CAD系统中的极小曲面及其优化方法的研究
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摘要
极小曲面问题的研究是微分几何领域巾最活跃的分支之一。由于极小曲面在工程领域有着广泛应用,因此将极小曲面引入CAGD(计算机辅助几何设计)领域具有重要意义。本文围绕着CAGD中的极小曲面造型这一主题,就以下几个方面给出了研究成果:
(1)极小曲面的控制网格表示与构造
控制网格是CAD系统中进行交互设计的重要工具。首先构造了空间{sint,cost,sinht,cosht,1,t,t~2,…,t~n}中的拟非均匀B样条基及相应的曲线曲面模型。由于节点情况比较复杂,我们利用行列式巧妙地把所有的节点情况统一到一个表达式之中。接着,基于该空间中的混合曲线曲面模型,给出了广义螺旋面的控制网格的几何构造方法。最后,利用混合曲线曲面模型给出了平面曲率线极小曲面的控制网格表示。这些结果为将这些极小曲面模型引入CAGD造型系统提供了一个有力工具,从而可以利用细分算法生成这些极小曲面,并在CAGD系统中与其他的曲而统一处理。
(2)高次参数多项式极小曲面的挖掘与性质
参数多项式形式是CAD系统中曲线曲面的标准形式。根据微分几何中的一个经典结论,给出了五次和六次调和参数多项式曲面为极小曲面的充分条件。基于这些条件,构造出了几类新的极小曲面,并深入研究了它们所具有的几何性质,如对称性、自交性、包含直线等。我们还进一步发现了两类新的共轭极小曲面,并实现了共轭极小曲面之间的动态变形。这些新的极小曲面不仅丰富了极小曲面的种类,而且可直接为当前的CAD系统所采用;最后,提出了关于任意次参数多项式极小曲面的存在性及其性质的三个猜想。
(3)基于线性PDE的极小曲面逼近造型
基于调和曲面与极小曲面的密切关系,首先利用方向导数研究了三角域上的调和B-B曲面的性质,给出了三角域上的B-B曲面为调和曲面的充要条件,并且证明了任何一个三角域上的B-B曲面调和面的控制网格均由它的第一层和第二层控制顶点完全决定。接着,研究了一类更广泛的负高斯曲率曲面的构造方法,得到了与调和曲面情形相类似的结果。最后,为了实现PDE曲面造型技术与CAD造型系统的数据交换,基于约束优化的思想,给出了PDE曲面的Bézier曲面逼近算法,并利用张量积Bézier曲面的细分性质对该算法进行了优化。
(4)基于平均曲率平方能量的Plateau-Bézier问题求解
Plateau-Bézier问题是Plateau问题在Bézier曲面形式下的推广,它以内部控制顶点为求解目标。我们从平均曲率为零出发,基于平均曲率平方能量来解决Plateau-Bézier问题。主要研究了矩形域上的张量积Bézier曲面和三角域上的B-B曲面两种情形,分别给出了其内部控制顶点所要满足的充要条件,并与基于Dirichlet能量的方法进行了比较,发现两者各有千秋。可以证明,若在给定的边界条件下,其所对应的极小曲面为等温参数多项式极小曲面,则按照本文方法所得到的Bézier曲面便是该等温参数多项式极小曲面。
(5)极小曲面造型中的边界优化问题
如何选择边界曲线以满足造型要求,是极小曲面造型中的一个重要问题。首先基于拉伸能量、弯曲能量和jerk能量,研究如下问题:给定部分控制顶点,如何构造其余的控制顶点使得所产生的Bézier曲线具有极小能量。推导出了待定控制顶点所要满足的充要条件。通过曲率图和曲率梳,对三种能量极小Bézier曲线进行了比较,并发现了四次能量最小Bézier曲线的共线性质。最后,对三次均匀B样条曲线进行了两个方面的扩展:首先,构造出了五次和六次调配函数,并以它们为基础定义了两种带形状参数的样条曲线,提高了曲线与其控制多边形的逼近程度;其次,构造出了两类三次和四次调配函数,并以此为基础定义了两类带局部形状参数的样条曲线,其优点是在保持曲线连续性不变的同时可以通过改变局部形状参数的取值对曲线进行局部调整。
(6)极小曲面在建筑设计中的应用
建筑设计是极小曲面的应用大户。首先基于曲面裁剪技术,讨论了本文所提出的几类新的极小曲面和三角域上的调和B-B曲面在张拉膜结构设计中的应用;最后,提出了旋转圆锥网格的概念,给出了它们的简单构造方法,并将其用于玻璃/钢结构设计。
The research on minimal surface problem is one of the most active fields of interest in differential geometry.As minimal surfaces have extensive applications in engineering,it is very meaningful to introduce minimal surface into the field of CAGD(Computer Aided Geometric Design).In this thesis,some creative contributions on the topic of minimal surface modeling are given as follows.
(1) Representation and construction of control mesh of minimal surfaces
Control mesh is an important tool for interactive design in CAD systems.We first construct the quasi non-uniform B-spline basis and the corresponding curve and surface models in the space spanned by {sin t,cos t,sinh t,cosh t,1,t,t~2,…,t~n}. Since the knot cases are complex,we unify all the knot cases into a formula using determinate technology.Based on the new hybrid curve and surface model,the geometric construction of control mesh of the generalized helicoid is proposed. Finally,we present the control mesh representation of minimal surfaces with planar lines of curvature.These results provide an efficient tool for introducing these minimal surfaces into CAGD modeling systems,then we can produce these minimal surfaces through subdivision algorithm,and do some geometry processing with other surface in a unified way in CAGD systems.
(2) Exploration and properties of parametric polynomial minimal surfaces of high degree
Parametric polynomial form is the standard form of curves and surfaces in CAD systems.Based on the classical result in differential geometry,we give the sufficient conditions of harmonic parametric polynomial surfaces of degree five and six being minimal surface.Prom these conditions,several kinds of new minimal surfaces are constructed.We study the interesting geometric properties of these new minimal surfaces,such as symmetry,self-intersection and containing straight lines.We also find two new kinds of conjugate minimal surfaces,and implement the dynamic deformation between the conjugate minimal surfaces. These new minimal surfaces not only enrich the category of minimal surfaces,but also can be integrated into the current CAD systems directly.Finally,we propose three conjectures about the existence and properties of parametric polynomial minimal surfaces of arbitrary degree.
(3) Approximate modeling of minimal surfaces based on linear PDE
Based on the close relationship between harmonic surfaces and minimal surfaces, the properties of the harmonic B-B surface over the triangular domain are first discussed using the direction derivatives.A sufficient and necessary condition of a B-B surface over the triangular domain being a harmonic surface is obtained.We have proved that the control net of an arbitrary harmonic B-B surface over the triangular domain is fully determined by the first and second layers of control points.Then,we study the construction method of a kinds of surfaces with negative Gaussian curvature,and the result is similar with the harmonic case.In order to exchange data between PDE modeling system and CAD systems,we present a novel algorithm for approximating PDE surface by tensor product Bézier surface based on constrained optimization.We also improve this algorithm by the subdivision property of Bézier surface.
(4) Solution of Plateau-Bézier problem based on squared mean curvature energy
Plateau-Bézier problem is the extension of Plateau problem in Bézier form. It makes inner control points as objective solution.From the condition of mean curvature being zero,we study this problem in the case of the tensor product Bézier surfaces and the B-B surfaces over triangular domain,and give the sufficient and necessary condition that the inner control points satisfy.We also compare this method with the Dirichlet method.We can prove that,if the minimal surface with respect to the given boundary curves is parametric polynomial minimal surface with isothermal parameter,then the surface obtained by squared mean curvature method is just the parametric polynomial minimal surfaces with isothermal parameter.
(5) Boundary optimization in minimal surface modeling
How to choose boundary curves to satisfy the user's requirement,is an important problem in minimal surface modeling.Firstly,based on the stretch energy, strain energy and jerk energy,we study the following problem:given partial control points of a Bézier curve,how to construct other control points such that the energy of the resulting Bézier curve is a minimum among all the energy of all Bézier curves with the same given control points.We derive the necessary and sufficient condition on the unknown control points for Bézier curves to have minimal energy.We compare the three kinds of energy-minimizing Bézier curves via curvature combs and curvature plots,and also present the collinear property of energy-minimizing quartic Bézier curves.Finally,we propose two extensions of the cubic uniform B-spline curves.First,two classes of polynomial blending functions of degree five and six are constructed.Based on the blending functions, two methods of generating piecewise polynomial curves with a shape parameter are given.It improves the approaching degree of the curves to their control polygon. Secondly,two classes of polynomial blending functions of degree three and four are presented.From these blending functions,we propose two kinds of spline curves with local shape parameter.The advantage of this method is that we can manipulate the shape of the curves locally by changing local shape parameter, while the continuity of the spline curve is unchanged.
(6) Application of minimal surface in architecture design
Architecture design is the main application field of minimal surface.Based on the surface trimming technology,we first discuss the application of the new minimal surfaces and the harmonic B-B surface over triangular domain proposed in this thesis in the design of membrane structure.Finally,we propose a new concept- conical mesh of revolution,and give the simple construction method of them.We also employ them in the design of glass/steel structure.
(1)极小曲面的控制网格表示与构造
控制网格是CAD系统中进行交互设计的重要工具。首先构造了空间{sint,cost,sinht,cosht,1,t,t~2,…,t~n}中的拟非均匀B样条基及相应的曲线曲面模型。由于节点情况比较复杂,我们利用行列式巧妙地把所有的节点情况统一到一个表达式之中。接着,基于该空间中的混合曲线曲面模型,给出了广义螺旋面的控制网格的几何构造方法。最后,利用混合曲线曲面模型给出了平面曲率线极小曲面的控制网格表示。这些结果为将这些极小曲面模型引入CAGD造型系统提供了一个有力工具,从而可以利用细分算法生成这些极小曲面,并在CAGD系统中与其他的曲而统一处理。
(2)高次参数多项式极小曲面的挖掘与性质
参数多项式形式是CAD系统中曲线曲面的标准形式。根据微分几何中的一个经典结论,给出了五次和六次调和参数多项式曲面为极小曲面的充分条件。基于这些条件,构造出了几类新的极小曲面,并深入研究了它们所具有的几何性质,如对称性、自交性、包含直线等。我们还进一步发现了两类新的共轭极小曲面,并实现了共轭极小曲面之间的动态变形。这些新的极小曲面不仅丰富了极小曲面的种类,而且可直接为当前的CAD系统所采用;最后,提出了关于任意次参数多项式极小曲面的存在性及其性质的三个猜想。
(3)基于线性PDE的极小曲面逼近造型
基于调和曲面与极小曲面的密切关系,首先利用方向导数研究了三角域上的调和B-B曲面的性质,给出了三角域上的B-B曲面为调和曲面的充要条件,并且证明了任何一个三角域上的B-B曲面调和面的控制网格均由它的第一层和第二层控制顶点完全决定。接着,研究了一类更广泛的负高斯曲率曲面的构造方法,得到了与调和曲面情形相类似的结果。最后,为了实现PDE曲面造型技术与CAD造型系统的数据交换,基于约束优化的思想,给出了PDE曲面的Bézier曲面逼近算法,并利用张量积Bézier曲面的细分性质对该算法进行了优化。
(4)基于平均曲率平方能量的Plateau-Bézier问题求解
Plateau-Bézier问题是Plateau问题在Bézier曲面形式下的推广,它以内部控制顶点为求解目标。我们从平均曲率为零出发,基于平均曲率平方能量来解决Plateau-Bézier问题。主要研究了矩形域上的张量积Bézier曲面和三角域上的B-B曲面两种情形,分别给出了其内部控制顶点所要满足的充要条件,并与基于Dirichlet能量的方法进行了比较,发现两者各有千秋。可以证明,若在给定的边界条件下,其所对应的极小曲面为等温参数多项式极小曲面,则按照本文方法所得到的Bézier曲面便是该等温参数多项式极小曲面。
(5)极小曲面造型中的边界优化问题
如何选择边界曲线以满足造型要求,是极小曲面造型中的一个重要问题。首先基于拉伸能量、弯曲能量和jerk能量,研究如下问题:给定部分控制顶点,如何构造其余的控制顶点使得所产生的Bézier曲线具有极小能量。推导出了待定控制顶点所要满足的充要条件。通过曲率图和曲率梳,对三种能量极小Bézier曲线进行了比较,并发现了四次能量最小Bézier曲线的共线性质。最后,对三次均匀B样条曲线进行了两个方面的扩展:首先,构造出了五次和六次调配函数,并以它们为基础定义了两种带形状参数的样条曲线,提高了曲线与其控制多边形的逼近程度;其次,构造出了两类三次和四次调配函数,并以此为基础定义了两类带局部形状参数的样条曲线,其优点是在保持曲线连续性不变的同时可以通过改变局部形状参数的取值对曲线进行局部调整。
(6)极小曲面在建筑设计中的应用
建筑设计是极小曲面的应用大户。首先基于曲面裁剪技术,讨论了本文所提出的几类新的极小曲面和三角域上的调和B-B曲面在张拉膜结构设计中的应用;最后,提出了旋转圆锥网格的概念,给出了它们的简单构造方法,并将其用于玻璃/钢结构设计。
The research on minimal surface problem is one of the most active fields of interest in differential geometry.As minimal surfaces have extensive applications in engineering,it is very meaningful to introduce minimal surface into the field of CAGD(Computer Aided Geometric Design).In this thesis,some creative contributions on the topic of minimal surface modeling are given as follows.
