三次插值PH样条与F-曲线拐点和奇点的研究
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摘要
PH曲线及样条和混合多项式曲线是近年来CAGD研究中的热点问题之一。本文围绕这两方面作了一些探讨研究。一、基于控制三角形周长最短设计平面三次PH样条;二、给出了平面三次F-曲线上拐点和奇异点存在的充要条件,并给出了在张纪文第一种三次F-Bezier定义下F-Bezier曲线的形状分类图。
第二章给出了平面三次PH样条基于控制三角形周长最短下的设计方法。主要采用以控制三角形周长最短为评价标准,PH样条在型值点以向量长度和转角方式给出的情况下,对各型值点处的切线斜率进行设计。并给出了控制多边形的周长表示和PH样条的弧长表示。
第三章给出了平面上广义F-曲线,三次F-曲线和张纪文第一种三次F-Bezier定义下F-Bezier曲线上的拐点和奇异点(包括尖点和重点)存在的充分必要条件。采用的主要方法是根据曲线的相对曲率定义了曲线的特征函数,通过适当的转化,使其成为实数域上的一个二次多项式函数,其实根(零点)对应于曲线上的拐点和奇异点。根据特征函数的零点分布规律容易得到对应曲线上的拐点和奇异点的分布规律。最后给出了F-Bezier曲线的形状分类图。由于F-曲线和C-曲线、H-曲线关系,实际上将C-曲线、H-曲线上拐点和奇异点分类用F-曲线统一起来。这些结果可用于检测F-曲线的拐点和奇异点,也可用在曲线造型设计和光顺插值中避免或排除多余拐点和奇异点。
This paper focuses on two problems,PH spline and hybrid polynomial curves, which are regarded as hot spots in recent CAGD research.First,we give the design of planar cubic PH-spline based on the shortest perimeter of control triangle.Second,we investigate the distribution of inflection points and singularities of planar cubic F-curve.And also we give the shape classification map of cubic F-Bezier curve which is the first form of F-Bezier defined by Zhang J.W.in 2005.
In Chapter two,we give the design of planar cubic PH-spline based on the shortest perimeter of control triangle.Using the minimum perimeter of control triangle as the evaluation of PH-spline,we design the slope of the tangent point.The data points is given by length and angle.Also we represent the perimeter of the control polygon and the arc length of the PH spline.
In Chapter three,we give the necessary and sufficient condition leading to inflection points and singularities(including cusps and loops) on planar generalized F-curve,planar cubic F-curve and cubic F-Bezier curve which is the first form of F-Bezier defined by Zhang J.W.in 2005.An eigenfunction of cubic F-curves according to the relative curvature of curves is introduced.Through the appropriate conversion,the eigenfunction can be made a quadratic polynomial function in real domain,whose zeros correspond to the points of inflexion and singularities of the curves above.Finally,the shape classification map of cubic F-Bezier curve is given.Thus the classification of inflection points and singularities on C-curves and H-curves are united by F-curve.These results can be used to detect the interior inflection point and singular points on F-curves and can also be used to avoid or exclude redundant inflection points and singularities.
第二章给出了平面三次PH样条基于控制三角形周长最短下的设计方法。主要采用以控制三角形周长最短为评价标准,PH样条在型值点以向量长度和转角方式给出的情况下,对各型值点处的切线斜率进行设计。并给出了控制多边形的周长表示和PH样条的弧长表示。
第三章给出了平面上广义F-曲线,三次F-曲线和张纪文第一种三次F-Bezier定义下F-Bezier曲线上的拐点和奇异点(包括尖点和重点)存在的充分必要条件。采用的主要方法是根据曲线的相对曲率定义了曲线的特征函数,通过适当的转化,使其成为实数域上的一个二次多项式函数,其实根(零点)对应于曲线上的拐点和奇异点。根据特征函数的零点分布规律容易得到对应曲线上的拐点和奇异点的分布规律。最后给出了F-Bezier曲线的形状分类图。由于F-曲线和C-曲线、H-曲线关系,实际上将C-曲线、H-曲线上拐点和奇异点分类用F-曲线统一起来。这些结果可用于检测F-曲线的拐点和奇异点,也可用在曲线造型设计和光顺插值中避免或排除多余拐点和奇异点。
This paper focuses on two problems,PH spline and hybrid polynomial curves, which are regarded as hot spots in recent CAGD research.First,we give the design of planar cubic PH-spline based on the shortest perimeter of control triangle.Second,we investigate the distribution of inflection points and singularities of planar cubic F-curve.And also we give the shape classification map of cubic F-Bezier curve which is the first form of F-Bezier defined by Zhang J.W.in 2005.
