极小曲面造型中的相关问题研究
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摘要
现代的产品设计和制造过程11]的很多问题都可以转化为曲线曲面造型问题,例如汽车外形设计、飞机机身设计、服装的设计和加工以及建筑物的外形设计等.而曲线曲面设计也是当前计算机辅助几何设计的核心的研究内容.特别是在实际应用中,满足某种性质和功能的曲线曲面的设计更是尤其重要的.本文重点研究了一类在微分几何和计算几何中尤为重要的曲面-极小曲面,同时也给出了一种简单的曲线细分格式,并考虑将极小曲面与细分格式这两种曲面造型方法有效地结合起来.具体工作主要包括以下几个方面：
1.多项式是计算几何辅助设计中曲线曲面的一种重要的表示形式,但是在CAGD领域中我们可以见到的参数多项式形式的极小曲面是非常少的,因此寻求参数形式的多项式极小曲面并将其引入到CAGD领域具有重要的意义.利用微分几何中的一个经典结论,我们给出了等温参数多项式极小曲面的一般形式,得到了其系数需要满足的充要条件,分析了它们具有的对称性质并给出了其共轭极小曲面的一般形式,同时也具体构造了6、7、8、9次的参数多项式极小曲面.
2.Bezier曲线曲面是计算机图形学和计算机辅助几何设计的一种基本的表示工具,而quasi-Bezier表示是Bezier表示的一种推广,它所张成的函数空问不仅包含了多项式空间也包括了三角函数空间和双曲函数空间.利用Dirichlet能量来代替面积泛函,我们考虑了在quasi-Bezier表示下的Plateau问题,即Plateau-quasi-Bezier司题,这时的边界曲线不仅可以是多项式曲线也可以圆弧曲线和悬链线.同时,我们也讨论了调和与双调和的quasi-Bezier曲面.
3.Plateau司题一直是极小曲面领域的一个重要问题.而在Plateau司题中,要求给定的曲线是曲面的整个边界,但在实际应用中我们获取的边界信息可能只是部分边界曲线.基于这一点,我们考虑了quasi-Plateau问题：在所有以给定的曲线为边界且定义在矩形域的参数曲面中,找一个而积最小的曲面,这里的‘给定的曲线’可以是曲面的整个边界亦可是部分边界.利用Dirichlet泛函来代替面积能量，并利用Ritz-Galerkin方法,这个问题转化为了一个简单的稀疏线性方程组的求解问题,并最终根据不同的边界条件归结为了四个算法.具体实例表明了我们的算法是简单有效的.
4.在许多的曲面设计问题中，比如汽车车身、飞机外壳以及机器零部件设计,都要求构造的曲面在给定的边界处满足一定的连续性条件.鉴于此,我们来考虑C1和C2的quasi-Plateau问题,即要求极小曲而不仅以给定的曲线为边界曲线,同时在这些曲线处也要满足C1或C2的连续性.同样地利用Dirichlet能量和Ritz-Galerkin方法，我们得到了对应于不同边界条件的六种算法来求得该问题在不同层的逼近解.具体实例表明了这些算法是简单有效的.
5.细分格式也是CAGD领域中曲线曲面的一种重要的表示方法.我们提出了一种形式简单的二进制六点的曲线细分格式,这种细分格式不仅形式简单,而且同时具有很多良好的几何性质,如多项式再生性、保凸性以及高阶连续性.
6.基于细分格式与极小曲面在曲面造型中的重要性,我们考虑将两者结合起来,利用Dirichlet能量,通过极小曲面的思想来优化细分曲面的初始网格使得构造的细分极限曲面是满足给定的初始边界控制顶点的拟极小曲面.得到了一般细分格式下的内部控制顶点需要满足的条件,同时通过Loop细分格式作用于几个具体的实例说明了我们的方法.