(1) Representation and construction of control mesh of minimal surfaces
Control mesh is an important tool for interactive design in CAD systems.We first construct the quasi non-uniform B-spline basis and the corresponding curve and surface models in the space spanned by {sin t,cos t,sinh t,cosh t,1,t,t~2,…,t~n}. Since the knot cases are complex,we unify all the knot cases into a formula using determinate technology.Based on the new hybrid curve and surface model,the geometric construction of control mesh of the generalized helicoid is proposed. Finally,we present the control mesh representation of minimal surfaces with planar lines of curvature.These results provide an efficient tool for introducing these minimal surfaces into CAGD modeling systems,then we can produce these minimal surfaces through subdivision algorithm,and do some geometry processing with other surface in a unified way in CAGD systems.
(2) Exploration and properties of parametric polynomial minimal surfaces of high degree
Parametric polynomial form is the standard form of curves and surfaces in CAD systems.Based on the classical result in differential geometry,we give the sufficient conditions of harmonic parametric polynomial surfaces of degree five and six being minimal surface.Prom these conditions,several kinds of new minimal surfaces are constructed.We study the interesting geometric properties of these new minimal surfaces,such as symmetry,self-intersection and containing straight lines.We also find two new kinds of conjugate minimal surfaces,and implement the dynamic deformation between the conjugate minimal surfaces. These new minimal surfaces not only enrich the category of minimal surfaces,but also can be integrated into the current CAD systems directly.Finally,we propose three conjectures about the existence and properties of parametric polynomial minimal surfaces of arbitrary degree.
(3) Approximate modeling of minimal surfaces based on linear PDE
Based on the close relationship between harmonic surfaces and minimal surfaces, the properties of the harmonic B-B surface over the triangular domain are first discussed using the direction derivatives.A sufficient and necessary condition of a B-B surface over the triangular domain being a harmonic surface is obtained.We have proved that the control net of an arbitrary harmonic B-B surface over the triangular domain is fully determined by the first and second layers of control points.Then,we study the construction method of a kinds of surfaces with negative Gaussian curvature,and the result is similar with the harmonic case.In order to exchange data between PDE modeling system and CAD systems,we present a novel algorithm for approximating PDE surface by tensor product Bézier surface based on constrained optimization.We also improve this algorithm by the subdivision property of Bézier surface.
(4) Solution of Plateau-Bézier problem based on squared mean curvature energy
Plateau-Bézier problem is the extension of Plateau problem in Bézier form. It makes inner control points as objective solution.From the condition of mean curvature being zero,we study this problem in the case of the tensor product Bézier surfaces and the B-B surfaces over triangular domain,and give the sufficient and necessary condition that the inner control points satisfy.We also compare this method with the Dirichlet method.We can prove that,if the minimal surface with respect to the given boundary curves is parametric polynomial minimal surface with isothermal parameter,then the surface obtained by squared mean curvature method is just the parametric polynomial minimal surfaces with isothermal parameter.
(5) Boundary optimization in minimal surface modeling
How to choose boundary curves to satisfy the user's requirement,is an important problem in minimal surface modeling.Firstly,based on the stretch energy, strain energy and jerk energy,we study the following problem:given partial control points of a Bézier curve,how to construct other control points such that the energy of the resulting Bézier curve is a minimum among all the energy of all Bézier curves with the same given control points.We derive the necessary and sufficient condition on the unknown control points for Bézier curves to have minimal energy.We compare the three kinds of energy-minimizing Bézier curves via curvature combs and curvature plots,and also present the collinear property of energy-minimizing quartic Bézier curves.Finally,we propose two extensions of the cubic uniform B-spline curves.First,two classes of polynomial blending functions of degree five and six are constructed.Based on the blending functions, two methods of generating piecewise polynomial curves with a shape parameter are given.It improves the approaching degree of the curves to their control polygon. Secondly,two classes of polynomial blending functions of degree three and four are presented.From these blending functions,we propose two kinds of spline curves with local shape parameter.The advantage of this method is that we can manipulate the shape of the curves locally by changing local shape parameter, while the continuity of the spline curve is unchanged.
(6) Application of minimal surface in architecture design
Architecture design is the main application field of minimal surface.Based on the surface trimming technology,we first discuss the application of the new minimal surfaces and the harmonic B-B surface over triangular domain proposed in this thesis in the design of membrane structure.Finally,we propose a new concept- conical mesh of revolution,and give the simple construction method of them.We also employ them in the design of glass/steel structure.
引文
[1] Arnal, A., Lluch, A., Monterde, J. Triangular Bezier surfaces of minimal area. In: Proceedings of the Int. Workshop on Computer Graphics and Geometric Modeling, CG GM'2003, Montreal. In: Lecture Notes in Com-put.Sci, 2003, vol.2669. Springer-Verlag, Berlin/New York, pp. 366-375.
[2] Aspert, N., Ebrahimi, T., Vandergheynst, P.. Non-linear subdivision using local spherical coordinates. Computer Aided Geometric Design, 2003, 20, 165 - 187.
[3] Au C.K.,Yuen M.M.F.. Unified Approach to NURBSCurve Shape Modification. Computer Aided Design, 1995, 27(2):85-93.
[4] Barr, AH., Global and local deformation of solid primitives, Computer Graphics (SIGGRAPH), 1984,18(3):21-30.
[5] Bezier, P. Numerical Control: Mathematics and Applications, John Wiley and Sons, New York, 1972.
[6] Bajaj, C, Ihm, I.. C~1 smoothing of polyhedral with implicit algebraic splines. Proceeding of SIGGRAPH'92, Computer Graphics, 1992, 26(2): 79-88.
[7] Bajaj, C. Surface fitting using implicit algebraic surface patches. In: Hans Hagen eds. SIAM Topics in surface modeling, 1992: 23-52.
[8] Bajaj, C., Ihm, I.. Algebraic surface design with Hermite interpolation. ACM Transactions on Graphics, 1992, 11(1): 61-91.
[9] Bajaj, C, Ihm, I., Warren, J.. Higher-order interpolation and least-square approximate using implicit algebraic surfaces. ACM Transaxtions on Graphics, 1993, 12(4): 327-347.
[10]Barsky B.A.Computer Graphics and Geometric Modeling Using Beta-Spline.Heidelberg Springer-Verlag,1988
[11]Blinn,J.F..A generalization of algebraic surface drawing.ACM Transaction on Graphics,1982,1(3):235-256.
[12]Bloor MIG,Wilson MJ.Generating blend surfaces using partial differential equations.Computer Aided Design,1989,21(3):165-171.
[13]Bloor MIG,Wilson MJ.Representing PDE surface in terms of B-splines.Computer Aided Design,1990,22:324-331.
[14]Bloor MIG,Wilson MJ.Spectral approximations to PDE surfaces.Computer Aided Design,1996,28(2):145-152.
[15]Bloor MIG,Wilson MJ.An analytic pseudo-spectral method to generate a regular 4-sided PDE surface patch.Computer Aided Geometric Design,2005,22(3):203-219.
[16]Boehm W..Inserting new knots into B-spline curve.Computer Aided Design,1980,12(4):199-201.
[17]Bohem,W.,Farin,G.Kahman,J..A survey of curve and surfaces methods in CAGD.Computer Aided Geometric Design,1984,1:1-60.
[18]Boehm,W.,Prautzsch,H..The insertion algorithm.Computer Aided Design,1985,17(2):58-59.
[19]Boehm,W..On the efficiency of knot insertion algorithms.Computer Aided Geometric Design,1985,2(1-3),141-143
[20]Boor,C.,Pinkus,A..The approximation of a totally positive band matrix by a strictly banded totally positive one.Lin.Alg.Appl.,1982,42:81-98.
[21]Bradley C.,Vichers G.W..Free-form surface reconstruction for machine vision rapid prototyping.Optical engineering,1993,32(9):2191-2199.
[22]Brunnett G,Hagen H,Santarelli P.Variational design of curves and surfaces.Surveys on Mathematics for Industry 1993;3:1-27.
[23]Carnicer,J.,Pena,J..Totally positive bases for shape preserving curve design and optimality of B-splines.Computer Aided Geometric Design,1994,11:635-656.
[24]Carnicer,J.,Pena,J.M..Total positivity and optimal bases,in:Gasca,M.and Micchelli,C.A.,eds..Total positivity and its Applications,Kluwer Academic,Dordrecht,1996:133-155.
[25]E.Catmull,J.Clark.Recursively generated B-spline surfaces on arbitrary topological meshes.Computer Aided Design,10(6):350-355,1978.
[26]G.M.Chaikin.An Algorithm for High Speed Curve Generation,Computer Vision,Graphics,and Image Processing,vol.3,no.4,346-349,1974.
[27]Chang,GZ.Bernstein polynomials via the shifting operator.American Mathematical Monthly,1984,91:634-638.
[28]常庚哲、吴俊恒.贝齐尔曲线曲面的数学基础及其计算,北京航空学院科学研究报告BH-B381,1978.
[29]Chang YK,Rockwood AP.A generalized de Casteljau approach to 3D free-Form deformation.Computer Graphics(SIGGRAP H '94),1994,257-260.
[30]Chen,F.,Deng,L..Interval implicitization of rational curves.Computer Aided Geometric Design,2004,21:401-415.
[31]Chen,Q.,Wang,G..A class of Bézier-like curves.Computer Aided Geometric Design,2003,20(1):29-39.
[32]陈文喻,汪国昭.Bézier曲面的精确自由变形.计算机辅助设计与图形学学报,2006,18(8),1160-1164.
[33]陈笑.有理Bézier极小曲面设计:Dirichlet方法的推广.杭州,浙江大学,硕士学位论文,2006.
[34]Cline,A.K..Scalar- and Planar-valued curve fitting using splines under tension.,Communications of the ACM,1974,17:218-233.
[35]Cohen,E.,Lyche,T.,Riesenfeld,R.F..Discrete B-spline and subdivision techniques in computer aided geometric design and computer graphics.Computer Graphics and Image Processing,1980,14(2):87-111.
[36]Cohen,E.,Riesenfeld,R.F..General matrix representations for Bézier and B-spline curves.Computers in Industry,1982,3(1-2):9-15.