In Chapter two,we give the design of planar cubic PH-spline based on the shortest perimeter of control triangle.Using the minimum perimeter of control triangle as the evaluation of PH-spline,we design the slope of the tangent point.The data points is given by length and angle.Also we represent the perimeter of the control polygon and the arc length of the PH spline.
In Chapter three,we give the necessary and sufficient condition leading to inflection points and singularities(including cusps and loops) on planar generalized F-curve,planar cubic F-curve and cubic F-Bezier curve which is the first form of F-Bezier defined by Zhang J.W.in 2005.An eigenfunction of cubic F-curves according to the relative curvature of curves is introduced.Through the appropriate conversion,the eigenfunction can be made a quadratic polynomial function in real domain,whose zeros correspond to the points of inflexion and singularities of the curves above.Finally,the shape classification map of cubic F-Bezier curve is given.Thus the classification of inflection points and singularities on C-curves and H-curves are united by F-curve.These results can be used to detect the interior inflection point and singular points on F-curves and can also be used to avoid or exclude redundant inflection points and singularities.
引文
[1]Zhang,J.W.,Krause,EL.,Zhang,H.Y..Unifying C-curves and H-curves by extending the calculation to complex numbers.Computer Aided Geometric Design,2005,22(9):865-883.
[2]Barnhill R.E.,Riesenfeld R.E.Computer Aided Geometric Design,1974.NewYork:Academic Press.
[3]王国瑾,汪国昭,郑建民.计算机辅助几何设计.2001.施普林格-高等教育出版社.
[4]Yang,Q.Y.,Wang,GZ..Inflection points and singularities on C-curves.Computer Aided Geometric Design,2004,21(2).
[5]Wang,G.Z.,Yang,Q.Y..Planar cubic hybrid hyperbolic polynomial curve and its shape classification.Progress in Natural Science,2004,Vol.14,No.1,January.
[6]Cao,J.,Wang,G.Z..Relation among C-curve characterization diagrams.Journal of Zhejiang University SCIENCE A,2007,8(10).
[7]Farouki,R.T..Exact offset procedures for simple solids.Computer Aided Geometric Design,1985,2(4):257-279.
[8]Forrest,A.R..Shape classifcation of the non-rational twisted cubic curve in terms of Bezier polygons.CAD Group Document,1970,No.52,University of Cambridge.
[9]Zhang,J.W..C-Bezier curves and surfaces.Graphical Models and Image Processing,1999,61(1).
[10]Zhang,J.W..Two different forms of CB-splines.Computer Aided Geometric Design,1997,14(1).
[11]Zhang,J.W..C-curves:an extension of cubic curves.Computer Aided Geometric Design,1996,13(3).
[12]Li,Y.M.,Cripps,R.J..Indentification of inflection points and cusps on rational curves.Computer Aided Geometric Design,1997,14(5).
[13]Manocha,D.and Canny,J.F..Detecting cusps and inflection points in curves.Comput.Aided Geom.Design 9(1),1992,1-24.
[14]杨勤民.三次混合多项式曲线和区间曲面的研究.2002.浙江大学硕士学位论文.
[15]Monterde,J..Singularities of rational Bezier curves.Computer Aided Geometric Design,,2001,18(8).
[16]Sakai,M..Inflection points and singularities on planar rational cubic curve segments.Computer Aided Geometric Design,1999,16(3).
[17]Kim,D.S..Hodograph approach to geometric characterization of parametric cubic curves.Computer-Aided Design,1993,25(10):644-654.
[18]Meek,D.S.and Walton,D.J..Shape determination of planar uniform B-spline segments.Computer-Aided Design,1990,22(7):434-441.
[19]张丰伟.代数三角二阶混合基的显式表示和区间的扩展.2008.浙江大学硕士学位论文.
[20]Bezier,P.E..Mathematical and practical possibilities of UNISURF.in:Barnhill,R.E.and Riesenfeld,R.F.,eds.Computer Aided Geometric Design,1974,Academic Press,127-152.