Many problems in the modern product design and manufacturing process can be translated into the problems of curve and surface modeling, such as auto appearance de-sign、design of airplane body、design and process of cloth、and the appearance design of buildings, and so on. The design of curve and surface is also the core content of Computer Aided Geometric Design. Especially in the real applications, the design of curve and surface meeting certain properties and functions is of particular importance. In this paper, our re-search focuses on the minimal surface, a kind of surface important in differential geometry and computational geometry, we also present a simple curve subdivision scheme, and con-sider to combine the research on minimal surface and subdivision scheme. The main work includes the following several aspects:
1. Polynomial is an important representation for curve and surface in computational ge-ometry design. However, we can only find very few parametric polynomial minimal surface in CAGD, therefore searching for the parametric polynomial minimal surface and introducing them into CAGD are of great significance. Using a classic result in differential geometry, we present the general formula of parametric polynomial minimal surface, obtain the sufficient and necessary conditions for the corresponding coefficients, analyze their symmetric prop-erties, give the conjugate minimal surfaces, and illustrate concrete examples for polynomial minimal surface of degree6、7、8、9.
2. Bezier curve and Bezier surface are basic representation tools in Computer Graph-ics and Computer Aided Geometric Design, while quasi-Bezier is a generalization of Bezier and the space it spans not only includes polynomial space also trigonometric space and hyperbolic space. Using Dirichlet energy, we consider the Plateau problem under the rep-resentation of quasi-Bezier, i.e., Plateau-quasi-Bezier problem. In this case, the boundary curve can not only be polynomial curve also circular curve and catenary. Moreover, we also discuss the harmonic and biharmonic quasi-Bezier surface.
3. Plateau problem is always an important problem for minimal surface. In this problem, the given curve is required to be the whole bound of the solution surface, however in applications the boundary information we get maybe only part boundary curve. Based on this, we consider the quasi-Plateau problem:find the surface of minimal area among all the surfaces bounded by the given curve and defined on the rectangular domain, the 'given curve'here can be the whole boundary or part boundary. Replacing the area functional with Dirichlet energy, and using the Ritz-Galerkin method, the quasi-Plateau problem changes into solving a simple sparse linear equations, and finally is summarized as four algorithms according to different boundary conditions. Examples demonstrate that our algorithms are simple and effective.
4. Many surface design problems, such as the design of car body、airplane hull and the machine parts, require the surface to satisfy certain continuity conditions at the given bound. Therefore, we consider the C1and C2quasi-Plateau problem, i.e., the surface of minimal area not only is bounded by the given curve, but satisfies the C1or C2continuity. Still using the Dirichlet energy and Ritz-Galerkin method, we obtain six algorithms corresponding to different boundary conditions to get the approximation surface in different levels. Examples demonstrate that these algorithms are simple and effective.
5. Subdivision scheme is also an important design method for curve and surface in CAGD. We present a simple binary six point curve subdivision method, this scheme not only has simple form, but also has many good geometric properties at the same time, such as polynomial reproduction%convexity preserving and high continuity.
6. Based on the importance of subdivision scheme and minimal surface, we consider to combine the both aspects to construct the quasi-minimal subdivision surface satisfying the given initial boundary control points. Using Dirichlet energy, we obtain the results on inner control points for general subdivision scheme, and illustrate our results by Loop subdivision scheme with several examples.
1.多项式是计算几何辅助设计中曲线曲面的一种重要的表示形式,但是在CAGD领域中我们可以见到的参数多项式形式的极小曲面是非常少的,因此寻求参数形式的多项式极小曲面并将其引入到CAGD领域具有重要的意义.利用微分几何中的一个经典结论,我们给出了等温参数多项式极小曲面的一般形式,得到了其系数需要满足的充要条件,分析了它们具有的对称性质并给出了其共轭极小曲面的一般形式,同时也具体构造了6、7、8、9次的参数多项式极小曲面.
2.Bezier曲线曲面是计算机图形学和计算机辅助几何设计的一种基本的表示工具,而quasi-Bezier表示是Bezier表示的一种推广,它所张成的函数空问不仅包含了多项式空间也包括了三角函数空间和双曲函数空间.利用Dirichlet能量来代替面积泛函,我们考虑了在quasi-Bezier表示下的Plateau问题,即Plateau-quasi-Bezier司题,这时的边界曲线不仅可以是多项式曲线也可以圆弧曲线和悬链线.同时,我们也讨论了调和与双调和的quasi-Bezier曲面.