[37]Choi,B.K.,Yoo,W.S.,Lee,C.S..Matrix representation for NURBS curves and surfaces.Computer Aided Design,1990,22(4):235-240.
[38]P.Concns,Numerical solution of the minimal surface equation,Math.Comput.1967,21,340-350.
[39]Coons,S.A..Surfaces for computer aided design of space Figures.Technical Report,MIT,1964,Project MAC-TR-255.
[40]Coons,S.A..Surfaces for computer aided design of space forms.Technical Report,MIT,1967,Project MAC-TR-411.
[41]Coquillart S.Extended free-form deformation:A sculpturing tool for 3D geometric modeling.Computer Graphics(SIGGRAP H'90),1990,24(4):187-196.
[42]Cosin,C.,Monterde,J..2002.Bézier surfaces of minimal area.In:Sloot,Tan,K.,Dongarra,Hoekstra(Eds.),Proceedings of the Int.Conf.of Computational Science,ICCS' 2002,Amsterdam,vol.2.In:Lecture Notes in Comput.Sci.,vol.2330.Springer-Verlag,pp.72-81.
[43]Cox,M.,G..The numerical evaluation of B-spline.Report No.NPLDNACS-4,National Laboratory,1971.
[44]Cutler B.,Whiting E.Constrained planar remeshing for achitecture.SGP 2006,Poster.
[45] Datta A., Soundaralakshmi S.. Fast parallel algorithm for distance transforms. Parallel and Distributed Processing Symposium, Proceedings 15th International, 2001: 1130-1134.
[46] de Boor, C On Calculation with B-spline. Journal of Approximation Theory, 1972, 6(1): 50-62.
[47] de Boor, C. Splines as Linear Combinations of B-splines. Academic Press, New York, 1976.
[48] de Boor, C. A practical guide to splines. Springer-Verlag, New York, 1978.
[49] Demko S, Hodges L, Naylor B. Construction of fractal objects with iterated functions systems. In Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH, 1985: 271-278.
[50] Dietz, R., Hoschek, J., Juttler, B.. An algebraic approach to curves and surfaces on the sphere and on other quadrics. Computer Aided Geometric Design, 1993, 10: 211-229.
[51] Ding, Q.L., Davies, B.J.. Surface Engineering Geometry for Computer Aided Design and Manufacture. Ellis Horwood, 1987.
[52] Do Carmo, M. 1976. Differential geometry of curves and surfaces. Prentice-Hall.
[53] DJ. Douglas. Solution of the problem of Plateau, Trans. AMS. 1931, 33, 263-321.
[54] D.Doo, M. Sabin. Analysis of the behaviour of recursive division surfaces near extraordinary points. Computer Aided Design, 1978 10(6):356-360.
[55] Du HX, Qin H. Dynamic PDE-based Surface design using geometric and physical constraints. Graphical Models, 2005, 67, 43-71.
[56] N. Dyn, D. Levin, J.A. Gregory. A Butterfly Subdivision Scheme for Surface Interpolation with Tension Control. ACM Transactions on Graphics, 1990, (9) 2,160-169.
[57]G.Dziuk,J.E.Hutchinson.The discrete Plateau problem,Math.Comp.1999,68(225):1-23.
[58]Farin,G..From Conics to NURBS:A Tutorial and Survey.IEEE Computer Graphics and Applications.1992,1(5):78-86.
[59]Farin,G..Curves and Surfaces for Computer Aided Geometric Design.4th edn.Academic Press,San Diego,1997.
[60]FeH.Federer,W.H.Fleming.Normal and integral currents,Ann.of Math.1960,72,458-520.
[61]FeH.Federer.Geometric Measure Theory,Springer-Verlag,New York,1969.
[62]FeH.Federer.The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension,Bull.AMS.76 1970,767-771.
[63]Feng JQ,Ma LZ,Peng QS.A new free-form deformations through the control of parametric surfaces.Computer and Graphics,1996,20(4):531-539.
[64]冯结青,彭群生,马利庄.沿参数曲面的均匀变形方法.软件学报,1998,9(4):263-267
[65]冯结青,马利庄,彭群生.一种应用参数曲面的动态自由变形方法.计算机学报,1997,20(7):600-606.
[66]Feng JQ,Pheng-Ann Heng,Tien-Tsin Wong.Accurate B-spline Deformation of Polygonal Object.Journal of Graphics Tools.1998,3(3):11-17.
[67]冯结青,彭群生.Bézier曲面的函数复合及其应用.软件学报,1999,10(12):1316-1321
[68]冯结青,赵豫红,万华根,郭建民,金小刚,彭群生.基于曲线和曲面控制的多边形物体变形反走样.计算机学报,2005,28(1),60-67
[69] Feng J., Peng Q.. Accelerating accurate B-spline free-form deformation through polynomial interpolation. Journal of Graphics Tools, 2000, 5(1): 1-8
[70] Feng, J., Shao, J., Jin, X., Peng, Q., and Forrest, A.R.. Multiresolution free-form deformation with subdivision surface of arbitrary topology. The Visual Computer, 2006, 22(1), 28-42.
[71] Fergusion, J.C. Multivariable curve interpolation. Report D2-22504, The Boeing Co. Seattle, Washington, 1963.
[72] Fergusion, J.C. Multivariable curve interpolation. Journal of ACM, 1964, 2: 221-228.
[73] Forrest, A.R.. Interactive interpolation and approximation by Bezier curve. The Computer Journal, 1972, 15(1): 71-79.
[74] Forsey D. R.,Bartels R H.. Surface fitting with hierarchical splines. ACM Trans. on Graphics, 1995, 14(2): 134-161.
[75] Giancarlo Amati. A multi-level filtering approach for fairing planar cubic B-spline curves. Computer Aided Geometric Design 2007, 24, 53 - 66.
[76] Goldman, R. N., Sederberg, T. W., Anderson, D. C. Vector elimination: A technique for the implicitization, inversion, and intersection of planar parametric rational polynomial curves. Computer Aided Geometric Design, 1984, 1: 327-356.
[77] Gomes, J., Darsa,L., Costa, B. and Velho, L.. Warping and Morphing of Graphical Objects. Morgan Kaufmann Publishers, 1998.
[78] Goodman, T.N.T.. Shape preserving interpolation by parametric rational cubic splines. in: International Series of Numerical Mathematics 86, Birkh iuser, Basel, 1988: 149-158.
[79] Goodman, T.N.T., Said H. B.. Shape preserving properties of the generalized Ball basis. Computer Aided Geometric Design, 1991, 8(2): 115-121.
[80]Goodman,T.N.T.,Said,H.B..Properties of generalized Ball curves and surfaces.Computer Aided Design,1991,23(8):554-560.
[81]Goodman,T.N.T.,Ong,B.H.and Unsworth K..Constrained interpolation using rational cubic splines,in:Farin G.,ed.,Nurbs for Curve and Surface Design,SIAM,1991:59-74.
[82]Goodman,T.N.T..Total positivity and the shape of curves,in:M.Gasca,C.A.Micchelli(Eds.),Total Positivity and Its Applications,Kluwer,Dordrecht,1996:157-186.
[83]Goodman,T.N.T.,Ong,B.H..Shape preserving interpolation by space curves.Report AA/963,University of Dundee,1996.
[84]Goodman,T.N.T.,Ong B.H..Shape preserving interpolation by space curves.Computer Aided Geometric Design,1997,16:1-17.
[85]Goodman,T.N.T..Automatic interpolation by fair shape-preserving C space curves.Computer Aided Geometric Design,1998,15(3):813-822.
[86]Goodman,T.N.T.,Sun Q..Total positivity and refinable functions with general dilation.Appl.Comput.Harmon.Anal.,2004,16:69-89.
[87]Gordon,W.J.,Riesenfeld,R.F..Bernstein Bézier methods for the Computer Aided Geometric Design of free-form curves and surfaces.Journal of ACM,1974,21:293-310.
[88]Gordon,W.J.,Riesenfeld,R.F..B-spline curves and surfaces.In:Barnhill,R.E.and Riesenfeld,R.F.eds.,Computer Aided Geometric Design.Academic Press,New York,1974:95-126.
[89]Grabowski,H.,Li,X..Coefficient formula and matrix of nonuniform B-spline functions.Computer-Aided Design,1992,24(12):637-642.
[90]Greissmair J,Purgathofer W..Deformation of solids with trivariate B-Splines.Computer Graphics Forum,1989,137-148.
[91]Gruber,F.,Glaeser,G..Magnetism and minimal surfaces-a different tool for surface design.Computational Aesthetics 2007-Eurographics Workshop on Computational Aesthetics in Graphics,Visualization and Imaging.D.W.Cunningham,G.Meyer,L.Neumann,A.Dunning,R.Paricio(eds.),81-88
[92]Halstead,M.,Barsky,B.,Klein,S.,Mandell,R..Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals.Proceedings of SIGGRAPH,1996:335-342.
[93]韩旭里,刘圣军.三次均匀B样条曲线的扩展.计算机辅助设计与图形学学报,2003,15(5):576-578
[94]He T,Wang S,Kaufman A.Wavelet-based volume morphing.Proceedings of Visualization'94,IEEE Computer Society Press,1994,85-92.
[95]何玲娜.代数双曲三角曲线的研究.浙江大学硕士学位论文,2004年4月.
[96]Hernández-Mederos,Victoria,Estrada-Sarlabous,Jorge.Sampling points on regular parametric curves with control of their distribution,Computer Aided Geometric Design,2003(20):363-382.
[97]Higashi M.,Kaneko K.,Hosaka M..Generation of high-quality curve and surface with smoothly varying curvature.Eurographics 88,p.79-92.
[98]Hiroshi,I..Minimal submanifolds in Riemannian spin manifolds with parallel spinor fields.J.of Geometry and Physics Volume,2002,41(3),258-273.
[99]Hoffmann,C.M..Implicit curves and surfaces in CAGD.IEEE Comput.Graph.Appl.,1993,13:79-88.
[100]Hoscheck J,Schneider F.1990.Spline conversion for trimmed rational Bezier and B-spline surfaces.Computer Aided Design,22(9),580-590.
[101]Hsu W.M,Hughes J.F.,Kaufman H..Direct manipulation of free-form deformations.Computer Graphics(SIGGRAPH '92),1992,26(2):177-184.
[102]Hu S.M.,Li Y.F.,Ju T.and Zhu X..Modifying the shape of NURBS surfaces with geometric constraints.Computer Aided Design,2001,33(12):903-912.
[103]Hu SM,Zhang H,Tai CL,Sun JG.Direct manipulation of FFD:efficient explicit solutions and decomposible multiple point constraints.The Visual Computer,2001,17:370-379.
[104]Hua J,Qin H.Free-form deformation via sketching and scalar fields.In Solid Modeling international(SM'03),2003,328-333.
[105]Hui K.C.Free Form design using axial curve pairs.Computer Aided Design,2002,34,583-555.
[106]S.Hyde,S.Andersson,K.Larsson,Z.Blum,T.Landh,S.Lidin,B.W.Ninham.The Language of Shape.The Role of Curvature in Condensed Matter:Physics,Chemistry and Biology,Elsevier,Amsterdam,1997.
[107]计忠平,刘利刚,王国瑾.无局部自交的轴变形新方法.计算计学报,2006,29(5):828-834.
[108]季洁,满家巨.任意边界下的极小曲面造型问题.计算机与现代化.2007,4:7-9.
[109]金文标,汪国昭.一类极小曲面的几何设计.计算机学报.1999,23(12):1277-1279.
[110]金文标,杨勋年.一种负高斯曲率的双三次Bézier曲面,高校应用数学学报.2000,15(2):206-210.
[111]金文标.若干特殊曲面的造型和应用.杭州,浙江大学,博士学位论文,1999.