[21]DeRose,T.D.and Stone,M..A geometric characterisation of parametric cubic curves.ACM Trans.Graphics 8,1989:147-163.
[22]Farin,G..curves and Surfaces for Computer Aided Geometric Design.Academic Press,1988,USA.
[23]Farin,G.and Sapodis,N..Curvature and the fairness of curves and surfaces.IEEE Computer Graphics and Application,1989,9(3.2):52-57.
[24]Forrest,A.R..Curves and Surfaces for computer-aided design.PhD. Dissertation, 1968,University of Cambridge, UK.
[25]Forrest, A.R.. The twisted cubic curve: A computer-aided geometric design approach. Computer-Aided Design,1980,12(3.4): 165-172.
[26]Li, Y.M. and Cripps, R.J.. Characterisation of planar rational cubic curves, in: 3rd SIAM Conference on Geometric Design, Tempe, Arizona ,1993.
[27]Li, Y.M. and Cripps, R.J.. Detecting and controlling inflection points on planar rational Bezier curves, in: Proceedings of 3rd International Conference on Computational Graphics and Visualisation Techniques, Algarve, Portugal, 1993,390-394.
[28]Liu, D.Y.. Rational Bezier curves. Acta Mathematicae Applicatae Sinica 8, 1985, 70-83.
[29]Mazure, M.L.. Chebyshev-Bernstein bases. Computer Aided Geometric Design, 1999,16(7):649-669.
[30]Patterson, R.R.. Parametric cubics as algebraic curves. Computer Aided Geometric Design, 1988,5:139-159.
[31]Primrose, E.J.. Planar Algebraic Curves. Macmillan, New York ,1955.
[32]Sederberg, T.W.. Degenerate parametric curves. Computer Aided Geometric Design, 1984,1: 301-307.
[33]Sederberg, T.W.. Improperly parametrised rational curves. Computer Aided Geometric Design, 1986, 3:67-75.
[34]Sederberg, T.W. and Meyers, R.J.. Loop detection in surface patch intersections. Computer Aided Geometric Design, 1988,5:161-171.
[35]Stone, M.C. and DeRose, T.D..A geometric characterization of parametric cubic curves.ACM Trans. Graphics ,1989,8(3.3):147-163.
[36]Su, B.Q. and Liu, D.Y.. An affine invariant and its application in computational geometry. Sci. Sinica(Ser.A) ,1983, 24(3):259-267.
[37]Su,B.Q.and Liu,D.Y..Computational Geometry -Curves and Surfaces Modeling.Chang Geng-zhe.Transl.(Academic Press,San Diego,CA.1989).
[38]Walker,R.J..Algebraic Curves,Princeton University Press,Princeton,NJ,1950.
[39]Wang,C.Y..Shape classification of the parametric cubic curve and parametric B-spline cubic curve.Computer-Aided Design,1981,13(4):199-206.
[40]Zhang,J.W..Two different forms of C-B-splines.Computer Aided Geometric Design,1997,14(1):31-41.
[41]Zhang,J.W..C-Bezier Curves and Surfaces.CVGIP:Graphical Models and Image Processing,1999,61(1):2-15.
[42]苏步青关于三次参数样条曲线的一个定理,应用数学学报,1977年第1期,49-54.
[43]苏步青关于三次参数样条曲线的一些注记,应用数学学报,1976年第1期,49-58.
[44]苏步青关于五次有理曲线的注记,应用数学学报,1977年第2期,80-89.
[45]苏步青有理整曲线的几个相对仿射不变量,复旦学报,自然科学版,1977年第2期,22-29.
[46]苏步青,刘鼎元论Bezier曲线的仿射不变量,计算数学,1980年第2期,289-298.
[47]Farouki,R.T.,Neff,C.A..Algebraic properties of plane offset curves.Computer Aided Geometric Design,1990,7(1-4):101-127.
[48]Farouki,R.T.,Sakkalis,T..Pythagorean hodographs.IBM Journal of Research and Development,1990,34(5):736-752.
[49]Lu W..Offset-rational parametric plane curves.Computer Aided Geometric Design,1995,12(6):601-616.
[50]Zheng J.M..On rational representation of offset curves.Chinese Science Bulletin,1995,40(1):9-10.