3.Plateau司题一直是极小曲面领域的一个重要问题.而在Plateau司题中,要求给定的曲线是曲面的整个边界,但在实际应用中我们获取的边界信息可能只是部分边界曲线.基于这一点,我们考虑了quasi-Plateau问题：在所有以给定的曲线为边界且定义在矩形域的参数曲面中,找一个而积最小的曲面,这里的‘给定的曲线’可以是曲面的整个边界亦可是部分边界.利用Dirichlet泛函来代替面积能量，并利用Ritz-Galerkin方法,这个问题转化为了一个简单的稀疏线性方程组的求解问题,并最终根据不同的边界条件归结为了四个算法.具体实例表明了我们的算法是简单有效的.
4.在许多的曲面设计问题中，比如汽车车身、飞机外壳以及机器零部件设计,都要求构造的曲面在给定的边界处满足一定的连续性条件.鉴于此,我们来考虑C1和C2的quasi-Plateau问题,即要求极小曲而不仅以给定的曲线为边界曲线,同时在这些曲线处也要满足C1或C2的连续性.同样地利用Dirichlet能量和Ritz-Galerkin方法，我们得到了对应于不同边界条件的六种算法来求得该问题在不同层的逼近解.具体实例表明了这些算法是简单有效的.
5.细分格式也是CAGD领域中曲线曲面的一种重要的表示方法.我们提出了一种形式简单的二进制六点的曲线细分格式,这种细分格式不仅形式简单,而且同时具有很多良好的几何性质,如多项式再生性、保凸性以及高阶连续性.
6.基于细分格式与极小曲面在曲面造型中的重要性,我们考虑将两者结合起来,利用Dirichlet能量,通过极小曲面的思想来优化细分曲面的初始网格使得构造的细分极限曲面是满足给定的初始边界控制顶点的拟极小曲面.得到了一般细分格式下的内部控制顶点需要满足的条件,同时通过Loop细分格式作用于几个具体的实例说明了我们的方法.
Many problems in the modern product design and manufacturing process can be translated into the problems of curve and surface modeling, such as auto appearance de-sign、design of airplane body、design and process of cloth、and the appearance design of buildings, and so on. The design of curve and surface is also the core content of Computer Aided Geometric Design. Especially in the real applications, the design of curve and surface meeting certain properties and functions is of particular importance. In this paper, our re-search focuses on the minimal surface, a kind of surface important in differential geometry and computational geometry, we also present a simple curve subdivision scheme, and con-sider to combine the research on minimal surface and subdivision scheme. The main work includes the following several aspects:
1. Polynomial is an important representation for curve and surface in computational ge-ometry design. However, we can only find very few parametric polynomial minimal surface in CAGD, therefore searching for the parametric polynomial minimal surface and introducing them into CAGD are of great significance. Using a classic result in differential geometry, we present the general formula of parametric polynomial minimal surface, obtain the sufficient and necessary conditions for the corresponding coefficients, analyze their symmetric prop-erties, give the conjugate minimal surfaces, and illustrate concrete examples for polynomial minimal surface of degree6、7、8、9.
2. Bezier curve and Bezier surface are basic representation tools in Computer Graph-ics and Computer Aided Geometric Design, while quasi-Bezier is a generalization of Bezier and the space it spans not only includes polynomial space also trigonometric space and hyperbolic space. Using Dirichlet energy, we consider the Plateau problem under the rep-resentation of quasi-Bezier, i.e., Plateau-quasi-Bezier problem. In this case, the boundary curve can not only be polynomial curve also circular curve and catenary. Moreover, we also discuss the harmonic and biharmonic quasi-Bezier surface.