[112]Joe B.Multiple-knot and rational cubic Beta-splines.ACM Transactions on Graphics,1989,8(2):100-120
[113]Ju T,Schaeffer S,Warren J.Mean value coordinates for closed triangular meshes.ACM Transaction on Graphics(SIGGRAPH '05),2005,24(3):561-566.
[114]J.M.Pena..Shape Preserving Representations in Computer Aided Geometric Design.Nova Science Publishers,Commack(New York),1999.
[115]J.Sánchez-Reyes.A Simple Technique for NURBS Shape Modification.IEEE Computer Graphics and Applications,Januaty-February,1997,17(1):52-59.
[116]R.D.Kamien Decomposition of the height function of Scherk's first surface,Applied Mathematics Letters 14(2001) 797-800
[117]Kobbelt,L..A variational approach to subdivision.Computer Aided Geometric Design,1996,13(8),743-761.
[118]Koch,P..Multivariate trigonometric B-splines.J.Approx.Theory,1988,54:162-168.
[119]Koch,P.E.,Lyche,T..Exponential B-splines in tension,in:C.K.Chui,L.L.Schumaker and J.D.Ward,eds.,Approximation Theory Ⅵ,Academic Press,New York,1989:361-364.
[120]Koch,P.E.,Lyche,T..Construction of exponential tension B-splines of arbitrary order.In:Laurent,P.J.,Le Méhauté,A.,Schumaker,L.L.(Eds.),Curves and Surfaces.Academic Press,New York,1991:255-258.
[121]Koch,P.,Lyche,T.,Neamtu,M.,Schumaker,L..Control curves and knot insertion for trigonometric splines.Adv.Comput.Math.,1995,3:405-424.
[122]KrishnamurhyV,Le voyM.Fitting smooth surfaces to dense polygon meshes.Computer Graphics,1996,30(4):313-324.
[123]Kvasov,B.,Sattayatham,P..GB-splines of arbitrary order.J.Comput.Appl.Math.,1999,104:63-88.
[124]Lazarus,F.,Coquillart,S.,Jancene,P..Axial deformations:an intuitive deformation technique.Computer Aided Design,1994,26(8):608-612.
[125]Lazarus,F.,Anne Verroust.Three-dimensional metamorphosis:a survey.The Visual Computer,1998 14(8/9):373-389.
[126]Lamousin H.J,Waggenspack W.J.Nurbs-based free-form deformations.IEEE Computer Graphics and Applications,1994,14(6):59-65.
[127]Lee S.,K.Chwa,S.Shin,G.Wolberg.Image Metamorphosis Using Snakes and Free-Form Deformations.In SIGGRAPH 95 Conference Proceeding,1995,439-448.
[128]Lee S.,Wolberg G.,Shin S.Y..Scatered data interpolation with multilevel B-splines.IEEE Trans.on Visualization and Computer Graphics.1997,3(3):228-244.
[129]Li Y.,Wang G..Two kinds of B-basis of the algebraic hyperbolic space.Journal of Zhejiang University SCIENCE 2005,6A(7):750-759.
[130]Liu Ligang,Wang Guojin.Explicit matrix representation for NURBS curves and surfaces.Computer Aided Geometric Design,2002,19(6):409-419.
[131]Liu,Y.,Pottmann,H.,Wallner,J.,Yang,Y.L.,Wang,W.P...Geometric modeling with conical meshes and developable surfaces.ACM Trans.Graphics,2006,25(3),681-689,Proc.SIGGRAPH 2006.
[132]刘计善.一类极小曲面造型设计的新方法.复旦学报(自然科学版).2007,46(2):192-197.
[133]刘鼎元.三次参数曲线段和三次Bézier曲线形状控制,应用数学学报,1981,5:158-166.
[134]刘鼎元.平面n次Bézier曲线的凸性定理,数学年刊,1982:45-55.
[135]刘鼎元.有理Bézier曲线.应用数学学报,1985,8(1):70-82.
[136]Loe K F.α-B-spline:a linear singular blending B-spline.The Visual Computer,1996,12(1):18-25.
[137]Charles Loop.Smooth subdivision surfaces based on triangles.Master's thesis,University of Utah,Department of Mathematics,1987.
[138]L(u|¨),Y.,Wang,G.,Yang,X..Uniform trigonometric polynomial B-spline curves.SCIENCE IN CHINA(Series F),2002,45:335-343.
[139]L(u|¨),Y.G.,Wang,G.Z.,Yang,X.N..Uniform hyperbolic polynomial B-Spline curves.Computer Aided Geometric Design,2002,19:379-393.
[140]路宁,宁涛,唐荣锡,张兆璞.Dirichlet自由变形方法及其在建立尺寸驱动人体模型中的应用.工程图学学报,2004,76-83.
[141]吕勇刚.CAGD自由曲线曲面造型中均匀样条的研究.浙江大学博士学位论文,2002年5月.
[142]Mainar,E.,Pena,J.,Sánchez-Reyes,J..Shape preserving alternatives to the rational Bézier model.Computer Aided Geometric Design,2001,18(1):37-60.
[143]Mainar,E.,Pena,J..Quadratic-cycloidal curves.Advances in Computational Mathematics,2004,20:161-175.
[144]满家巨,汪国昭.正螺面与悬链面的表示与构造.计算机辅助设计与图形学学报.2005,17(3):431-436.
[145]满家巨,汪国昭.等温参数多项式极小曲面.计算机学报.2002,25(2):197-201.
[146]满家巨,汪国昭.基于有限元方法的极小曲面造型.计算机学报.2003,26(4):507-510.
[147]满家巨,汪国昭.B样条函数极小曲面造型.软件学报.2003,14(4):824-829
[148]Martin R R,Stephenson P C.Sweeping of the three dimensional objects.Computer Aided Design,1990,22(4):223-234.
[149]Meeks,William H.,Ⅲ;Yau,Shing Tung The classical Plateau problem and the topology of 3-manifolds.Minimal submanifolds and geodesics(Proc.Japan-United States Sem.,Tokyo,1977),pp.101-102,North-Holland,Amsterdam-New York,1979.
[150]Meier,H,Nowacki,H.Interpolating curves with gradual changes in curvature.Computer Aided Geometric Design 1987;4:297-305.
[151]Yongwei Miao,Huahao Shou,Jieqing Feng,Qunsheng Peng,A.Robin Forrest.Bézier Surfaces of Minimal Internal Energy.IMA Conference on the Mathematics of Surfaces 2005:318-335
[152]Moccozet L,Magnenat NT.Dirichlet free-form deformations and their application to hand simulation.In Computer Animation '97,1997,93-102.IEEE Computer Society.
[153]Monterde,J..The Plateau-Bézier problem.In:The Proceedings of the X Conf.on Mathematics of Surfaces,Leeds,UK.In:Lecture Notes in Comput.Sci.,vol.2768.Springer-Verlag,Berlin/New York,2003,262-273.
[154]Monterde,J.Bézier surfaces of minimal area:The Dirichlet approach.Computer Aided Geometric Design.2004,21,117-136.
[155]Monterde J,Ugail H.On harmonic and biharmonic Bézier surfaces.Computer Aided Geometric Design,2004,21(7),697-715.
[156]Monterde J,Ugail H.A general 4th-order PDE method to generate Bézier surfaces from the boundary.Computer Aided Geometric Design,2006,23(2),208-225.
[157]MoC.B.Morrey.Multiple integrals in the calculus of variations,Springer-Verlag,New York,1966.
[158]Moreton HP,Séquin CH.Minimum variation curves and surfaces for computer aided geometric design.In Designing Fair Curves and Surfaces-Shape Quality in Geometric Modeling and Computer Aided Design,Siam,1994.
[159] Nielson, G.M., Franke, R.. A method for construction of surfaces under tension. Rocky Mountain J. Math., 1984, 14: 203-221.
[160] Nielson G M. Scattered data modeling. IEEE Computer Graphics and Application, 1993, 13(1): 61-70.
[161] Nishimura, H., Hirai, M., Kawai, T., Kawata, T., Shirakawa, I., Omura, K.. Object modeling by distribution function and a method of image generation. Transactions IECE Japan, Part D, 1985, J68-D(4): 718-725.
[162] Nitsche, J.C.C. . Lectures on minimal surfaces, vol. 1. Cambridge Univ. Press, Cambridge.
[163] Osserman, R. A survey of minimal surfaces, Dover publ. 2nd ed., New York.
[164] Patrikalakis, N.M.. Approximation conversion of rational splines. Computer Aided Geometric Design, 1989, 6: 155-165.
[165] J.F.Paul, C.Djurdje, L.Alan, J.Klinowski. Exact computation of the triply periodic D-minimal surface. Chemical Physics Letters, 1999, 314( 5-6), 543-551.
[166] Piegl, L., Tiller, W.. Curve and surface constructions using rational B-splines. Computer Aided Design, 1987, 19(9): 485-498.
[167] Piegl, L.. On NURBS: A survey. IEEE Computer Graphics and Application, 1991, 11(1): 55-71.
[168] Piegl, L., Tiller, W.. The NURBS book. Springer Verlag, 1995.
[169] Piegl, L., Tiller, W.. Computing the derivative of NURBS with respect to a knot. Comput Aided Geometric Design, 1998, 15: 925-934.
[170] Piegl, L., Tiller, W.. Surface approximation to scanned data. Visual Comput, 2000, 16: 386-395.
[171] U. Pinkall, K. Polthier. Computing discrete minimal surface and their conjugates. Exp. Math, 1993, 2(1):15-36.
[172]Pottmann,H..The geometry of Tchebycheffian spines.Computer Aided Geometric Design,1993,10:181-210.
[173]Pottmann,H.,Wagner,M.G..Helix splines as example of affine Tchebycheffian splines,Advance in Computational Mathematics,1994,2:123-142.
[174]H.Pottmann,S.Brell-Cokcan,and J.Wallner.Discrete surfaces for architectural design.In P.Chenin,T.Lyche,and L.L.Schumaker,editors,Curves and Surface Design:Avignon 2006,213-234.Nashboro Press,2007,ISBN 978-0-9728482-7-5.
[175]Pottmann,H.,Liu,Y.,Wallner,J.,Bobenko,A.,Wang,W.P..Geometry of multi-layer freeform structures for architecture,ACM Transactions on Graphics,2007,26(3),1-11.
[176]Pottmann,H.,Liu,Y..2007.Discrete surfaces in isotropic geometry.Mathematics of Surfaces Ⅻ,341-363.
[177]齐东旭.分形及计算机生成.北京:科学出版社,1994.
[178]覃廉,关履泰.基于NURBS的极小曲面造型.高等学校计算数学学报.2005,16(S1):175-181.
[179]Renka,R.J..Interpolatory tension splines with automatic selection of tension factors.SIAM J.Sci.Stat.Comp.1987,8:393-415.
[180]Rentrop,P..An algorithm for the computation of the exponential spline.Numer.Math.1980,35:81-93.
[181]Rado,On Plateau's problem,Ann.of Math.1930,31(2),457-469.
[182]Ruprecht D.,Nagel R.,Muller H..Spatial free-form deformation with scattered data interpolation methods.Computer and Graphics,1995,19:63-72.
[183]Sanchez-Reyes J..A simple technique for NURBS shape modification.IEEE Comput Graphics Appl 1997,17:52-61.
[184]Sambandan K,Kedem K,Wang K K.Generalized planar sweeping of polygons,Journal of Manufacturing Systems,1992,11(4):246-257.
[185]Schoenberg,I.J..Contributions to the problem of approximation of equidistant data by analytic functions.Quart.Appl.Math.,1946,4:45-99.
[186]Schoenberg,I.J..On spline function.In:Shisha,O.eds.Inequalities,Academic Press,New York,1967.
