G~2四次Bézier插值曲线的构造
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摘要
在CAD/CAM造型系统中,随着曲线曲面造型技术的发展,人们提出了许多相关的理论。CAGD主要研究以复杂方式自由变化的曲线曲面,即所谓的自由型曲线曲面。Bezier曲线和曲面广泛应用于CAGD和计算机图形学中,并且在形状设计方面有很多好的性质。所以对Bezier曲线或者曲面的设计和形状修改是曲线曲面设计的一个重要的问题。当生成Bezier曲线或曲面时,我们往往因为设计的需要而对它做形状的修改。然而实际的Bezier曲线或曲面的形状修改问题往往可以归结为数学优化问题,即可以通过用解方程的方式解决这类问题。
参数曲线曲面造型按用户提供的初始信息不同可分为两类:一类是自由设计方法,它只要求设计者根据构思给出一些控制点和控制参数来定义曲线和曲面,然后在设计过程中允许改变这些控制点和参数来调整曲线和曲面的形状,直至它们符合设计要求为止。另一类是插值或逼近法(工程上统称为拟合法),其特点是给定一组离散点,要求生成的曲线或曲面要么通过所有这些点(成为插值曲线或曲面),要么以一定的精度贴近这些点(称为逼近曲线和曲面)。这两类方法生成的曲线曲面的形状都受参数化的影响。参数化既决定了所表示曲线曲面的形状,也决定了该曲线曲面上的点与其参数域内的点(即参数值)之间的一种对应关系。由此可见,参数化方法和插值逼近技术是曲线曲面造型的基础问题,具有重要的理论价值和实际意义。
另外,在参数曲线曲面的设计应用中,单独的一段Bezier曲线曲面表示能力有限,特别是对飞机、汽车等现实生活中千姿百态的自由曲线曲面形状,由于它们形状复杂,光顺性要求极高,需要分段、分片进行表示。用户首先经常遇到的一个问题就是延长以后的曲线不能满足精度要求,需要重新调整。
针对以上的问题本文提出了一种构造G~2连续的四次Bezier插值曲线的新方法,主要以曲线的二阶几何连续和曲线的能量为约束条件,进行形状优化设计。根据CAGD中几何连续性能客观准确地度量参数样条曲线连接光滑度的特点,用在每相邻的两个型值点之间增加一个控制点和两个自由度构造一段Bezier曲线的方法,研究了两条Bezier曲线在G~2连续下的光滑连接。同时给出了相关的实例。与已有的构造G~2连续曲线的方法相比,新方法构造的曲线灵活性更强且具有局部可调整性。用新方法构造G~2连续曲线可提供更多的自由度,用于提高设计和构造曲线的灵活性并且很好的控制曲线的形状。
In CAD/CAM modeling system,with the development of Surface modeling, more theories were proposed.Computer-Aided-Geometric-Design mainly researches on free curve and surface.And Bezier curve and surfaces are one kind of the most commonly used parametric curves in CAGD and Computer Graphics.It is an important problem to modify the shape of the Bezier curve and surface.Developing more convenient techniques for designing and modifying Bezier curve/surface is an important problem.When the Bezier curve/surface is generated,we always want to modify the shape of the curve/surface in order to satisfy the design.The solution of the practical problem often leads to solving optimization problem.
According to the initial information,curves and surfaces modeling can be divided into two methods.One is free form designing,based on control points and parameters,designers can define curves or surfaces and modify interactively until the shapes satisfy the design objective.The other is the technology of interpolation and approximation.Curves and surfaces reconstructed from the given points by interpolation are called as interpolation curves and surfaces or by approximation called as approximation curves and surfaces.These two methods are all influenced by parameterization.So curves and surfaces parameterization and the technologies of interpolation and approximation are the foundation of geometric modeling and have important theory value and practical meaning.
Besides,in the design of curves and surfaces one separate section of Bezier curves and surfaces has limited capacity,especially the various free curves and surfaces like automobile,airplane surface shapes,many sections are needed because of their complicated shapes and highly demanding smoothing.First of all a problem which users often encountered is that the extended curve unable to meet the accuracy requirements and it need to be re-adjusted.