3. Plateau problem is always an important problem for minimal surface. In this problem, the given curve is required to be the whole bound of the solution surface, however in applications the boundary information we get maybe only part boundary curve. Based on this, we consider the quasi-Plateau problem:find the surface of minimal area among all the surfaces bounded by the given curve and defined on the rectangular domain, the 'given curve'here can be the whole boundary or part boundary. Replacing the area functional with Dirichlet energy, and using the Ritz-Galerkin method, the quasi-Plateau problem changes into solving a simple sparse linear equations, and finally is summarized as four algorithms according to different boundary conditions. Examples demonstrate that our algorithms are simple and effective.
4. Many surface design problems, such as the design of car body、airplane hull and the machine parts, require the surface to satisfy certain continuity conditions at the given bound. Therefore, we consider the C1and C2quasi-Plateau problem, i.e., the surface of minimal area not only is bounded by the given curve, but satisfies the C1or C2continuity. Still using the Dirichlet energy and Ritz-Galerkin method, we obtain six algorithms corresponding to different boundary conditions to get the approximation surface in different levels. Examples demonstrate that these algorithms are simple and effective.
5. Subdivision scheme is also an important design method for curve and surface in CAGD. We present a simple binary six point curve subdivision method, this scheme not only has simple form, but also has many good geometric properties at the same time, such as polynomial reproduction%convexity preserving and high continuity.
6. Based on the importance of subdivision scheme and minimal surface, we consider to combine the both aspects to construct the quasi-minimal subdivision surface satisfying the given initial boundary control points. Using Dirichlet energy, we obtain the results on inner control points for general subdivision scheme, and illustrate our results by Loop subdivision scheme with several examples.
引文
[1]BEZIER P. Numerical Control:Mathematics and Applications[M]. New York:John Wiley and Sons,1972.
[2]BEZIER P. Mathematical and practical possibilities of UNISURF[M]. In:Barnhill, R and Riesenfeld, R.F., eds., Computer Aided Geometric Design, New York:Academic Press,1974,127-152.
[3]BEZIER P. The mathematical basis of the UNISURF CAD systems[M]. Butterworths, England,1986.
[4]PIEGL L. Modifying the shape of rational B-splines, part 1:curves [J]. Computer-Aided Design,1989,21:509-518.
[5]PIEGL L. Modifying the shape of rational B-splines, part 2:surfaces[J]. Computer-Aided Design,1989,21:538-546.
[6]PIEGL L. On NURBS:a survey [J]. IEEE Computer Graphics and Application.1991, 10:55-71.
[7]PIEGL L, TILLER W. Curve and surface constructions using rational B-splines [J]. Computer-Aided Design,1987,19:485-498.
[8]FARIN G. NURBS for Curves and Surface Design[M]. SIAM, Philadelphia, PA,1991.
[9]BAJAJ C L, CHEN J, XU G. Modelling with cubic A-patches[J]. ACM Transactions on Graphics,1995,14(2):103-133.
[10]LI J G, HOSCHEK J, HARTMANN E. Gnn-1-function splines for interpolation and approximation of curves, surfaces and solids[J]. Computer Aided Geometric Design, 1990,7(1/4):209-220.
[11]CHEN C, CHEN F, FENG Y Y. Blending quadric surfaces with piecewise algebraic surfaces[J]. Graphical Models,2001,63(4):212-227.
[12]ZHU C G, WANG R H, SHI X, et al. Functional splines with different degrees of smoothness and their applications [J]. Computer-Aided Design,2008,40(5):616-624.
[13]PALUSZNY M, PATTERSON R R. A family of tangent continuous cubic algebraic splines[J]. ACM Transactions on Graphics,1993,12(3):209-232.
[14]HARTMANN E. Blending of implicit surfaces with functional splines[J]. Computer-Aided Design,1990,22(10):500-506.
[15]WU T R, ZHOU Y S. On blending of severalquadratic algebraic surfaces[J]. Computer Aided Geometric Design,2000,17(8):759-766.
[16]张三元.隐式曲线、曲面的几何不变量及几何连续性[J].计算机学报,1999,22(7)：774-776.
[17]FENG Y, CHEN F, DENG J, et al. Construction piecewise algebraic blending sur-faces[J]. Geometric Computation, Lecture Notes Ser. Comput.,11, River Edge, NJ: World Sci. Publishing,2004:34-64.