[187]Schoenberg,I.J..Spline interpolation and best quadrature formulae.Bull.Amer.Math.Soc.1964,70(1),143-148.
[188]Schumaker,L.L..Spline Functions:Basic Theory.John Wiley Sons,New York,1981:363-499.
[189]Sederberg T.W,Parry S.R.Free-form deformation of solid geometric models.Computer Graphics(SIGGRAPH '86),1986,20(4):151-160.
[190]Sederberg,T.W..Techniques for cubic algebraic surfaces,part one.IEEE Computer Graphics and its Applications,1990,10(4):14-25.
[191]Sederberg,T.W..Techniques for cubic algebraic surfaces,part two.IEEE Computer Graphics and its Applications,1990,10(5):12-21.
[192]Sederberg T.,Zheng J.,Swell D.,et al..Non-Uniform recursive subdivision surfaces.Cohen Med Computer Graphics,1998,32(4):387-394.
[193]Séquin,C.H...CAD Tools for Aesthetic Engineering.Computer Aided Design,2005,37(7),737-750.
[194]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.
[195]Singh K,Fiume E..Wires:A geometric deformation technique.Computer Graphics(SIGGRAPH '98),1998,405-414.
[196]Song WH,Yang XN,Free Form deformation with weighted T-splines.The visual computer,2005,21(3):139-151.
[197]Spath,H..Two-dimensional exponential splines.Computing,1971,7:364-369.
[198]Tai C L,Wang G J.Interpolation with slackness and continuity control and convexity-preservation using singular blending.Journal of Computational and Applied Mathematics,2004,173(4):337-361
[199]苏步青.计算几何的兴起.自然杂志,1978,7:409-412.
[200]苏步青.论Bézier曲线的仿射不变量,计算数学,1980.
[201]苏步青,刘鼎元.计算几何.上海科学技术出版社,1981.
[202]唐荣锡.CAD/CAM技术.北京航天航空大学出版社,北京,1994
[203]Tiller,W..Rational B-splines for curve and surface representation.IEEE Computer Graphics and its Application,1983,3(6):61-69.
[204]Tiller,W..Geometric modeling using non-uniform rational B-splines:mathematical techniques.SIGGRAH tutorial notes,ACM,New York,1986.
[205]Tiller,W..Knot-remove algorithms for blURBS curves and surfaces.Computer Aided Design,1992,24(8):445-453.
[206]Turk,G.,O'Brien,J..Shape Transformation Using Variational Implicit Functions.Computer Graphics(SIGGRAPH),1999:335-342.
[207]Ugail H,Bloor MIG,Wilson MJ.Techniques for interactive design using the PDE method.ACM Transactions on Graphics,1999,18(2):195-212.
[208]Ugail H.On the spine ofa PDE surface.In:Wilson MJ,Martin RR,editors.Mathematics of Surfaces X.Berlin:Springer;2003,p366-376.
[209]Veltkamp RC,Wesselink W.Modeling 3D curves of minimal energy.Eurographics 95,p.97-110.
[210]Versprille,K.J..Computer aided design application of rational B-spline approximation form.Ph.D.thesis,Syracuse University,Syracuse,N.Y.,1975.
[211]Wallner J,Pottmann H,Hofer M.Fair webs.The Visual Computer 2007;23:83-94.
[212]Walz,G..Identities for trigonometric B-splines with an application to curve design.BIT,1997,37(1):189-201.
[213]Wang Guojin.Generating NURBS curves by envelopes.Computing,1992,48(3-4):275-289.
[214]Wang,G.,Chen,Q..NUAT B-spline curves.Computer Aided Geometric Design,2004,21:193-205.
[215]Wang J.Z,Jiang T.Z.Nonrigid registration of brain MRI using NURBS.Pattern Recognition Letters 2007,28,214-223.
[216]王国瑾.Bézier曲线曲面的离散求交方法.浙江大学学报,计算几何专辑,1984:108-119.
[217]王国瑾,汪国昭,郑建民.计算机辅助几何设计,高等教育出版社,北京,2001.
[218]王文涛,汪国昭.带形状参数的均匀B样条.计算机辅助设计与图形学学报,2004,16(6):783-788.
[219]王文涛,汪国昭.带形状参数的三角多项式均匀B样条.计算机学报,2005,28(7):1192-1198.
[220]王文涛,汪国昭.带形状参数的双曲多项式均匀B样条.软件学报,2005,16(4):625-633.
[221]汪国昭,沈金福.有理Bézier曲线的离散和几何性质.浙江大学学报,1985,19(3):123-130.
[222]汪国昭.有理Bézier曲线的的包络性质全国计算几何与样条函数学术会议,计算几何论文集,1986,6.
[223]Wang XF,Cheng F,Barsky BA.Energy and B-spline interproximation.Computer Aided Design,1997;29:485-496.
[224]Wang,Y..Periodic surface modeling for computer aided nano design.Computer Aided Design,2007,39(3),179-189.
[225]Weir D J,Milroy M J,Bradley C.et al..Reverse engineering physical models employing wrap around B-Spline surfaces and quadrics.Journal of Engineering Manufacture,1996,210(B):147-157.
[226]Wyvill,G.C.,McPheeters,Wyvill,B..Data structure for soft objects.The Visual Computer,1986,2:227-234.
[227]Wyvill,B.,Wyvill,G..Field functions for implicit surface.The Visual Computer,1989,5:75-82.
[228]Xu Gang,Wang Guozhao.AHT Bézier surfaces and NUAHT B-spline curves.Journal of Computer Science and Technology,2007,22(4):597-607.
[229]Xu Gang,Wang Guozhao.Control mesh representation of a class of minimal surfaces.Journal of Zhejiang University SCIENCE A,2006,7(9):1544-1549.
[230]Xu Gang,Wang Guozhao.Harmonic-type Bézier surfaces over rectangular and triangular domain.Journal of Information and Computational Science,2006,3(2):325-332.
[231]徐岗,汪国昭.Doo-Sabin细分算法在动态模式下的推广.计算机辅助设计与图形学学报.2006,18(3),341-346.
[232]徐岗,汪国昭.三角域上的调和B-B曲面.计算机学报.2006,29(12):2180-2185.
[233]徐岗,汪国昭.一类极小曲面的控制网格表示.计算机辅助设计与图形学学报.2007,19(2),240-244.
[234]徐岗,汪国昭.带局部形状参数的三次均匀B样条曲线的扩展.计算机研究与发展.2007,44(6):1032-1037.
[235]徐岗,汪国昭.一类新的极小曲面及其应用.第三届全国几何设计与计算大会论文集,中国兰州,2007年7月,56-60.
[236]徐岗,汪国昭.PDE曲面的Bézier逼近.软件学报.2007,18(11):2974-2920.
[237]徐岗,汪国昭.参数多项式极小曲面.第一届全国计算机数学学术会议,中国南昌,2007年11月.
[238]Xu,Guoliang,Zhang,Qin.Minimal mean-curvature-variation sufaces and their applications in surface modeling.GMP 2006,Lecture Notes in Computer Science,2006,4077,357-370.
[239]Yang,Q.,Wang,G..Inflection points and singularities on C-curves.Computer Aided Geometric Design,2004,21,207-213.
[240]Yang X.N.Surface interpolation of meshes by geometric subdivision,Computer Aided Design,2005,37(5):497-508
[241]Yang X.N.Normal based subdivision scheme for curve design,Computer Aided Geometric Design,2006,23(3):243-260
[242]Yates,R.C.."Envelopes." A Handbook on Curves and Their Properties.Ann Arbor,MI:J.W.Edwards,1952:75-80.
[243]Yoon,S.H.,Kim,M.S..Sweep-based Free form deformation.Computer Graph Forum(Eurographics 2006),2006,25(3),487-496
[244]Zhang,J..C-curves:An extension of cubic curves.Computer Aided Geometric Design,1996,13(3):199-217.
[245]Zhang,J..Two different forms of C-B-splines.Computer Aided Geometric Design,1997,14(1):31-41.
[246]Zhang,J..C-Bézier curves and surfaces.Graphical Models and Image Processing,1999,61(1):2-15.
[247]Zhang J,Krause F,Zhang H.Unifying C-curves and H-curves by extending the calculation to complex numbers.Computer Aided Geometric Design,2005,22(9):865-883.
[248]Zhang J,Krause F.Extend cubic uniform B-splines by unified trigonometric and hyperbolic basis.Graphic Models,2005,67(2):100-119.
[249]Zhang JJ,You LH.Surface representation using second,fourth and mixed order partial differential equations.International conference on Shape Modeling and Applications,Genova,Italy,5-7 May,2001.
[250]Zheng J.M.,Chan K.W.,GibsonI..A new approach for direct manipulation of free-form curve.Comput Graphics Forum,1998,17:327-34.
[251]Zhou,Kambhamettu C..Extending superquadrics with exponent modeling and reconstruction.Graphical Models,2001,63:1-20.
[252]曾芳玲,陈效群,冯玉瑜.二次曲线的多项式逼近.计算机辅助设计与图形学学报,2003,15(5):547-551.
[253]张宏鑫.复杂形体建模与绘制的离散方法研究.浙江大学博士学位论文,2002年4月.
[254]朱心雄.自由曲线曲面造型技术.科学出版社,2000.
[2] Aspert, N., Ebrahimi, T., Vandergheynst, P.. Non-linear subdivision using local spherical coordinates. Computer Aided Geometric Design, 2003, 20, 165 - 187.
[3] Au C.K.,Yuen M.M.F.. Unified Approach to NURBSCurve Shape Modification. Computer Aided Design, 1995, 27(2):85-93.
[4] Barr, AH., Global and local deformation of solid primitives, Computer Graphics (SIGGRAPH), 1984,18(3):21-30.
[5] Bezier, P. Numerical Control: Mathematics and Applications, John Wiley and Sons, New York, 1972.
[6] Bajaj, C, Ihm, I.. C~1 smoothing of polyhedral with implicit algebraic splines. Proceeding of SIGGRAPH'92, Computer Graphics, 1992, 26(2): 79-88.
[7] Bajaj, C. Surface fitting using implicit algebraic surface patches. In: Hans Hagen eds. SIAM Topics in surface modeling, 1992: 23-52.
[8] Bajaj, C., Ihm, I.. Algebraic surface design with Hermite interpolation. ACM Transactions on Graphics, 1992, 11(1): 61-91.
[9] Bajaj, C, Ihm, I., Warren, J.. Higher-order interpolation and least-square approximate using implicit algebraic surfaces. ACM Transaxtions on Graphics, 1993, 12(4): 327-347.
[10]Barsky B.A.Computer Graphics and Geometric Modeling Using Beta-Spline.Heidelberg Springer-Verlag,1988
[11]Blinn,J.F..A generalization of algebraic surface drawing.ACM Transaction on Graphics,1982,1(3):235-256.
[12]Bloor MIG,Wilson MJ.Generating blend surfaces using partial differential equations.Computer Aided Design,1989,21(3):165-171.
[13]Bloor MIG,Wilson MJ.Representing PDE surface in terms of B-splines.Computer Aided Design,1990,22:324-331.
[14]Bloor MIG,Wilson MJ.Spectral approximations to PDE surfaces.Computer Aided Design,1996,28(2):145-152.
[15]Bloor MIG,Wilson MJ.An analytic pseudo-spectral method to generate a regular 4-sided PDE surface patch.Computer Aided Geometric Design,2005,22(3):203-219.
[16]Boehm W..Inserting new knots into B-spline curve.Computer Aided Design,1980,12(4):199-201.
[17]Bohem,W.,Farin,G.Kahman,J..A survey of curve and surfaces methods in CAGD.Computer Aided Geometric Design,1984,1:1-60.
[18]Boehm,W.,Prautzsch,H..The insertion algorithm.Computer Aided Design,1985,17(2):58-59.