[51]Lu W.,Pottmann, H.. Pipe surfaces with rational spine curve are rational. Computer Aided Geometric Design, 1996,13(7): 621-628.
[52]Lu W.. Rationality of the offsets to algebraic curves and surfaces. Applied Mathematics, Journal of Chinese Universities, 1994, 9B(3): 265-278.
[53]Lu W.. Rational parameterization of quadrics and their offsets. Computing, 1996, 57(2): 135-147.
[54]Pottmann, H., Lu W., Ravani, B.. Rational ruled surfaces and their offsets. CVGIP: Graphical Models and Image Processing, 1996, 58(6): 544-552.
[55]Michael S. Floater..On the deviation of a parametric cubic spline interpolant from its data polygon.Computer Aided Geometric Design, 2008,25(3): 148-156.
[2]Barnhill R.E.,Riesenfeld R.E.Computer Aided Geometric Design,1974.NewYork:Academic Press.
[3]王国瑾,汪国昭,郑建民.计算机辅助几何设计.2001.施普林格-高等教育出版社.
[4]Yang,Q.Y.,Wang,GZ..Inflection points and singularities on C-curves.Computer Aided Geometric Design,2004,21(2).
[5]Wang,G.Z.,Yang,Q.Y..Planar cubic hybrid hyperbolic polynomial curve and its shape classification.Progress in Natural Science,2004,Vol.14,No.1,January.
[6]Cao,J.,Wang,G.Z..Relation among C-curve characterization diagrams.Journal of Zhejiang University SCIENCE A,2007,8(10).
[7]Farouki,R.T..Exact offset procedures for simple solids.Computer Aided Geometric Design,1985,2(4):257-279.
[8]Forrest,A.R..Shape classifcation of the non-rational twisted cubic curve in terms of Bezier polygons.CAD Group Document,1970,No.52,University of Cambridge.
[9]Zhang,J.W..C-Bezier curves and surfaces.Graphical Models and Image Processing,1999,61(1).
[10]Zhang,J.W..Two different forms of CB-splines.Computer Aided Geometric Design,1997,14(1).
[11]Zhang,J.W..C-curves:an extension of cubic curves.Computer Aided Geometric Design,1996,13(3).
[12]Li,Y.M.,Cripps,R.J..Indentification of inflection points and cusps on rational curves.Computer Aided Geometric Design,1997,14(5).
[13]Manocha,D.and Canny,J.F..Detecting cusps and inflection points in curves.Comput.Aided Geom.Design 9(1),1992,1-24.
[14]杨勤民.三次混合多项式曲线和区间曲面的研究.2002.浙江大学硕士学位论文.
[15]Monterde,J..Singularities of rational Bezier curves.Computer Aided Geometric Design,,2001,18(8).
[16]Sakai,M..Inflection points and singularities on planar rational cubic curve segments.Computer Aided Geometric Design,1999,16(3).
[17]Kim,D.S..Hodograph approach to geometric characterization of parametric cubic curves.Computer-Aided Design,1993,25(10):644-654.
[18]Meek,D.S.and Walton,D.J..Shape determination of planar uniform B-spline segments.Computer-Aided Design,1990,22(7):434-441.
[19]张丰伟.代数三角二阶混合基的显式表示和区间的扩展.2008.浙江大学硕士学位论文.
[20]Bezier,P.E..Mathematical and practical possibilities of UNISURF.in:Barnhill,R.E.and Riesenfeld,R.F.,eds.Computer Aided Geometric Design,1974,Academic Press,127-152.
[21]DeRose,T.D.and Stone,M..A geometric characterisation of parametric cubic curves.ACM Trans.Graphics 8,1989:147-163.
[22]Farin,G..curves and Surfaces for Computer Aided Geometric Design.Academic Press,1988,USA.
[23]Farin,G.and Sapodis,N..Curvature and the fairness of curves and surfaces.IEEE Computer Graphics and Application,1989,9(3.2):52-57.
[24]Forrest,A.R..Curves and Surfaces for computer-aided design.PhD. Dissertation, 1968,University of Cambridge, UK.
[25]Forrest, A.R.. The twisted cubic curve: A computer-aided geometric design approach. Computer-Aided Design,1980,12(3.4): 165-172.