[18]HARTMANN E. Implicit Gn-blending of vertices[J]. Computer-Aided Design,2001, 18(3):267-285.
[19]ZHANG S, BAO H, WEI B. Cubic algebraic curves based on geometric constraints[J]. Computer Aided Geometric Design,2001,4(18):299-307.
[20]FARIN G. Curves and surfaces for computer aided geometric design:A practical guide[M]. Fifth edition, Morgan Kaufmann, San Francisco,2002.
[21]PIEGL L, TILLER W. The NURBS book[M]. Section edition, Springer-Verlag,1997.
[22]ZHANG J W. C-curves:An extension of cubic curves[J]. Computer Aided Geometric-Design,1996,13(3):199-217.
[23]ZHANG J W, KRAUSE F-L. Extending cubic uniform B-splinos by unified trigono-metric and hyperbolic basis[J]. Graphical Models,2005,67(2):100-119.
[24]CHEN Q, WANG G. A class of Bezier-like curves[J]. Computer Aided Geometric De-sign,2003,20:29-39.
[25]LU Y G, WANG G Z, YANG X N. Uniform hyperbolic polynomial B-spline curves[J]. Computer Aided Geometric Design,2002,19(6):379-393.
[26]WANG G Z, FANG M E. Unified and extened form of three types of splines[J]. Journal of Computational and Applied Mathematics,2008,216(2):418-508.
[27]WANG G Z, LI Y J. Optimal properties of the uniform algebraic trigonometric B-splines[J]. Computer Aided Geometric Design,2006,23(2):226-238.
[28]WANG G Z, YANG Q M. Planar cubic hyperbolic polynomial curve and its shape classification[J]. Progress in Natural Science,2004,14(1):41-46.
[29]MAINAR E, PENA. A basis of C-Bezier splines with optimal properties [J]. Computer Aided Geometric Design,2002,19(4):291-295.
[30]王仁宏,施锡泉,罗钟铉,苏志勋.多元样条函数及其应用[M].北京：科学出版社,1994.
[31]王仁宏,李崇君,朱春钢.计算几何教程[M].北京：科学出版社,2008.
[32]王仁宏.数值逼近[M].北京：高等教育出版社,1999.
[33]李宝军.指数多项式曲线细分重构与插值细分曲面快速计算[D].大连：大连理工大学,2008.
[34]郑彩玲.偏微分方程过渡面构造与曲面控制问题的研究[D].大连：大连理工大学,2005.
[35]QIN H, TERZOPOULOS D. D-NURBS:a physics-based framework for geometric de-sign[J]. IEEE Transactions on Visualization and Computer Graphics,1996,2(1):85-96.
[36]王志国.曲线曲面形状修改和变形关键技术研究[D].南京：南京航空航天大学,2006.
[37]周晓杰.曲线曲面造型中的变形方法研究与应用[D].大连：大连理工大学,2008.
[38]PLATEAU.实验和理论流体静力学[M].1873.
[39]MONGE.画法几何学[M].1794.
[40]DO CARMO M. Differential geometry of curves and surfaces[M]. Prentice-Hall, Engle-wood Cliffs,1976.
[41]DOUGLAS J. Solution of the problem of Plateau[J]. Transactions of the AMS,1931, 33:263-321.
[42]RATO T. On Plateau's problem[J]. Annals of Mathematics,1930,31(2):457-469.
[43]王子航.利用四点插值细分法构造光顺极小曲面[D].大连：大连理工大学,2006.
[44]SEQUIN C H. CAD Tools for Aesthetic Engineering[J]. Computer-Aided Design,2005, 7:737-750.
[45]SULLIVAN J. The aesthetic value of optimal geometry[M]. In:Emmer, M. (ed.) The Visual Mind II,547-563. MIT Press, Cambridge,2005.
[46]WANG Y. Periodic surface modeling for computer aided nano design[J]. Computer-Aided Design,2007,3:179-189.
