京大学出版社,2001:322~352
[5]苏步青,刘鼎元.初等微分几何[M].上海:上海科学技术出版社,1985:177~220
[6] Bezier P E.Numerical control-Mathematics and applications[M].Forrest trans.Wiley,London,1972.
[7] de Boor, C.On calculating with B-splines[J].Journal of Approximation Theory, 1972, 6:50~62
[8] Piegl, L.Curve fitting algorithm for rough cutting[J].Computer Aided Design, 1986, 18:79~82
[9] Tiller,W.Rational B-splines for curve and surface representation [J]. IEEE Computer Graphics and its Application, 1983, 3:61~69
[10] Piegl, L.and Tiller,W..Curve and Surface construction using B-splines [J].Computer Aided Design, 1987, 19:487-498
[11] Ken S . Animation rotation with quaternion curves[J] . ACM SIGGRAPH ,1985,(19):245~254
[12]张怀.球面Bézier曲线的生成及其性质[D]:[博士学位论文].上海:复旦大学,1994.
[13]张怀.限制在光滑曲面上的C1曲线插值方法[J].计算机辅助设计与图形学学报,1997,9(5):385~390
[14]王小平,周儒荣,余湛悦等.参数曲面上的插值与混合[J].软件学报,2004,15(3):451~460
[15]蔺宏伟,王国瑾.光滑曲面上的G1插值曲线[J].计算机辅助设计与图形学学报,2003,15(5):541~546
[16]王小平,周儒荣,王志国等.利用Coons曲面构造曲面上的G1插值曲线[J].东南大学学报(英文版),2004,20(2):187~190
[17] Hartmann E.G2interpolation and blending on surfaces [J].The Visual Computer,1996,12(4): 181~192.
[18]胡淼、杨勋年.保测地曲率的曲面曲线设计[J].计算机辅助设计与图形学学报,2005, 17(5):981~985
[19 ] Li J,Hoschek J ,Hartmann E.Gn-1 functional splines for interpolation and approximation of Curves,surfaces and solids [J].Computer Aided Geometric Design ,1990,7(2) : 209~220.
[20]唐月红,李卫国,孙本华.基于曲面上光滑插值方法的飞机表面Cp值重建[J].航空学报,2001,22(3):250~252
[21]徐国良.曲面上数据的C1有理插值[J].计算数学, 1997,19 (4) : 431~437.
[22] Pottmann H.Interpolation on surfaces using minimum norm networks[J].CAGD, 1992, 9 (1) :51~67
[23]李晓武,雷毅.平面曲线向自由曲面映射的探讨[A].见:第1届全国几何设计与计算学术会议论文集[C].青岛:石油大学出版社,2002.252-256
[24]宋琨,张渭源.用贝塞尔曲线拟合服装结构曲线的处理方法[J].国际纺织导报,2001,4:76~79
[25]徐良宏,孟勇,陈铁.给定两端点及端点处切方向和曲率的空间Bézier曲线的插值问题[J].数值计算与计算机应用,2001,2:81~86
[26] Ravi Kumar G V V , Prabha S , et al.Geodesic curve computations onsurfaces[J].Computer- Aided Geometric Design,2003 , 20 (2):119～133
[27] Azariadis P N , Aspragathos N A.Geodesic curvature preservation in surface flattening through constrained global optimization[J].Computer-Aided Design,2001,33 (8):581～591
[28] Do Carmo M P.Differential Geometry of Curves and Surfaces[M].Prentice-Hall, Englewood Cliffs,1976
[29] Hotz I,Hagen H,Visualizing geodesics[A].In: Proceedings IEEE Visualization.Salt Lake City,2000.311～318
[30] H Tam , H Xu , Z-Zhou.Iso-planar interpolation for the machining of implicit surfaces [J].Computer-Aided Design, 2002, 34 (2) : 125～136
[31]王仲才,欧阳崇珍.可展曲面上的曲线展平线的一般计算方法[J].数学的实践与认识,1976,1:21~25
[32]王仲才,欧阳崇珍,裘伯铭.平面曲线的弯曲线的一般计算方法[J].南昌大学学报,1979,2:20~29
[33]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
[34]满家巨,胡事民,雍俊海等.B样条曲线的节点去除与光顺[J].软件学报,2001,12(1): 143~147
[35] Kjellander J A P.Smoothing of cubic parametric splines[J].CAD, 1983,15(3): 175~179.
[36] Kjellander J A P.Smoothing of cubic parametric surface [J].CAD, 1983, 15(5): 288~293.
[37]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
[38]忻云龙,双三次样条函数及其在曲面光顺中的应用[J],复旦学报,自然科学版,1977, 63-68
[39]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
[40]刘舸,李柏林.一种新的曲线光顺优化模型[J].西南交通大学学报,2002,37(5):584~587
[41]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
[42]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,2003,32:92~94
[43]穆国旺,宋秀琴.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,2005,6:118~121
[44]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(12):1109~1113
[45]戴霞.计算机辅助几何设计中的曲线光顺算法研究[D]:[硕士学位论文].陕西:西北工业大学,2004
[36]刘定焜.曲线曲面光顺的理论研究与实现[D]:[硕士学位论文].陕西:西北工业大学,2001
[47]袁琦睦,林意,刘飞.一种新的自由曲线的造型方法[J].东华大学学报,2005,31(4):121~124
[48]吴晓勤,唐运梅.曲率连续的三角B样条曲线与曲面[J].计算机应用与软件,2005,22(1):118~120
[49]孙倩.一类含参数均匀三角样条及其插值曲线[D]:[硕士学位论文].安徽:合肥工业大学,2006