According to the issue,this paper presents a new method for constructing a G~2 quartic Bezier interpolation curve,referring to the curves' G~2 continuity and energy constraint to optimize the shape.According to the fact that geometric continuity in CAGD can accurately and objectively measure the degree of smooth for the connection between two spline parameter curves,by adding a control point and two degree of freedom on each sub-interval to construct a segment of the Bezier curve, we research the smooth connection between Bezier curves in G~2 continuity and proposes some examples.Compared with the existing methods of constructing G~2 curves,the curve by the new method has the properties that it has better flexibility and is locally adjustable.Moreover,G~2 curve by the new method offers additional degrees of freedom which can be used to increase the flexibility of construction and design to control the shape of the curve and therefore make the shape more desirable.
参数曲线曲面造型按用户提供的初始信息不同可分为两类:一类是自由设计方法,它只要求设计者根据构思给出一些控制点和控制参数来定义曲线和曲面,然后在设计过程中允许改变这些控制点和参数来调整曲线和曲面的形状,直至它们符合设计要求为止。另一类是插值或逼近法(工程上统称为拟合法),其特点是给定一组离散点,要求生成的曲线或曲面要么通过所有这些点(成为插值曲线或曲面),要么以一定的精度贴近这些点(称为逼近曲线和曲面)。这两类方法生成的曲线曲面的形状都受参数化的影响。参数化既决定了所表示曲线曲面的形状,也决定了该曲线曲面上的点与其参数域内的点(即参数值)之间的一种对应关系。由此可见,参数化方法和插值逼近技术是曲线曲面造型的基础问题,具有重要的理论价值和实际意义。
另外,在参数曲线曲面的设计应用中,单独的一段Bezier曲线曲面表示能力有限,特别是对飞机、汽车等现实生活中千姿百态的自由曲线曲面形状,由于它们形状复杂,光顺性要求极高,需要分段、分片进行表示。用户首先经常遇到的一个问题就是延长以后的曲线不能满足精度要求,需要重新调整。
针对以上的问题本文提出了一种构造G~2连续的四次Bezier插值曲线的新方法,主要以曲线的二阶几何连续和曲线的能量为约束条件,进行形状优化设计。根据CAGD中几何连续性能客观准确地度量参数样条曲线连接光滑度的特点,用在每相邻的两个型值点之间增加一个控制点和两个自由度构造一段Bezier曲线的方法,研究了两条Bezier曲线在G~2连续下的光滑连接。同时给出了相关的实例。与已有的构造G~2连续曲线的方法相比,新方法构造的曲线灵活性更强且具有局部可调整性。用新方法构造G~2连续曲线可提供更多的自由度,用于提高设计和构造曲线的灵活性并且很好的控制曲线的形状。
In CAD/CAM modeling system,with the development of Surface modeling, more theories were proposed.Computer-Aided-Geometric-Design mainly researches on free curve and surface.And Bezier curve and surfaces are one kind of the most commonly used parametric curves in CAGD and Computer Graphics.It is an important problem to modify the shape of the Bezier curve and surface.Developing more convenient techniques for designing and modifying Bezier curve/surface is an important problem.When the Bezier curve/surface is generated,we always want to modify the shape of the curve/surface in order to satisfy the design.The solution of the practical problem often leads to solving optimization problem.
According to the initial information,curves and surfaces modeling can be divided into two methods.One is free form designing,based on control points and parameters,designers can define curves or surfaces and modify interactively until the shapes satisfy the design objective.The other is the technology of interpolation and approximation.Curves and surfaces reconstructed from the given points by interpolation are called as interpolation curves and surfaces or by approximation called as approximation curves and surfaces.These two methods are all influenced by parameterization.So curves and surfaces parameterization and the technologies of interpolation and approximation are the foundation of geometric modeling and have important theory value and practical meaning.
Besides,in the design of curves and surfaces one separate section of Bezier curves and surfaces has limited capacity,especially the various free curves and surfaces like automobile,airplane surface shapes,many sections are needed because of their complicated shapes and highly demanding smoothing.First of all a problem which users often encountered is that the extended curve unable to meet the accuracy requirements and it need to be re-adjusted.