[47]JIN W, WANG G. Geometric modeling using minimal surfaces[J]. Chinese Journa of Computers,1999,22(12):1276-1280.
[48]金文标,汪国昭.一类极小曲面的几何设计[J].计算机学报,1999,23(12)：1277-1279.
[49]金文标.若干特殊曲面的造型和应用[D].杭州：浙江大学,1999.
[50]满家巨,汪国昭.正螺面与悬链面的表示和构造[J].计算机辅助设计与图形学学报,2005,17(3)：431-436.
[51]XU G, WANG G Z. AHT Bezier surfaces and NUAHT B-spline curves[J]. Journal of Computer Science and Technology,2007,22(4):597-607.
[52]JIN W B. Modeling and applications of some special surfaces[D]. Hangzhou:Zhejiang University,1999 (in Chinese with English abstract).
[53]MAN J J, WANG G Z. Polynomial minimal surface in isothermal parameter[J]. Journal of Computers,2002,25(2):197-201.
[54]XU G, WANG G Z. Quintic parametric polynomial minimal surfaces and their prop-erties[J]. Differential Geometry and its Applications,2010,28:697-704.
[55]XU G, WANG G Z. Parametric polynomial minimal surfaces of degree six with isother-mal parameter[C]//Proc. of Geometric Modeling and Processing (GMP 2008),2008, LNCS 4975,329-343.
[56]徐岗,汪国昭.计算机辅助几何设计中极小曲面造型的研究进展[J].系统科学与数学,2008,28(8)：984-992.
[57]MAN J J, WANG G Z. Approximating to nonparameterzied minimal surface with B-spline surface[J]. Journal of Software,2003,14(4):824-829.
[58]MAN J J, WANG G Z. Minimal surface modeling using finite element method[J]. Chinese Journal of Computers,2003,26(4):507-510.
[59]DZIUK G. An algorithm for evolutionary surfaces[J].Numerische Mathematik,1991, 58:603-611.
[60]CONCUS P. Numerical solution of the minimal surface equation[J]. Mathematics of Computation,1967,21:340-350.
[61]PINKALL U, POLTHIER K. Computing discrete minimal surface and their conju-gates[J]. Exp. Math,1993,2(1):15-36.
[62]MONTERDE J. Bezier surfaces of minimal area:The Dirichlet approach[J]. Computer Aided Geometric Design,2004,21:117-136.
[63]MONTERDE J. The Plateau-Bezier Problem In:Wilson, M. J., Martin, R.R. (eds.) Mathematics of Surfaces[M]. LNCS, vol.2768,262-273. Springer, Heidelberg (2003).
[64]COSIN C, MONTERDE J. Bezier Surfaces of Minimal Area[M]. In:Sloot, P.M.A., Tan, C.J.K., Dongarra, J., Hoekstra, A.G. (eds.) ICCS 2002. LNCS, vol.2330,72-81. Springer, Heidelberg,2002.
[65]FARIN G, HANSFORD D. Discrete Coons patches[J]. Computer Aided Geometric Design 1999,16:691-700.
[66]ARNAL A, LLUCH A, MONTERDE J. Triangular Bezier surfaces of minimal area[C]//Proceedings of the Int. Workshop on Computer Graphics and Geometric Mod-elling, CG GM'2003, Montreal. Leture Notes in Comput. Sci,2003, Springer-Verlag, Berlin/New York,366-375.
[67]刘计善.一类极小曲面造型设计的新方法[J].复旦学报(自然科学版),2007,46(2)：192-197.
[68]吴花精灵,王国瑾.推广Dirichlet方法用于B样条极小曲面设计[J].软件学报,2011,22(12)：3015-3022.
[69]吴花精灵,王国瑾.插值边界曲线的NURBS近似极小曲面设计[J].软件学报,2010,21：67-74.
[70]覃廉,关履泰.基于NURBS的极小曲面造型[J].高等学校计算数学学报.2005,16(S1)：175-181.
[71]CHEN X. Design of rational Bezier minimal surface:an extension of Dirichlet ap-proach [D].杭州：浙江大学,2006.