[26]Li, Y.M. and Cripps, R.J.. Characterisation of planar rational cubic curves, in: 3rd SIAM Conference on Geometric Design, Tempe, Arizona ,1993.
[27]Li, Y.M. and Cripps, R.J.. Detecting and controlling inflection points on planar rational Bezier curves, in: Proceedings of 3rd International Conference on Computational Graphics and Visualisation Techniques, Algarve, Portugal, 1993,390-394.
[28]Liu, D.Y.. Rational Bezier curves. Acta Mathematicae Applicatae Sinica 8, 1985, 70-83.
[29]Mazure, M.L.. Chebyshev-Bernstein bases. Computer Aided Geometric Design, 1999,16(7):649-669.
[30]Patterson, R.R.. Parametric cubics as algebraic curves. Computer Aided Geometric Design, 1988,5:139-159.
[31]Primrose, E.J.. Planar Algebraic Curves. Macmillan, New York ,1955.
[32]Sederberg, T.W.. Degenerate parametric curves. Computer Aided Geometric Design, 1984,1: 301-307.
[33]Sederberg, T.W.. Improperly parametrised rational curves. Computer Aided Geometric Design, 1986, 3:67-75.
[34]Sederberg, T.W. and Meyers, R.J.. Loop detection in surface patch intersections. Computer Aided Geometric Design, 1988,5:161-171.
[35]Stone, M.C. and DeRose, T.D..A geometric characterization of parametric cubic curves.ACM Trans. Graphics ,1989,8(3.3):147-163.
[36]Su, B.Q. and Liu, D.Y.. An affine invariant and its application in computational geometry. Sci. Sinica(Ser.A) ,1983, 24(3):259-267.
[37]Su,B.Q.and Liu,D.Y..Computational Geometry -Curves and Surfaces Modeling.Chang Geng-zhe.Transl.(Academic Press,San Diego,CA.1989).
[38]Walker,R.J..Algebraic Curves,Princeton University Press,Princeton,NJ,1950.
[39]Wang,C.Y..Shape classification of the parametric cubic curve and parametric B-spline cubic curve.Computer-Aided Design,1981,13(4):199-206.
[40]Zhang,J.W..Two different forms of C-B-splines.Computer Aided Geometric Design,1997,14(1):31-41.
[41]Zhang,J.W..C-Bezier Curves and Surfaces.CVGIP:Graphical Models and Image Processing,1999,61(1):2-15.
[42]苏步青关于三次参数样条曲线的一个定理,应用数学学报,1977年第1期,49-54.
[43]苏步青关于三次参数样条曲线的一些注记,应用数学学报,1976年第1期,49-58.
[44]苏步青关于五次有理曲线的注记,应用数学学报,1977年第2期,80-89.
[45]苏步青有理整曲线的几个相对仿射不变量,复旦学报,自然科学版,1977年第2期,22-29.
[46]苏步青,刘鼎元论Bezier曲线的仿射不变量,计算数学,1980年第2期,289-298.
[47]Farouki,R.T.,Neff,C.A..Algebraic properties of plane offset curves.Computer Aided Geometric Design,1990,7(1-4):101-127.
[48]Farouki,R.T.,Sakkalis,T..Pythagorean hodographs.IBM Journal of Research and Development,1990,34(5):736-752.
[49]Lu W..Offset-rational parametric plane curves.Computer Aided Geometric Design,1995,12(6):601-616.
[50]Zheng J.M..On rational representation of offset curves.Chinese Science Bulletin,1995,40(1):9-10.
[51]Lu W.,Pottmann, H.. Pipe surfaces with rational spine curve are rational. Computer Aided Geometric Design, 1996,13(7): 621-628.
[52]Lu W.. Rationality of the offsets to algebraic curves and surfaces. Applied Mathematics, Journal of Chinese Universities, 1994, 9B(3): 265-278.
[53]Lu W.. Rational parameterization of quadrics and their offsets. Computing, 1996, 57(2): 135-147.
[54]Pottmann, H., Lu W., Ravani, B.. Rational ruled surfaces and their offsets. CVGIP: Graphical Models and Image Processing, 1996, 58(6): 544-552.
[55]Michael S. Floater..On the deviation of a parametric cubic spline interpolant from its data polygon.Computer Aided Geometric Design, 2008,25(3): 148-156.