[19]Boehm,W..On the efficiency of knot insertion algorithms.Computer Aided Geometric Design,1985,2(1-3),141-143
[20]Boor,C.,Pinkus,A..The approximation of a totally positive band matrix by a strictly banded totally positive one.Lin.Alg.Appl.,1982,42:81-98.
[21]Bradley C.,Vichers G.W..Free-form surface reconstruction for machine vision rapid prototyping.Optical engineering,1993,32(9):2191-2199.
[22]Brunnett G,Hagen H,Santarelli P.Variational design of curves and surfaces.Surveys on Mathematics for Industry 1993;3:1-27.
[23]Carnicer,J.,Pena,J..Totally positive bases for shape preserving curve design and optimality of B-splines.Computer Aided Geometric Design,1994,11:635-656.
[24]Carnicer,J.,Pena,J.M..Total positivity and optimal bases,in:Gasca,M.and Micchelli,C.A.,eds..Total positivity and its Applications,Kluwer Academic,Dordrecht,1996:133-155.
[25]E.Catmull,J.Clark.Recursively generated B-spline surfaces on arbitrary topological meshes.Computer Aided Design,10(6):350-355,1978.
[26]G.M.Chaikin.An Algorithm for High Speed Curve Generation,Computer Vision,Graphics,and Image Processing,vol.3,no.4,346-349,1974.
[27]Chang,GZ.Bernstein polynomials via the shifting operator.American Mathematical Monthly,1984,91:634-638.
[28]常庚哲、吴俊恒.贝齐尔曲线曲面的数学基础及其计算,北京航空学院科学研究报告BH-B381,1978.
[29]Chang YK,Rockwood AP.A generalized de Casteljau approach to 3D free-Form deformation.Computer Graphics(SIGGRAP H '94),1994,257-260.
[30]Chen,F.,Deng,L..Interval implicitization of rational curves.Computer Aided Geometric Design,2004,21:401-415.
[31]Chen,Q.,Wang,G..A class of Bézier-like curves.Computer Aided Geometric Design,2003,20(1):29-39.
[32]陈文喻,汪国昭.Bézier曲面的精确自由变形.计算机辅助设计与图形学学报,2006,18(8),1160-1164.
[33]陈笑.有理Bézier极小曲面设计:Dirichlet方法的推广.杭州,浙江大学,硕士学位论文,2006.
[34]Cline,A.K..Scalar- and Planar-valued curve fitting using splines under tension.,Communications of the ACM,1974,17:218-233.
[35]Cohen,E.,Lyche,T.,Riesenfeld,R.F..Discrete B-spline and subdivision techniques in computer aided geometric design and computer graphics.Computer Graphics and Image Processing,1980,14(2):87-111.
[36]Cohen,E.,Riesenfeld,R.F..General matrix representations for Bézier and B-spline curves.Computers in Industry,1982,3(1-2):9-15.
[37]Choi,B.K.,Yoo,W.S.,Lee,C.S..Matrix representation for NURBS curves and surfaces.Computer Aided Design,1990,22(4):235-240.
[38]P.Concns,Numerical solution of the minimal surface equation,Math.Comput.1967,21,340-350.
[39]Coons,S.A..Surfaces for computer aided design of space Figures.Technical Report,MIT,1964,Project MAC-TR-255.
[40]Coons,S.A..Surfaces for computer aided design of space forms.Technical Report,MIT,1967,Project MAC-TR-411.
[41]Coquillart S.Extended free-form deformation:A sculpturing tool for 3D geometric modeling.Computer Graphics(SIGGRAP H'90),1990,24(4):187-196.
[42]Cosin,C.,Monterde,J..2002.Bézier surfaces of minimal area.In:Sloot,Tan,K.,Dongarra,Hoekstra(Eds.),Proceedings of the Int.Conf.of Computational Science,ICCS' 2002,Amsterdam,vol.2.In:Lecture Notes in Comput.Sci.,vol.2330.Springer-Verlag,pp.72-81.
[43]Cox,M.,G..The numerical evaluation of B-spline.Report No.NPLDNACS-4,National Laboratory,1971.
[44]Cutler B.,Whiting E.Constrained planar remeshing for achitecture.SGP 2006,Poster.
[45] Datta A., Soundaralakshmi S.. Fast parallel algorithm for distance transforms. Parallel and Distributed Processing Symposium, Proceedings 15th International, 2001: 1130-1134.
[46] de Boor, C On Calculation with B-spline. Journal of Approximation Theory, 1972, 6(1): 50-62.
[47] de Boor, C. Splines as Linear Combinations of B-splines. Academic Press, New York, 1976.
[48] de Boor, C. A practical guide to splines. Springer-Verlag, New York, 1978.
[49] Demko S, Hodges L, Naylor B. Construction of fractal objects with iterated functions systems. In Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH, 1985: 271-278.
[50] Dietz, R., Hoschek, J., Juttler, B.. An algebraic approach to curves and surfaces on the sphere and on other quadrics. Computer Aided Geometric Design, 1993, 10: 211-229.
[51] Ding, Q.L., Davies, B.J.. Surface Engineering Geometry for Computer Aided Design and Manufacture. Ellis Horwood, 1987.
[52] Do Carmo, M. 1976. Differential geometry of curves and surfaces. Prentice-Hall.
[53] DJ. Douglas. Solution of the problem of Plateau, Trans. AMS. 1931, 33, 263-321.
[54] D.Doo, M. Sabin. Analysis of the behaviour of recursive division surfaces near extraordinary points. Computer Aided Design, 1978 10(6):356-360.
[55] Du HX, Qin H. Dynamic PDE-based Surface design using geometric and physical constraints. Graphical Models, 2005, 67, 43-71.
[56] N. Dyn, D. Levin, J.A. Gregory. A Butterfly Subdivision Scheme for Surface Interpolation with Tension Control. ACM Transactions on Graphics, 1990, (9) 2,160-169.
[57]G.Dziuk,J.E.Hutchinson.The discrete Plateau problem,Math.Comp.1999,68(225):1-23.
[58]Farin,G..From Conics to NURBS:A Tutorial and Survey.IEEE Computer Graphics and Applications.1992,1(5):78-86.
[59]Farin,G..Curves and Surfaces for Computer Aided Geometric Design.4th edn.Academic Press,San Diego,1997.
[60]FeH.Federer,W.H.Fleming.Normal and integral currents,Ann.of Math.1960,72,458-520.
[61]FeH.Federer.Geometric Measure Theory,Springer-Verlag,New York,1969.
[62]FeH.Federer.The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension,Bull.AMS.76 1970,767-771.
[63]Feng JQ,Ma LZ,Peng QS.A new free-form deformations through the control of parametric surfaces.Computer and Graphics,1996,20(4):531-539.
[64]冯结青,彭群生,马利庄.沿参数曲面的均匀变形方法.软件学报,1998,9(4):263-267
[65]冯结青,马利庄,彭群生.一种应用参数曲面的动态自由变形方法.计算机学报,1997,20(7):600-606.
[66]Feng JQ,Pheng-Ann Heng,Tien-Tsin Wong.Accurate B-spline Deformation of Polygonal Object.Journal of Graphics Tools.1998,3(3):11-17.
[67]冯结青,彭群生.Bézier曲面的函数复合及其应用.软件学报,1999,10(12):1316-1321
[68]冯结青,赵豫红,万华根,郭建民,金小刚,彭群生.基于曲线和曲面控制的多边形物体变形反走样.计算机学报,2005,28(1),60-67
[69] Feng J., Peng Q.. Accelerating accurate B-spline free-form deformation through polynomial interpolation. Journal of Graphics Tools, 2000, 5(1): 1-8
[70] Feng, J., Shao, J., Jin, X., Peng, Q., and Forrest, A.R.. Multiresolution free-form deformation with subdivision surface of arbitrary topology. The Visual Computer, 2006, 22(1), 28-42.
[71] Fergusion, J.C. Multivariable curve interpolation. Report D2-22504, The Boeing Co. Seattle, Washington, 1963.
[72] Fergusion, J.C. Multivariable curve interpolation. Journal of ACM, 1964, 2: 221-228.
[73] Forrest, A.R.. Interactive interpolation and approximation by Bezier curve. The Computer Journal, 1972, 15(1): 71-79.
[74] Forsey D. R.,Bartels R H.. Surface fitting with hierarchical splines. ACM Trans. on Graphics, 1995, 14(2): 134-161.
[75] Giancarlo Amati. A multi-level filtering approach for fairing planar cubic B-spline curves. Computer Aided Geometric Design 2007, 24, 53 - 66.
[76] Goldman, R. N., Sederberg, T. W., Anderson, D. C. Vector elimination: A technique for the implicitization, inversion, and intersection of planar parametric rational polynomial curves. Computer Aided Geometric Design, 1984, 1: 327-356.
[77] Gomes, J., Darsa,L., Costa, B. and Velho, L.. Warping and Morphing of Graphical Objects. Morgan Kaufmann Publishers, 1998.
[78] Goodman, T.N.T.. Shape preserving interpolation by parametric rational cubic splines. in: International Series of Numerical Mathematics 86, Birkh iuser, Basel, 1988: 149-158.
[79] Goodman, T.N.T., Said H. B.. Shape preserving properties of the generalized Ball basis. Computer Aided Geometric Design, 1991, 8(2): 115-121.
[80]Goodman,T.N.T.,Said,H.B..Properties of generalized Ball curves and surfaces.Computer Aided Design,1991,23(8):554-560.
[81]Goodman,T.N.T.,Ong,B.H.and Unsworth K..Constrained interpolation using rational cubic splines,in:Farin G.,ed.,Nurbs for Curve and Surface Design,SIAM,1991:59-74.
[82]Goodman,T.N.T..Total positivity and the shape of curves,in:M.Gasca,C.A.Micchelli(Eds.),Total Positivity and Its Applications,Kluwer,Dordrecht,1996:157-186.
[83]Goodman,T.N.T.,Ong,B.H..Shape preserving interpolation by space curves.Report AA/963,University of Dundee,1996.
[84]Goodman,T.N.T.,Ong B.H..Shape preserving interpolation by space curves.Computer Aided Geometric Design,1997,16:1-17.
[85]Goodman,T.N.T..Automatic interpolation by fair shape-preserving C space curves.Computer Aided Geometric Design,1998,15(3):813-822.
[86]Goodman,T.N.T.,Sun Q..Total positivity and refinable functions with general dilation.Appl.Comput.Harmon.Anal.,2004,16:69-89.
[87]Gordon,W.J.,Riesenfeld,R.F..Bernstein Bézier methods for the Computer Aided Geometric Design of free-form curves and surfaces.Journal of ACM,1974,21:293-310.
[88]Gordon,W.J.,Riesenfeld,R.F..B-spline curves and surfaces.In:Barnhill,R.E.and Riesenfeld,R.F.eds.,Computer Aided Geometric Design.Academic Press,New York,1974:95-126.
[89]Grabowski,H.,Li,X..Coefficient formula and matrix of nonuniform B-spline functions.Computer-Aided Design,1992,24(12):637-642.
[90]Greissmair J,Purgathofer W..Deformation of solids with trivariate B-Splines.Computer Graphics Forum,1989,137-148.
[91]Gruber,F.,Glaeser,G..Magnetism and minimal surfaces-a different tool for surface design.Computational Aesthetics 2007-Eurographics Workshop on Computational Aesthetics in Graphics,Visualization and Imaging.D.W.Cunningham,G.Meyer,L.Neumann,A.Dunning,R.Paricio(eds.),81-88
[92]Halstead,M.,Barsky,B.,Klein,S.,Mandell,R..Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals.Proceedings of SIGGRAPH,1996:335-342.
[93]韩旭里,刘圣军.三次均匀B样条曲线的扩展.计算机辅助设计与图形学学报,2003,15(5):576-578
[94]He T,Wang S,Kaufman A.Wavelet-based volume morphing.Proceedings of Visualization'94,IEEE Computer Society Press,1994,85-92.
[95]何玲娜.代数双曲三角曲线的研究.浙江大学硕士学位论文,2004年4月.
[96]Hernández-Mederos,Victoria,Estrada-Sarlabous,Jorge.Sampling points on regular parametric curves with control of their distribution,Computer Aided Geometric Design,2003(20):363-382.
[97]Higashi M.,Kaneko K.,Hosaka M..Generation of high-quality curve and surface with smoothly varying curvature.Eurographics 88,p.79-92.
[98]Hiroshi,I..Minimal submanifolds in Riemannian spin manifolds with parallel spinor fields.J.of Geometry and Physics Volume,2002,41(3),258-273.
[99]Hoffmann,C.M..Implicit curves and surfaces in CAGD.IEEE Comput.Graph.Appl.,1993,13:79-88.
[100]Hoscheck J,Schneider F.1990.Spline conversion for trimmed rational Bezier and B-spline surfaces.Computer Aided Design,22(9),580-590.
[101]Hsu W.M,Hughes J.F.,Kaufman H..Direct manipulation of free-form deformations.Computer Graphics(SIGGRAPH '92),1992,26(2):177-184.
[102]Hu S.M.,Li Y.F.,Ju T.and Zhu X..Modifying the shape of NURBS surfaces with geometric constraints.Computer Aided Design,2001,33(12):903-912.
[103]Hu SM,Zhang H,Tai CL,Sun JG.Direct manipulation of FFD:efficient explicit solutions and decomposible multiple point constraints.The Visual Computer,2001,17:370-379.
[104]Hua J,Qin H.Free-form deformation via sketching and scalar fields.In Solid Modeling international(SM'03),2003,328-333.
[105]Hui K.C.Free Form design using axial curve pairs.Computer Aided Design,2002,34,583-555.
[106]S.Hyde,S.Andersson,K.Larsson,Z.Blum,T.Landh,S.Lidin,B.W.Ninham.The Language of Shape.The Role of Curvature in Condensed Matter:Physics,Chemistry and Biology,Elsevier,Amsterdam,1997.
[107]计忠平,刘利刚,王国瑾.无局部自交的轴变形新方法.计算计学报,2006,29(5):828-834.
[108]季洁,满家巨.任意边界下的极小曲面造型问题.计算机与现代化.2007,4:7-9.
[109]金文标,汪国昭.一类极小曲面的几何设计.计算机学报.1999,23(12):1277-1279.
[110]金文标,杨勋年.一种负高斯曲率的双三次Bézier曲面,高校应用数学学报.2000,15(2):206-210.
[111]金文标.若干特殊曲面的造型和应用.杭州,浙江大学,博士学位论文,1999.
[112]Joe B.Multiple-knot and rational cubic Beta-splines.ACM Transactions on Graphics,1989,8(2):100-120
[113]Ju T,Schaeffer S,Warren J.Mean value coordinates for closed triangular meshes.ACM Transaction on Graphics(SIGGRAPH '05),2005,24(3):561-566.
[114]J.M.Pena..Shape Preserving Representations in Computer Aided Geometric Design.Nova Science Publishers,Commack(New York),1999.
[115]J.Sánchez-Reyes.A Simple Technique for NURBS Shape Modification.IEEE Computer Graphics and Applications,Januaty-February,1997,17(1):52-59.
[116]R.D.Kamien Decomposition of the height function of Scherk's first surface,Applied Mathematics Letters 14(2001) 797-800
[117]Kobbelt,L..A variational approach to subdivision.Computer Aided Geometric Design,1996,13(8),743-761.
[118]Koch,P..Multivariate trigonometric B-splines.J.Approx.Theory,1988,54:162-168.
[119]Koch,P.E.,Lyche,T..Exponential B-splines in tension,in:C.K.Chui,L.L.Schumaker and J.D.Ward,eds.,Approximation Theory Ⅵ,Academic Press,New York,1989:361-364.
[120]Koch,P.E.,Lyche,T..Construction of exponential tension B-splines of arbitrary order.In:Laurent,P.J.,Le Méhauté,A.,Schumaker,L.L.(Eds.),Curves and Surfaces.Academic Press,New York,1991:255-258.
[121]Koch,P.,Lyche,T.,Neamtu,M.,Schumaker,L..Control curves and knot insertion for trigonometric splines.Adv.Comput.Math.,1995,3:405-424.
[122]KrishnamurhyV,Le voyM.Fitting smooth surfaces to dense polygon meshes.Computer Graphics,1996,30(4):313-324.
[123]Kvasov,B.,Sattayatham,P..GB-splines of arbitrary order.J.Comput.Appl.Math.,1999,104:63-88.
[124]Lazarus,F.,Coquillart,S.,Jancene,P..Axial deformations:an intuitive deformation technique.Computer Aided Design,1994,26(8):608-612.
[125]Lazarus,F.,Anne Verroust.Three-dimensional metamorphosis:a survey.The Visual Computer,1998 14(8/9):373-389.
[126]Lamousin H.J,Waggenspack W.J.Nurbs-based free-form deformations.IEEE Computer Graphics and Applications,1994,14(6):59-65.
[127]Lee S.,K.Chwa,S.Shin,G.Wolberg.Image Metamorphosis Using Snakes and Free-Form Deformations.In SIGGRAPH 95 Conference Proceeding,1995,439-448.
[128]Lee S.,Wolberg G.,Shin S.Y..Scatered data interpolation with multilevel B-splines.IEEE Trans.on Visualization and Computer Graphics.1997,3(3):228-244.
[129]Li Y.,Wang G..Two kinds of B-basis of the algebraic hyperbolic space.Journal of Zhejiang University SCIENCE 2005,6A(7):750-759.
[130]Liu Ligang,Wang Guojin.Explicit matrix representation for NURBS curves and surfaces.Computer Aided Geometric Design,2002,19(6):409-419.
[131]Liu,Y.,Pottmann,H.,Wallner,J.,Yang,Y.L.,Wang,W.P...Geometric modeling with conical meshes and developable surfaces.ACM Trans.Graphics,2006,25(3),681-689,Proc.SIGGRAPH 2006.
[132]刘计善.一类极小曲面造型设计的新方法.复旦学报(自然科学版).2007,46(2):192-197.
[133]刘鼎元.三次参数曲线段和三次Bézier曲线形状控制,应用数学学报,1981,5:158-166.
[134]刘鼎元.平面n次Bézier曲线的凸性定理,数学年刊,1982:45-55.
[135]刘鼎元.有理Bézier曲线.应用数学学报,1985,8(1):70-82.
[136]Loe K F.α-B-spline:a linear singular blending B-spline.The Visual Computer,1996,12(1):18-25.
[137]Charles Loop.Smooth subdivision surfaces based on triangles.Master's thesis,University of Utah,Department of Mathematics,1987.
[138]L(u|¨),Y.,Wang,G.,Yang,X..Uniform trigonometric polynomial B-spline curves.SCIENCE IN CHINA(Series F),2002,45:335-343.
[139]L(u|¨),Y.G.,Wang,G.Z.,Yang,X.N..Uniform hyperbolic polynomial B-Spline curves.Computer Aided Geometric Design,2002,19:379-393.
[140]路宁,宁涛,唐荣锡,张兆璞.Dirichlet自由变形方法及其在建立尺寸驱动人体模型中的应用.工程图学学报,2004,76-83.
[141]吕勇刚.CAGD自由曲线曲面造型中均匀样条的研究.浙江大学博士学位论文,2002年5月.
[142]Mainar,E.,Pena,J.,Sánchez-Reyes,J..Shape preserving alternatives to the rational Bézier model.Computer Aided Geometric Design,2001,18(1):37-60.
[143]Mainar,E.,Pena,J..Quadratic-cycloidal curves.Advances in Computational Mathematics,2004,20:161-175.
[144]满家巨,汪国昭.正螺面与悬链面的表示与构造.计算机辅助设计与图形学学报.2005,17(3):431-436.
[145]满家巨,汪国昭.等温参数多项式极小曲面.计算机学报.2002,25(2):197-201.
[146]满家巨,汪国昭.基于有限元方法的极小曲面造型.计算机学报.2003,26(4):507-510.
[147]满家巨,汪国昭.B样条函数极小曲面造型.软件学报.2003,14(4):824-829
[148]Martin R R,Stephenson P C.Sweeping of the three dimensional objects.Computer Aided Design,1990,22(4):223-234.
[149]Meeks,William H.,Ⅲ;Yau,Shing Tung The classical Plateau problem and the topology of 3-manifolds.Minimal submanifolds and geodesics(Proc.Japan-United States Sem.,Tokyo,1977),pp.101-102,North-Holland,Amsterdam-New York,1979.
[150]Meier,H,Nowacki,H.Interpolating curves with gradual changes in curvature.Computer Aided Geometric Design 1987;4:297-305.
[151]Yongwei Miao,Huahao Shou,Jieqing Feng,Qunsheng Peng,A.Robin Forrest.Bézier Surfaces of Minimal Internal Energy.IMA Conference on the Mathematics of Surfaces 2005:318-335
[152]Moccozet L,Magnenat NT.Dirichlet free-form deformations and their application to hand simulation.In Computer Animation '97,1997,93-102.IEEE Computer Society.
[153]Monterde,J..The Plateau-Bézier problem.In:The Proceedings of the X Conf.on Mathematics of Surfaces,Leeds,UK.In:Lecture Notes in Comput.Sci.,vol.2768.Springer-Verlag,Berlin/New York,2003,262-273.
[154]Monterde,J.Bézier surfaces of minimal area:The Dirichlet approach.Computer Aided Geometric Design.2004,21,117-136.
[155]Monterde J,Ugail H.On harmonic and biharmonic Bézier surfaces.Computer Aided Geometric Design,2004,21(7),697-715.
[156]Monterde J,Ugail H.A general 4th-order PDE method to generate Bézier surfaces from the boundary.Computer Aided Geometric Design,2006,23(2),208-225.
[157]MoC.B.Morrey.Multiple integrals in the calculus of variations,Springer-Verlag,New York,1966.
[158]Moreton HP,Séquin CH.Minimum variation curves and surfaces for computer aided geometric design.In Designing Fair Curves and Surfaces-Shape Quality in Geometric Modeling and Computer Aided Design,Siam,1994.
[159] Nielson, G.M., Franke, R.. A method for construction of surfaces under tension. Rocky Mountain J. Math., 1984, 14: 203-221.
[160] Nielson G M. Scattered data modeling. IEEE Computer Graphics and Application, 1993, 13(1): 61-70.
[161] Nishimura, H., Hirai, M., Kawai, T., Kawata, T., Shirakawa, I., Omura, K.. Object modeling by distribution function and a method of image generation. Transactions IECE Japan, Part D, 1985, J68-D(4): 718-725.
[162] Nitsche, J.C.C. . Lectures on minimal surfaces, vol. 1. Cambridge Univ. Press, Cambridge.
[163] Osserman, R. A survey of minimal surfaces, Dover publ. 2nd ed., New York.
[164] Patrikalakis, N.M.. Approximation conversion of rational splines. Computer Aided Geometric Design, 1989, 6: 155-165.
[165] J.F.Paul, C.Djurdje, L.Alan, J.Klinowski. Exact computation of the triply periodic D-minimal surface. Chemical Physics Letters, 1999, 314( 5-6), 543-551.
[166] Piegl, L., Tiller, W.. Curve and surface constructions using rational B-splines. Computer Aided Design, 1987, 19(9): 485-498.
[167] Piegl, L.. On NURBS: A survey. IEEE Computer Graphics and Application, 1991, 11(1): 55-71.
[168] Piegl, L., Tiller, W.. The NURBS book. Springer Verlag, 1995.
[169] Piegl, L., Tiller, W.. Computing the derivative of NURBS with respect to a knot. Comput Aided Geometric Design, 1998, 15: 925-934.
[170] Piegl, L., Tiller, W.. Surface approximation to scanned data. Visual Comput, 2000, 16: 386-395.
[171] U. Pinkall, K. Polthier. Computing discrete minimal surface and their conjugates. Exp. Math, 1993, 2(1):15-36.
[172]Pottmann,H..The geometry of Tchebycheffian spines.Computer Aided Geometric Design,1993,10:181-210.
[173]Pottmann,H.,Wagner,M.G..Helix splines as example of affine Tchebycheffian splines,Advance in Computational Mathematics,1994,2:123-142.
[174]H.Pottmann,S.Brell-Cokcan,and J.Wallner.Discrete surfaces for architectural design.In P.Chenin,T.Lyche,and L.L.Schumaker,editors,Curves and Surface Design:Avignon 2006,213-234.Nashboro Press,2007,ISBN 978-0-9728482-7-5.
[175]Pottmann,H.,Liu,Y.,Wallner,J.,Bobenko,A.,Wang,W.P..Geometry of multi-layer freeform structures for architecture,ACM Transactions on Graphics,2007,26(3),1-11.
[176]Pottmann,H.,Liu,Y..2007.Discrete surfaces in isotropic geometry.Mathematics of Surfaces Ⅻ,341-363.
[177]齐东旭.分形及计算机生成.北京:科学出版社,1994.
[178]覃廉,关履泰.基于NURBS的极小曲面造型.高等学校计算数学学报.2005,16(S1):175-181.
[179]Renka,R.J..Interpolatory tension splines with automatic selection of tension factors.SIAM J.Sci.Stat.Comp.1987,8:393-415.
[180]Rentrop,P..An algorithm for the computation of the exponential spline.Numer.Math.1980,35:81-93.
[181]Rado,On Plateau's problem,Ann.of Math.1930,31(2),457-469.
[182]Ruprecht D.,Nagel R.,Muller H..Spatial free-form deformation with scattered data interpolation methods.Computer and Graphics,1995,19:63-72.
[183]Sanchez-Reyes J..A simple technique for NURBS shape modification.IEEE Comput Graphics Appl 1997,17:52-61.
[184]Sambandan K,Kedem K,Wang K K.Generalized planar sweeping of polygons,Journal of Manufacturing Systems,1992,11(4):246-257.
[185]Schoenberg,I.J..Contributions to the problem of approximation of equidistant data by analytic functions.Quart.Appl.Math.,1946,4:45-99.
[186]Schoenberg,I.J..On spline function.In:Shisha,O.eds.Inequalities,Academic Press,New York,1967.
[187]Schoenberg,I.J..Spline interpolation and best quadrature formulae.Bull.Amer.Math.Soc.1964,70(1),143-148.
[188]Schumaker,L.L..Spline Functions:Basic Theory.John Wiley Sons,New York,1981:363-499.
[189]Sederberg T.W,Parry S.R.Free-form deformation of solid geometric models.Computer Graphics(SIGGRAPH '86),1986,20(4):151-160.
[190]Sederberg,T.W..Techniques for cubic algebraic surfaces,part one.IEEE Computer Graphics and its Applications,1990,10(4):14-25.
[191]Sederberg,T.W..Techniques for cubic algebraic surfaces,part two.IEEE Computer Graphics and its Applications,1990,10(5):12-21.
[192]Sederberg T.,Zheng J.,Swell D.,et al..Non-Uniform recursive subdivision surfaces.Cohen Med Computer Graphics,1998,32(4):387-394.
[193]Séquin,C.H...CAD Tools for Aesthetic Engineering.Computer Aided Design,2005,37(7),737-750.
[194]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.
[195]Singh K,Fiume E..Wires:A geometric deformation technique.Computer Graphics(SIGGRAPH '98),1998,405-414.
[196]Song WH,Yang XN,Free Form deformation with weighted T-splines.The visual computer,2005,21(3):139-151.
[197]Spath,H..Two-dimensional exponential splines.Computing,1971,7:364-369.
[198]Tai C L,Wang G J.Interpolation with slackness and continuity control and convexity-preservation using singular blending.Journal of Computational and Applied Mathematics,2004,173(4):337-361
[199]苏步青.计算几何的兴起.自然杂志,1978,7:409-412.
[200]苏步青.论Bézier曲线的仿射不变量,计算数学,1980.
[201]苏步青,刘鼎元.计算几何.上海科学技术出版社,1981.
[202]唐荣锡.CAD/CAM技术.北京航天航空大学出版社,北京,1994
[203]Tiller,W..Rational B-splines for curve and surface representation.IEEE Computer Graphics and its Application,1983,3(6):61-69.
[204]Tiller,W..Geometric modeling using non-uniform rational B-splines:mathematical techniques.SIGGRAH tutorial notes,ACM,New York,1986.
[205]Tiller,W..Knot-remove algorithms for blURBS curves and surfaces.Computer Aided Design,1992,24(8):445-453.
[206]Turk,G.,O'Brien,J..Shape Transformation Using Variational Implicit Functions.Computer Graphics(SIGGRAPH),1999:335-342.
[207]Ugail H,Bloor MIG,Wilson MJ.Techniques for interactive design using the PDE method.ACM Transactions on Graphics,1999,18(2):195-212.
[208]Ugail H.On the spine ofa PDE surface.In:Wilson MJ,Martin RR,editors.Mathematics of Surfaces X.Berlin:Springer;2003,p366-376.
[209]Veltkamp RC,Wesselink W.Modeling 3D curves of minimal energy.Eurographics 95,p.97-110.
[210]Versprille,K.J..Computer aided design application of rational B-spline approximation form.Ph.D.thesis,Syracuse University,Syracuse,N.Y.,1975.
[211]Wallner J,Pottmann H,Hofer M.Fair webs.The Visual Computer 2007;23:83-94.
[212]Walz,G..Identities for trigonometric B-splines with an application to curve design.BIT,1997,37(1):189-201.
[213]Wang Guojin.Generating NURBS curves by envelopes.Computing,1992,48(3-4):275-289.
[214]Wang,G.,Chen,Q..NUAT B-spline curves.Computer Aided Geometric Design,2004,21:193-205.
[215]Wang J.Z,Jiang T.Z.Nonrigid registration of brain MRI using NURBS.Pattern Recognition Letters 2007,28,214-223.
[216]王国瑾.Bézier曲线曲面的离散求交方法.浙江大学学报,计算几何专辑,1984:108-119.
[217]王国瑾,汪国昭,郑建民.计算机辅助几何设计,高等教育出版社,北京,2001.
[218]王文涛,汪国昭.带形状参数的均匀B样条.计算机辅助设计与图形学学报,2004,16(6):783-788.
[219]王文涛,汪国昭.带形状参数的三角多项式均匀B样条.计算机学报,2005,28(7):1192-1198.
[220]王文涛,汪国昭.带形状参数的双曲多项式均匀B样条.软件学报,2005,16(4):625-633.
[221]汪国昭,沈金福.有理Bézier曲线的离散和几何性质.浙江大学学报,1985,19(3):123-130.
[222]汪国昭.有理Bézier曲线的的包络性质全国计算几何与样条函数学术会议,计算几何论文集,1986,6.
[223]Wang XF,Cheng F,Barsky BA.Energy and B-spline interproximation.Computer Aided Design,1997;29:485-496.
[224]Wang,Y..Periodic surface modeling for computer aided nano design.Computer Aided Design,2007,39(3),179-189.
[225]Weir D J,Milroy M J,Bradley C.et al..Reverse engineering physical models employing wrap around B-Spline surfaces and quadrics.Journal of Engineering Manufacture,1996,210(B):147-157.
[226]Wyvill,G.C.,McPheeters,Wyvill,B..Data structure for soft objects.The Visual Computer,1986,2:227-234.
[227]Wyvill,B.,Wyvill,G..Field functions for implicit surface.The Visual Computer,1989,5:75-82.
[228]Xu Gang,Wang Guozhao.AHT Bézier surfaces and NUAHT B-spline curves.Journal of Computer Science and Technology,2007,22(4):597-607.
[229]Xu Gang,Wang Guozhao.Control mesh representation of a class of minimal surfaces.Journal of Zhejiang University SCIENCE A,2006,7(9):1544-1549.
[230]Xu Gang,Wang Guozhao.Harmonic-type Bézier surfaces over rectangular and triangular domain.Journal of Information and Computational Science,2006,3(2):325-332.
[231]徐岗,汪国昭.Doo-Sabin细分算法在动态模式下的推广.计算机辅助设计与图形学学报.2006,18(3),341-346.
[232]徐岗,汪国昭.三角域上的调和B-B曲面.计算机学报.2006,29(12):2180-2185.
[233]徐岗,汪国昭.一类极小曲面的控制网格表示.计算机辅助设计与图形学学报.2007,19(2),240-244.
[234]徐岗,汪国昭.带局部形状参数的三次均匀B样条曲线的扩展.计算机研究与发展.2007,44(6):1032-1037.
[235]徐岗,汪国昭.一类新的极小曲面及其应用.第三届全国几何设计与计算大会论文集,中国兰州,2007年7月,56-60.
[236]徐岗,汪国昭.PDE曲面的Bézier逼近.软件学报.2007,18(11):2974-2920.
[237]徐岗,汪国昭.参数多项式极小曲面.第一届全国计算机数学学术会议,中国南昌,2007年11月.
[238]Xu,Guoliang,Zhang,Qin.Minimal mean-curvature-variation sufaces and their applications in surface modeling.GMP 2006,Lecture Notes in Computer Science,2006,4077,357-370.
[239]Yang,Q.,Wang,G..Inflection points and singularities on C-curves.Computer Aided Geometric Design,2004,21,207-213.
[240]Yang X.N.Surface interpolation of meshes by geometric subdivision,Computer Aided Design,2005,37(5):497-508
[241]Yang X.N.Normal based subdivision scheme for curve design,Computer Aided Geometric Design,2006,23(3):243-260
[242]Yates,R.C.."Envelopes." A Handbook on Curves and Their Properties.Ann Arbor,MI:J.W.Edwards,1952:75-80.
[243]Yoon,S.H.,Kim,M.S..Sweep-based Free form deformation.Computer Graph Forum(Eurographics 2006),2006,25(3),487-496
[244]Zhang,J..C-curves:An extension of cubic curves.Computer Aided Geometric Design,1996,13(3):199-217.
[245]Zhang,J..Two different forms of C-B-splines.Computer Aided Geometric Design,1997,14(1):31-41.
[246]Zhang,J..C-Bézier curves and surfaces.Graphical Models and Image Processing,1999,61(1):2-15.
[247]Zhang J,Krause F,Zhang H.Unifying C-curves and H-curves by extending the calculation to complex numbers.Computer Aided Geometric Design,2005,22(9):865-883.
[248]Zhang J,Krause F.Extend cubic uniform B-splines by unified trigonometric and hyperbolic basis.Graphic Models,2005,67(2):100-119.
[249]Zhang JJ,You LH.Surface representation using second,fourth and mixed order partial differential equations.International conference on Shape Modeling and Applications,Genova,Italy,5-7 May,2001.
[250]Zheng J.M.,Chan K.W.,GibsonI..A new approach for direct manipulation of free-form curve.Comput Graphics Forum,1998,17:327-34.
[251]Zhou,Kambhamettu C..Extending superquadrics with exponent modeling and reconstruction.Graphical Models,2001,63:1-20.
[252]曾芳玲,陈效群,冯玉瑜.二次曲线的多项式逼近.计算机辅助设计与图形学学报,2003,15(5):547-551.
[253]张宏鑫.复杂形体建模与绘制的离散方法研究.浙江大学博士学位论文,2002年4月.
[254]朱心雄.自由曲线曲面造型技术.科学出版社,2000.