According to the issue,this paper presents a new method for constructing a G~2 quartic Bezier interpolation curve,referring to the curves' G~2 continuity and energy constraint to optimize the shape.According to the fact that geometric continuity in CAGD can accurately and objectively measure the degree of smooth for the connection between two spline parameter curves,by adding a control point and two degree of freedom on each sub-interval to construct a segment of the Bezier curve, we research the smooth connection between Bezier curves in G~2 continuity and proposes some examples.Compared with the existing methods of constructing G~2 curves,the curve by the new method has the properties that it has better flexibility and is locally adjustable.Moreover,G~2 curve by the new method offers additional degrees of freedom which can be used to increase the flexibility of construction and design to control the shape of the curve and therefore make the shape more desirable.
引文
[1]施法中 计算机辅助几何设计与非均匀有理B样条 高等教育出版社2001
[2]张彩明,杨兴强等计算机图形学科学出版社2005
[3]Schoenberg I J.Contributions to the problem of approximation of equidistant data by analytic functions.Quart,Appl.Math,1946,4:45-99
[4]Schoenberg I J.Contributions to the problem of application of equidistant data by analytic functions.Quart,Appl.Math,1946,45-99
[5]Farin G.Curves and surfaces for computer aided geometric design:A practical guide.New York:Academic Press,1989,111
[6]Ahlberg J H,Nilson E N,walsh J L.The theory of splines and their applications.New York:Academic Press,1967,51
[7]de Boor C.A practical guide to splines.New York:Springer Verlag,1978,318
[8]Faux I D,Pratt M J.Computational geometry for design and manufacture.Chichester:Ellis Horwood Ltd,1979,176
[9]Su B,Liu D.Computer geometry.Shanghai:Shanghai Academic Press,1982,29-32(in Chinese)
(苏步青,刘鼎元计算几何上海:上海科学技术出版社,1982,29-32)
[10]王仁宏数值逼近北京:高等教育出版社1999
[11]王幼宁,宋家宏,苏效乐 空间曲线段间的光滑连接 北京师范大学学报(自然科学版),2001.2,37(1):16-18
[12]Ren-Hongwang,Jing-Xinwang.poisedness of interpolation problem for splines.Applied Numerical Mathematics,54(2005):95-103
[13]I.J.Schoenberg,A.Whiteny.On Po'lya frenquency function,(3):The positivity of translation determinantswith an application to the interpolation problem by spline curves,Trans.Amer.Math.Soc.74(1953):246-259
[14]Bezier P.E.The mathematical basis of the UNISURF CAD system,Butternorth,London,1986.
[15]Bezier P.E.Numerical control-Mathematics and Applications.York,1972
[16]BezierP.E Mathematical and practical possibilities of UNISURF,Computer Aided Geometric Design,in:Barnhill R.E.and Riesenfeld R.E.,eds.Academic Press,New York,1974,127-152
[17]Bezier P.E Numerical contro-Mathematics and applications,Forrest trans.wiley,London,1972
[18]Gordonw.J,Ricsenfeld R.E Bernstein-Bezier methods for the CAD of free-form curves and surfaces.JACM,1974,21(2):293-310
[19]Piegl L.The sphere as a rational Bezier surfaces.CAGD,1986,3:45-52
[20]Farin G.Algorithms for rational Bezier curves.CAD,1983,15:73-77
[21]TillerW.Rational B2 splines for curves and surface representation.IEEE Computer Graphics and Applications,1983,3(01):61- 69
[22]Gerald Farin.Curves and surfaces for computer aided geometric design.A Practical Guide Fourth Edition,Elsevier Science Ltd,1997,96,
[23]梁友栋,叶修梓 曲线几何连续性及其应用[J]计算数学,1989,04:60-70
[24]姜献峰,梁友栋 有理Bezier曲线的几何连续条件及其应用[J]高等学校计算数学学报,1992,04:52-65
[25]张三元 基于代数曲线的G2连续的曲线造型方法[J]计算机学报,2000,23(2):153-157