[72]COX J, CHARLESWORTH W, ANDERSON D. Domain composition methods for as-sociating geometric modelling with finite element modeling[C]//Proceeding of the ACM SIGGRAPH, Symposium on Solid Modeling Foundation and CAD/CAM application, Austin, TX,1991,5-7.
[73]CELNIKE G, GOSSARD D. Deformable curve and surface finite elements for free-form shape design[J]. Proc. SIGGRAPH,1991,257-266.
[74]季洁,满家巨.任意边界下的极小曲面造型问题[J].计算机与现代化.2007,4：7-9.
[75]LI Y, WANG G. Two kinds of B-basis of the algebraic hyperbolic space[J]. Journal of Zhejiang University,2005,6A(7):750-759.
[76]RIESENFELD R F. On Chaikin's algorithm[J]. Computer Graphics and Image Pro-cessing,1975,4:153-174.
[77]FORREST A R. Curves and surfaces for computer sided design[D]. Cambridge:Uni-versity of Cambridge,1968.
[78]CATMULL E, CLARK J. Recursively generated B-spline surfaces on arbitrary topo-logical meshes[J]. Computer-Aided Design,1978,10(6):350-355.
[79]RIESENFELD R F. An algorithm for high speed curve generation [J]. Computer Graph-ics and Image Processing,1974,4:346-349.
[80]DEROSE T. Necessary and sufficient conditions for tangent plane continuity of Bozior surfaces[J]. CAGD, Special issue curves and surfaces in CAGD'89,1990,7(4):165-179.
[81]DOO D, SABIN M. Behavior of recursive division surfaces near extraordinary points[J]. Computer-Aided Design,1978,10(6):356-360.
[82]DYN NIRA, LEVIN DAVID, GREGORY J A. A butterfly subdivision scheme for surface interpolatory with tension control [J]. ACM Transactions on Graphics,1990, 9(2):160-169.
[83]KOBBELT LEIF.(?)-subdivision[C]//In Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH,2000.
[84]LOOP C. Smooth subdivision surfaces based on triangles[D]. Utah:University of Utah, Department of Mathematics,1987.
[85]NITSCHE J C C. Lectures on minimal surfaces[M]. Cambridge University Press, Cam-bridge,1989.
[86]STRUWE M. Plateau's problem and the calculus of variations[M]. Princeton University Press,1988.
[87]OSSERMAN R. A survey of minimal surfaces[M].Dover publ.2nd ed.,New York,1986.
[88]MONTERDE J, UGAIL H. A general 4th-order PDE method to generate Bezier sur-faces from the boundary[J]. Computer Aided Geometric Design,2006,23:208-225.
[89]CENTELLA P, MONTERDE J, MORENO E, OSET R. Two C1-methods to generate Bezier surfaces from the boundary[J]. Computer Aided Geometric Design,2009,20: 152-173.
[90]PAN Q, XU G L. Construction of minimal subdivision surface with a given boundary[J]. Computer-Aided Design,2011,43:374-380.
[91]XU G, WANG G Z. Parametric polynomial minimal surfaces of arbitrary degree[R]. INRIA Research Report 00507790,2010.
[92]FRANK MORGAN. Minimal surfaces, crystals, and norms on Rn[C]//Proc.7th Annual Symp.on Computational Geom., June,1991, N. Conway, NH.
[93]TRASDAHL (?), R(?)NQUIST E M.High order numerical approximation of minimal surfaces[J]. Journal of Computational Physics,2011,230:4795-4810.
[94]POLTHIER K. Polyhedral surfaces of constant mean curvature[M]. Habilitationsschrift, Technische Universitat Berlin,2002.
[95]MONTERDE J, UGAIL H. On harmonic and biharmonic Bezier surfaces[J]. Computer Aided Geometric Design,2004,7:697-715.
[96]XU G, WANG G Z. Harmonic-type Bezicr surface over rectangular and triangualr doinain[J]. Journal of Information and Computational Science,2006,3:325-332.
[97]WU H J L, WANG G J. Extending and correcting some research results on minimal and harmonic surfaces[J]. