拥有指数参数的新三角基
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摘要
目的为了使构造的曲线拥有传统Bézier曲线的良好性质,同时还具备形状可调性、逼近性、保形性以及实用性。方法首先在拟扩展切比雪夫空间的框架下,构造了一类具有全正性的拟三次三角Bernstein基函数,并给出了该基函数的性质;基于此基函数,构造了相应的拟三次三角Bézier曲线,分析了其曲线的性质,得到了生成曲线的割角算法以及C1,C2光滑拼接条件,同时还提出了一种估计曲线逼近控制多边形程度的三角Bernstein算子;接着在拟三次三角Bernstein基函数的基础上提出一种三角域上带3个指数参数的拟三次三角Bernstein-Bézier基,基于此基生成了一种三角域上的拟三次三角Bernstein-Bézier曲面,该曲面可以构建边界为椭圆弧、抛物线弧以及圆弧的曲面,此外,还提出一种实用的de-Casteljau-type算法,同时还给出了连接两个曲面的G1连续条件。结果实验表明,本文在拟扩展切比雪夫空间中构造的具有全正性的曲线曲面,能够灵活地进行形状调整,而且具有良好的逼近性以及适用性。结论本文在拟扩展切比雪夫空间的框架下构造了一类具有全正性的基函数,并以此基函数进行曲线曲面构造。实验表明本文构造的曲线具备传统三次Bézier曲线的所有优良性质,而且具有灵活的形状可调性。随着参数的增大,所生成的曲线能够更加逼近控制多边形,模拟控制多边形的行为。此外,本文在三角域上构造的曲面能够生成边界为椭圆弧的曲面。综上,本文提出的基函数满足几何工业的需要,是一种实用的方法。
Objective The construction of basis functions has consistently been a difficult point of computer-aided geometric design( CAGD). The construction of a class of practical basis functions often plays a decisive role in the development of the geometric industry. The traditional Bézier curves and B-splines have been widely used in CAGD. However,when the control points are determined,the generated curve is relatively fixed with respect to the control points and has a certain rigidity. Although the proposed rational B-spline curve can adjust the curve by adjusting the weight factor,the rational methods have difficulty in predicting the influence of the weight factor on the curve due to its own shortcomings. Researchers have exerted efforts in the past two decades to solve this problem. However,most of the improved methods have the basic properties of the traditional Bézier method and the B-spline method,such as affine invariance,convex hull,non-negativity,geometric invariance and flexible shape adjustability. Moreover,the proposed curve can accurately represent special curves used in engineering,such as conic and hyperbolic curves. However,most of the literature does not discuss the variation diminishing the generated curve. The curve with vanishing variation must have convexity. The curve with the total positivity must have diminishing variation. Therefore,the total positivity of the basis functions indicates that these functions are suitable for geometric design. We can easily obtain rectangular patches with shape parameters through these new curves. However,the Bernstein-Bézier patch over the triangular domain is not a tensor product patch exactly. Therefore,we cannot obtain triangular surfaces with an adjustable shape using the method of tensor product. Surface modeling over triangular domain is important for many applications. Thus,the practical methods for generating surfaces over a triangular domain must be explored. The blossom property in quasi extended Chebyshev space is used to construct a group of optimal normalized entirely positive basis for curve and surface construction. This method enables the extended curve and surface,thereby maintaining the good nature of the traditional Bézier and B-spline methods while preserving shape,shape adjustability,and practicability. Method A class of cubic trigonometric quasi Bernstein basis functions with total positivity is constructed under the framework of the extended Chebyshev space,and the properties of the basis functions are provided.The corresponding curve is presented based on this basis function. The properties of the curve are analyzed. The cutting algorithm of the curve and the smooth connecting conditions are obtained. A trigonometric quasi Bernstein operator for estimating the degree of the control polygon is also proposed. Then,based on the cubic trigonometric quasi Bernstein basis function,a class of trigonometric polynomial basis functions with three shape parameters over the triangular domain is proposed. A type of triangular polynomial patch over the triangular domain is proposed based on this basis functions. This patch can be used to construct patches whose boundaries are elliptical arcs,parabolic arcs,and arcs. A practical de-Casteljau-type algorithm is proposed to calculate the proposed triangular polynomial surface efficiently and stably. In addition,G1 continuous conditions for joining two triangular polynomial patches are provided. Result Experimental results show that the proposed total positivity patch in the frame of Chebyshev space not only can adjust the shape flexibly but also has shape preservation and good approximation. Conclusion We construct a class of basis functions with total positivity under the framework of the extended Chebysh ev space,and construct the curve and surface with this basis function. Experimental results show that the curve constructed in this study has all the excellent properties of a traditional cubic Bézier curve and has flexible shape adjustability. As the parameters increase,the generated curve can be closer to the control polygon,thereby simulating its behavior. In addition,the surface constructed on the triangular domain can generate the surface whose boundaries are elliptical arcs. A de Casteljau-type algorithm for calculating the surface is also provided. In summary,the proposed basis function satisfies the requirements of the geometric industry and is a practical method.
Objective The construction of basis functions has consistently been a difficult point of computer-aided geometric design( CAGD). The construction of a class of practical basis functions often plays a decisive role in the development of the geometric industry. The traditional Bézier curves and B-splines have been widely used in CAGD. However,when the control points are determined,the generated curve is relatively fixed with respect to the control points and has a certain rigidity. Although the proposed rational B-spline curve can adjust the curve by adjusting the weight factor,the rational methods have difficulty in predicting the influence of the weight factor on the curve due to its own shortcomings. Researchers have exerted efforts in the past two decades to solve this problem. However,most of the improved methods have the basic properties of the traditional Bézier method and the B-spline method,such as affine invariance,convex hull,non-negativity,geometric invariance and flexible shape adjustability. Moreover,the proposed curve can accurately represent special curves used in engineering,such as conic and hyperbolic curves. However,most of the literature does not discuss the variation diminishing the generated curve. The curve with vanishing variation must have convexity. The curve with the total positivity must have diminishing variation. Therefore,the total positivity of the basis functions indicates that these functions are suitable for geometric design. We can easily obtain rectangular patches with shape parameters through these new curves. However,the Bernstein-Bézier patch over the triangular domain is not a tensor product patch exactly. Therefore,we cannot obtain triangular surfaces with an adjustable shape using the method of tensor product. Surface modeling over triangular domain is important for many applications. Thus,the practical methods for generating surfaces over a triangular domain must be explored. The blossom property in quasi extended Chebyshev space is used to construct a group of optimal normalized entirely positive basis for curve and surface construction. This method enables the extended curve and surface,thereby maintaining the good nature of the traditional Bézier and B-spline methods while preserving shape,shape adjustability,and practicability. Method A class of cubic trigonometric quasi Bernstein basis functions with total positivity is constructed under the framework of the extended Chebyshev space,and the properties of the basis functions are provided.The corresponding curve is presented based on this basis function. The properties of the curve are analyzed. The cutting algorithm of the curve and the smooth connecting conditions are obtained. A trigonometric quasi Bernstein operator for estimating the degree of the control polygon is also proposed. Then,based on the cubic trigonometric quasi Bernstein basis function,a class of trigonometric polynomial basis functions with three shape parameters over the triangular domain is proposed. A type of triangular polynomial patch over the triangular domain is proposed based on this basis functions. This patch can be used to construct patches whose boundaries are elliptical arcs,parabolic arcs,and arcs. A practical de-Casteljau-type algorithm is proposed to calculate the proposed triangular polynomial surface efficiently and stably. In addition,G1 continuous conditions for joining two triangular polynomial patches are provided. Result Experimental results show that the proposed total positivity patch in the frame of Chebyshev space not only can adjust the shape flexibly but also has shape preservation and good approximation. Conclusion We construct a class of basis functions with total positivity under the framework of the extended Chebysh ev space,and construct the curve and surface with this basis function. Experimental results show that the curve constructed in this study has all the excellent properties of a traditional cubic Bézier curve and has flexible shape adjustability. As the parameters increase,the generated curve can be closer to the control polygon,thereby simulating its behavior. In addition,the surface constructed on the triangular domain can generate the surface whose boundaries are elliptical arcs. A de Casteljau-type algorithm for calculating the surface is also provided. In summary,the proposed basis function satisfies the requirements of the geometric industry and is a practical method.
引文
[1]Farin G.Curves and Surfaces for CAGD[M].5th ed.San Francisco:Morgan Kaufmann,2001.
[2]Hoschek J,Lasser D.Fundamentals of Computer Aided Geometric Design[M].Natick,MA:AK Peters,1993.
[3]Piegl L,Tiller W.The NURBS Book[M].New York:Springer,1995.
[4]Wang W T,Wang G Z.Bézier curves with shape parameter[J].Journal of Zhejiang University-Science A,2005,6(6):497-501.[DOI:10.1631/jzus.2005.A0497]
[5]Chen J,Wang G J.A new type of the generalized Bézier curves[J].Applied Mathematics-A Journal of Chinese Universities,2011,26(1):47-56.[DOI:10.1007/s11766-011-2453-8]
[6]Qin X Q,Hu G,Zhang N J,et al.A novel extension to the polynomial basis functions describing Bézier curves and surfaces of degree n with multiple shape parameters[J].Applied Mathematics and Computation,2013,223:1-16.[DOI:10.1016/j.amc.2013.07.073]
[7]Xu G,Wang G Z.Extended cubic uniform B-spline andα-B-spline[J].Acta Automatica Sinica,2008,34(8):980-984.[DOI:10.1016/S1874-1029(08)60047-6]
[8]Chen Q Y,Wang G Z.A class of Bézier-like curves[J].Computer Aided Geometric Design,2003,20(1):29-39.[DOI:10.1016/S0167-8396(03)00003-7]
[9]Han X L,Zhu Y P.Curve construction based on five trigonometric blending functions[J].BIT Numerical Mathematics,2012,52(4):953-979.[DOI:10.1007/s10543-012-0386-0]
[10]Su B Y,Tan J Q.A family of quasi-cubic blended splines and applications[J].Journal of Zhejiang University-Science A,2006,7(9):1550-1560.[DOI:10.1631/jzus.2006.A1550]
[11]Zhang J X,Tan J Q.Extensions of hyperbolic Bézier curves[J].Journal of Engineering Graphics,2011,32(1):31-38.[张锦秀,檀结庆.代数双曲Bézier曲线的扩展[J].工程图学学报,2011,32(1):31-38.][DOI:10.3969/j.issn.1003-0158.2011.01.007]
[12]Wu H Y,Zuo H.Quadratic non-uniform hyperbolic B-spline curves with multiple shape parameters[J].Journal of ComputerAided Design&Computer Graphics,2007,19(7):876-883.[邬弘毅,左华.多形状参数的二次非均匀双曲B-样条曲线[J].计算机辅助设计与图形学学报,2007,19(7):876-883.][DOI:10.3321/j.issn:1003-9775.2007.07.011]
[13]Han X L,Zhu Y P.Total positivity of the cubic trigonometric Bézier basis[J].Journal of Applied Mathematics,2014,2014:#198745.[DOI:10.1155/2014/198745]
[14]Wang K,Zhang G C,Gong J H.Constructing trigonometric polynomial curves and surfaces with two parameters[J].Journal of Image and Graphics,2018,23(12):1910-1924.[汪凯,张贵仓,龚进慧.带两个参数的三角多项式曲线曲面构造[J].中国图象图形学报,2018,23(12):1910-1924.][DOI:10.11834/jig.180328]
[15]Kvasov B I.Methods of Shape-Preserving Spline Approximation[M].Singapore:World Scientific Publishing,2000.
[16]Carnicer J M,Pe1a J M.Shape preserving representations and optimality of the Bernstein basis[J].Advances in Computational Mathematics,1993,1(2):173-196.[DOI:10.1007/BF02071384]
[17]Carnicer J M,Pe1a J M.Totally positive bases for shape preserving curve design and optimality of B-splines[J].Computer Aided Geometric Design,1994,11(6):633-654.[DOI:10.1016/0167-8396(94)90056-6]
[18]Cao J,Wang G Z.An extension of Bernstein-Bézier surface over the triangular domain[J].Progress in Natural Science,2007,17(3):352-357.
[19]Shen W Q,Wang G Z.Triangular domain extension of linear Bernstein-like trigonometric polynomial basis[J].Journal of Zhejiang University-Science C,2010,11(5):356-364.[DOI:10.1631/jzus.C0910347]
[20]Wei Y W,Shen W Q,Wang G Z.Triangular domain extension of algebraic trigonometric Bézier-like basis[J].Applied Mathematics-A Journal of Chinese Universities,2011,26(2):151-160.[DOI:10.1007/s11766-011-2672-z]
[21]Yang L Q,Zeng X M.Bézier curves and surfaces with shape parameters[J].International Journal of Computer Mathematics,2009,86(7):1253-1263.[DOI:10.1080/0020716070182-1715]
[22]Yan L L,Liang J F.An extension of the Bézier model[J].Applied Mathematics and Computation,2011,218(6):2863-2879.[DOI:10.1016/j.amc.2011.08.030]
[23]Zhu Y P,Han X L.A class ofαβγBernstein-Bézier basis functions over triangular domain[J].Applied Mathematics and Computation,2013,220:446-454.[DOI:10.1016/j.amc.2013.06.043]
[24]Zhang J W.C-curves:an extension of cubic curves[J].Computer Aided Geometric Design,1996,13(3):199-217.[DOI:10.1016/0167-8396(95)00022-4]
[25]Mazure M L.Which spaces for design?[J].Numerische Mathematik,2008,110(3):357-392.[DOI:10.1007/s00211-008-0164-8]
[26]Mazure M L.Quasi-Chebyshev splines with connection matrices:application to variable degree polynomial splines[J].Computer Aided Geometric Design,2001,18(3):287-298.[DOI:10.1016/S0167-8396(01)00031-0]
[27]Mazure M L.On dimension elevation in quasi extended Chebyshev spaces[J].Numerische Mathematik,2008,109(3):459-475.[DOI:10.1007/s00211-007-0133-7]
[28]Mazure M L.On a general new class of quasi Chebyshevian splines[J].Numerical Algorithms,2011,58(3):399-438.[DOI:10.1007/s11075-011-9461-x]
[29]Mazure M L.Quasi Extended Chebyshev spaces and weight functions[J].Numerische Mathematik,2011,118(1):79-108.[DOI:10.1007/s00211-010-0312-9]
[30]Bosner T,Rogina M.Variable degree polynomial splines are Chebyshev splines[J].Advances in Computational Mathematics,2013,38(2):383-400.[DOI:10.1007/s10444-011-9242-z]
[31]Ibraheem F,Hussain M,Hussain M Z.Positive data visualization using trigonometric function[J].Journal of Applied Mathematics,2012,2012:#247120.[DOI:10.1155/2012/247120]
[32]Hussain M,Saleem S.C1rational quadratic trigonometric spline[J].Egyptian Informatics Journal,2013,14(3):211-220.[DOI:10.1016/j.eij.2013.09.002]
[33]Hussain M Z,Hussain M,Waseem A.Shape-preserving trigonometric functions[J].Computational and Applied Mathematics,2014,33(2):411-431.[DOI:10.1007/s40314-013-0071-1]
[34]Ibraheem F,Hussain M,Hussain M Z.Monotone data visualization using rational trigonometric spline interpolation[J].The Scientific World Journal,2014,2014:#602453.[DOI:10.1155/2014/602453]
[35]Zhao K.On the extension of the second main theorem of Nevanlinna[J].Journal of Hunan Agricultural University,1997(03):275-279.[赵坤.Nevanlinna第二基本定理的推广(英文)[J].湖南农业大学学报,1997(03):275-279.]
[36]Zhu Y P.Research on theory and methods for geometric modeling based on basis functions possessing shape parameters[D].Changsha:Central South University,2014.[朱远鹏.基函数中带形状参数的几何造型理论与方法研究[D].长沙:中南大学,2014.]
[37]Carnicer J M,Mainar E,Pe1a J M.A Bernstein-like operator for a mixed algebraic-trigonometric space[C].XXI Congreso de Ecuacuines Difernciales y Aplicaciones,XI Congreso de Matematica Aplicada,Ciudad Real,2009:1-7.
[38]Cooper S,Waldron S.The eigenstructure of the Bernstein operator[J].Journal of Approximation Theory,2000,105(1):133-165.[DOI:10.1006/jath.2000.3464]
[39]Salomon D.The Computer Graphics Manual[M].New York:Springer,2011:719-721.
[2]Hoschek J,Lasser D.Fundamentals of Computer Aided Geometric Design[M].Natick,MA:AK Peters,1993.
[3]Piegl L,Tiller W.The NURBS Book[M].New York:Springer,1995.
[4]Wang W T,Wang G Z.Bézier curves with shape parameter[J].Journal of Zhejiang University-Science A,2005,6(6):497-501.[DOI:10.1631/jzus.2005.A0497]
[5]Chen J,Wang G J.A new type of the generalized Bézier curves[J].Applied Mathematics-A Journal of Chinese Universities,2011,26(1):47-56.[DOI:10.1007/s11766-011-2453-8]
[6]Qin X Q,Hu G,Zhang N J,et al.A novel extension to the polynomial basis functions describing Bézier curves and surfaces of degree n with multiple shape parameters[J].Applied Mathematics and Computation,2013,223:1-16.[DOI:10.1016/j.amc.2013.07.073]
[7]Xu G,Wang G Z.Extended cubic uniform B-spline andα-B-spline[J].Acta Automatica Sinica,2008,34(8):980-984.[DOI:10.1016/S1874-1029(08)60047-6]
[8]Chen Q Y,Wang G Z.A class of Bézier-like curves[J].Computer Aided Geometric Design,2003,20(1):29-39.[DOI:10.1016/S0167-8396(03)00003-7]
[9]Han X L,Zhu Y P.Curve construction based on five trigonometric blending functions[J].BIT Numerical Mathematics,2012,52(4):953-979.[DOI:10.1007/s10543-012-0386-0]
[10]Su B Y,Tan J Q.A family of quasi-cubic blended splines and applications[J].Journal of Zhejiang University-Science A,2006,7(9):1550-1560.[DOI:10.1631/jzus.2006.A1550]
[11]Zhang J X,Tan J Q.Extensions of hyperbolic Bézier curves[J].Journal of Engineering Graphics,2011,32(1):31-38.[张锦秀,檀结庆.代数双曲Bézier曲线的扩展[J].工程图学学报,2011,32(1):31-38.][DOI:10.3969/j.issn.1003-0158.2011.01.007]
[12]Wu H Y,Zuo H.Quadratic non-uniform hyperbolic B-spline curves with multiple shape parameters[J].Journal of ComputerAided Design&Computer Graphics,2007,19(7):876-883.[邬弘毅,左华.多形状参数的二次非均匀双曲B-样条曲线[J].计算机辅助设计与图形学学报,2007,19(7):876-883.][DOI:10.3321/j.issn:1003-9775.2007.07.011]
[13]Han X L,Zhu Y P.Total positivity of the cubic trigonometric Bézier basis[J].Journal of Applied Mathematics,2014,2014:#198745.[DOI:10.1155/2014/198745]
[14]Wang K,Zhang G C,Gong J H.Constructing trigonometric polynomial curves and surfaces with two parameters[J].Journal of Image and Graphics,2018,23(12):1910-1924.[汪凯,张贵仓,龚进慧.带两个参数的三角多项式曲线曲面构造[J].中国图象图形学报,2018,23(12):1910-1924.][DOI:10.11834/jig.180328]
[15]Kvasov B I.Methods of Shape-Preserving Spline Approximation[M].Singapore:World Scientific Publishing,2000.
[16]Carnicer J M,Pe1a J M.Shape preserving representations and optimality of the Bernstein basis[J].Advances in Computational Mathematics,1993,1(2):173-196.[DOI:10.1007/BF02071384]
[17]Carnicer J M,Pe1a J M.Totally positive bases for shape preserving curve design and optimality of B-splines[J].Computer Aided Geometric Design,1994,11(6):633-654.[DOI:10.1016/0167-8396(94)90056-6]
[18]Cao J,Wang G Z.An extension of Bernstein-Bézier surface over the triangular domain[J].Progress in Natural Science,2007,17(3):352-357.
[19]Shen W Q,Wang G Z.Triangular domain extension of linear Bernstein-like trigonometric polynomial basis[J].Journal of Zhejiang University-Science C,2010,11(5):356-364.[DOI:10.1631/jzus.C0910347]
[20]Wei Y W,Shen W Q,Wang G Z.Triangular domain extension of algebraic trigonometric Bézier-like basis[J].Applied Mathematics-A Journal of Chinese Universities,2011,26(2):151-160.[DOI:10.1007/s11766-011-2672-z]
[21]Yang L Q,Zeng X M.Bézier curves and surfaces with shape parameters[J].International Journal of Computer Mathematics,2009,86(7):1253-1263.[DOI:10.1080/0020716070182-1715]
[22]Yan L L,Liang J F.An extension of the Bézier model[J].Applied Mathematics and Computation,2011,218(6):2863-2879.[DOI:10.1016/j.amc.2011.08.030]
[23]Zhu Y P,Han X L.A class ofαβγBernstein-Bézier basis functions over triangular domain[J].Applied Mathematics and Computation,2013,220:446-454.[DOI:10.1016/j.amc.2013.06.043]
[24]Zhang J W.C-curves:an extension of cubic curves[J].Computer Aided Geometric Design,1996,13(3):199-217.[DOI:10.1016/0167-8396(95)00022-4]
[25]Mazure M L.Which spaces for design?[J].Numerische Mathematik,2008,110(3):357-392.[DOI:10.1007/s00211-008-0164-8]
[26]Mazure M L.Quasi-Chebyshev splines with connection matrices:application to variable degree polynomial splines[J].Computer Aided Geometric Design,2001,18(3):287-298.[DOI:10.1016/S0167-8396(01)00031-0]
[27]Mazure M L.On dimension elevation in quasi extended Chebyshev spaces[J].Numerische Mathematik,2008,109(3):459-475.[DOI:10.1007/s00211-007-0133-7]
[28]Mazure M L.On a general new class of quasi Chebyshevian splines[J].Numerical Algorithms,2011,58(3):399-438.[DOI:10.1007/s11075-011-9461-x]
[29]Mazure M L.Quasi Extended Chebyshev spaces and weight functions[J].Numerische Mathematik,2011,118(1):79-108.[DOI:10.1007/s00211-010-0312-9]
[30]Bosner T,Rogina M.Variable degree polynomial splines are Chebyshev splines[J].Advances in Computational Mathematics,2013,38(2):383-400.[DOI:10.1007/s10444-011-9242-z]
[31]Ibraheem F,Hussain M,Hussain M Z.Positive data visualization using trigonometric function[J].Journal of Applied Mathematics,2012,2012:#247120.[DOI:10.1155/2012/247120]
[32]Hussain M,Saleem S.C1rational quadratic trigonometric spline[J].Egyptian Informatics Journal,2013,14(3):211-220.[DOI:10.1016/j.eij.2013.09.002]
[33]Hussain M Z,Hussain M,Waseem A.Shape-preserving trigonometric functions[J].Computational and Applied Mathematics,2014,33(2):411-431.[DOI:10.1007/s40314-013-0071-1]
[34]Ibraheem F,Hussain M,Hussain M Z.Monotone data visualization using rational trigonometric spline interpolation[J].The Scientific World Journal,2014,2014:#602453.[DOI:10.1155/2014/602453]
[35]Zhao K.On the extension of the second main theorem of Nevanlinna[J].Journal of Hunan Agricultural University,1997(03):275-279.[赵坤.Nevanlinna第二基本定理的推广(英文)[J].湖南农业大学学报,1997(03):275-279.]
[36]Zhu Y P.Research on theory and methods for geometric modeling based on basis functions possessing shape parameters[D].Changsha:Central South University,2014.[朱远鹏.基函数中带形状参数的几何造型理论与方法研究[D].长沙:中南大学,2014.]
[37]Carnicer J M,Mainar E,Pe1a J M.A Bernstein-like operator for a mixed algebraic-trigonometric space[C].XXI Congreso de Ecuacuines Difernciales y Aplicaciones,XI Congreso de Matematica Aplicada,Ciudad Real,2009:1-7.
[38]Cooper S,Waldron S.The eigenstructure of the Bernstein operator[J].Journal of Approximation Theory,2000,105(1):133-165.[DOI:10.1006/jath.2000.3464]
[39]Salomon D.The Computer Graphics Manual[M].New York:Springer,2011:719-721.