[26]彭丰富,韩旭里一类C-2连续分段四次代数样条[J]计算机辅助设计与图形学学报,2006,18(09):1420-1425
[27]张三元一种G-2连续的二次曲线样条插值方法[J]计算机辅助设计与图形学学报,2000,12(06):419-422
[28]张彩明,汪嘉业可调整C-2四次Bezier插值曲线的构造[J]计算机学报,2004.27(12):1665-1671
[29]Kjellander JAP.Smoothing of cubic parametric splines.Computer Aided Design,1983,15(3):175-179
[30]Cheng.F.and Barsky.B.A..Interproximation:interpolation and approximation using cubic spline curves.Computer Aided Design,1991,10:700-706
[31]Cheng.F.,Barsky.B.A..Interproximation using cubic B-spline curves,In Modeling in Computer Graphics,eds.B.Falcidieno,T.L.Kunii.Springer-Verlag,Berlin,1993,pp:359-374
[32]Lee,E.T.Y.Energy,fairness and a counter example.Computer Aided Design,1990,22(1):37-40
[33]Hosaka M.,Kimura K.Synthesis methods of curves and surfaces in CAD.In IEEE Conference Proceedings:Interactive Techniques in Computer Aided Design.Bologna,Italy,1978,pp:151-155
[34]Kallay M.and Ravani B.Optimal twist vectors as a tool for interpolating a network of curveswith a minimum energy surface.Computer-Aided Geometric Design,1990,7:465-473
[35]Wang X,Cheng F,Barsky B.Energy and B-spline interproximation.Computer Aided Design,1997,29(7):485-496.
[36]Akima H.A new method of interpolation and smooth curve fitting based on local procedures.J.ACM,1970,17(4):589-602
[37]Zhang C,Cheng F,Miura K T.A method for determining knots in parametric curve interpolation.Computer Aided Geometric Design,1998,15:399-417
[38]Hyman JM.Accurate monotonicity preserving cubic interpolation.SIAM J.Sci.Statist.Comput.1983,4(4):645-654
[39]Delbourgo R,Gregory J A.Shape preserving piecewise rational quadratic interpolation.SIAM J.Sci.Statist.Comput.1985,6(4):967-976
[40]Fritsch F N,Carlson R E.Monotone piecewise cubic interpolation.SIAM J.Numer,Anal,1980,17(2):238-246
[41]Brodlie Kw.A review of methods for curve and function drawing In:Brodlie K W.Mathematical methods in computer graphics and design.London:Academic Press,1980,1-37
[42]Adleman L.Molecular.Computation of solutions to combinatorial problems.Science,1994,266(09):1021- 1024
[43]潘日晶 满足数点切向约束的二次B样条插值曲线计算机学报,2007,30(12): 2132-2141
[2]张彩明,杨兴强等计算机图形学科学出版社2005
[3]Schoenberg I J.Contributions to the problem of approximation of equidistant data by analytic functions.Quart,Appl.Math,1946,4:45-99
[4]Schoenberg I J.Contributions to the problem of application of equidistant data by analytic functions.Quart,Appl.Math,1946,45-99
[5]Farin G.Curves and surfaces for computer aided geometric design:A practical guide.New York:Academic Press,1989,111
[6]Ahlberg J H,Nilson E N,walsh J L.The theory of splines and their applications.New York:Academic Press,1967,51
[7]de Boor C.A practical guide to splines.New York:Springer Verlag,1978,318
[8]Faux I D,Pratt M J.Computational geometry for design and manufacture.Chichester:Ellis Horwood Ltd,1979,176
[9]Su B,Liu D.Computer geometry.Shanghai:Shanghai Academic Press,1982,29-32(in Chinese)
(苏步青,刘鼎元计算几何上海:上海科学技术出版社,1982,29-32)
[10]王仁宏数值逼近北京:高等教育出版社1999
[11]王幼宁,宋家宏,苏效乐 空间曲线段间的光滑连接 北京师范大学学报(自然科学版),2001.2,37(1):16-18
[12]Ren-Hongwang,Jing-Xinwang.poisedness of interpolation problem for splines.Applied Numerical Mathematics,54(2005):95-103
[13]I.J.Schoenberg,A.Whiteny.On Po'lya frenquency function,(3):The positivity of translation determinantswith an application to the interpolation problem by spline curves,Trans.Amer.Math.Soc.74(1953):246-259
[14]Bezier P.E.The mathematical basis of the UNISURF CAD system,Butternorth,London,1986.
[15]Bezier P.E.Numerical control-Mathematics and Applications.York,1972
[16]BezierP.E Mathematical and practical possibilities of UNISURF,Computer Aided Geometric Design,in:Barnhill R.E.and Riesenfeld R.E.,eds.Academic Press,New York,1974,127-152
[17]Bezier P.E Numerical contro-Mathematics and applications,Forrest trans.wiley,London,1972
[18]Gordonw.J,Ricsenfeld R.E Bernstein-Bezier methods for the CAD of free-form curves and surfaces.JACM,1974,21(2):293-310
[19]Piegl L.The sphere as a rational Bezier surfaces.CAGD,1986,3:45-52
[20]Farin G.Algorithms for rational Bezier curves.CAD,1983,15:73-77
[21]TillerW.Rational B2 splines for curves and surface representation.IEEE Computer Graphics and Applications,1983,3(01):61- 69
[22]Gerald Farin.Curves and surfaces for computer aided geometric design.A Practical Guide Fourth Edition,Elsevier Science Ltd,1997,96,
[23]梁友栋,叶修梓 曲线几何连续性及其应用[J]计算数学,1989,04:60-70
[24]姜献峰,梁友栋 有理Bezier曲线的几何连续条件及其应用[J]高等学校计算数学学报,1992,04:52-65
[25]张三元 基于代数曲线的G2连续的曲线造型方法[J]计算机学报,2000,23(2):153-157
[26]彭丰富,韩旭里一类C-2连续分段四次代数样条[J]计算机辅助设计与图形学学报,2006,18(09):1420-1425
[27]张三元一种G-2连续的二次曲线样条插值方法[J]计算机辅助设计与图形学学报,2000,12(06):419-422
[28]张彩明,汪嘉业可调整C-2四次Bezier插值曲线的构造[J]计算机学报,2004.27(12):1665-1671
[29]Kjellander JAP.Smoothing of cubic parametric splines.Computer Aided Design,1983,15(3):175-179
[30]Cheng.F.and Barsky.B.A..Interproximation:interpolation and approximation using cubic spline curves.Computer Aided Design,1991,10:700-706
[31]Cheng.F.,Barsky.B.A..Interproximation using cubic B-spline curves,In Modeling in Computer Graphics,eds.B.Falcidieno,T.L.Kunii.Springer-Verlag,Berlin,1993,pp:359-374
[32]Lee,E.T.Y.Energy,fairness and a counter example.Computer Aided Design,1990,22(1):37-40
[33]Hosaka M.,Kimura K.Synthesis methods of curves and surfaces in CAD.In IEEE Conference Proceedings:Interactive Techniques in Computer Aided Design.Bologna,Italy,1978,pp:151-155
[34]Kallay M.and Ravani B.Optimal twist vectors as a tool for interpolating a network of curveswith a minimum energy surface.Computer-Aided Geometric Design,1990,7:465-473
[35]Wang X,Cheng F,Barsky B.Energy and B-spline interproximation.Computer Aided Design,1997,29(7):485-496.
[36]Akima H.A new method of interpolation and smooth curve fitting based on local procedures.J.ACM,1970,17(4):589-602
[37]Zhang C,Cheng F,Miura K T.A method for determining knots in parametric curve interpolation.Computer Aided Geometric Design,1998,15:399-417
[38]Hyman JM.Accurate monotonicity preserving cubic interpolation.SIAM J.Sci.Statist.Comput.1983,4(4):645-654
[39]Delbourgo R,Gregory J A.Shape preserving piecewise rational quadratic interpolation.SIAM J.Sci.Statist.Comput.1985,6(4):967-976
[40]Fritsch F N,Carlson R E.Monotone piecewise cubic interpolation.SIAM J.Numer,Anal,1980,17(2):238-246
[41]Brodlie Kw.A review of methods for curve and function drawing In:Brodlie K W.Mathematical methods in computer graphics and design.London:Academic Press,1980,1-37
[42]Adleman L.Molecular.Computation of solutions to combinatorial problems.Science,1994,266(09):1021- 1024
[43]潘日晶 满足数点切向约束的二次B样条插值曲线计算机学报,2007,30(12): 2132-2141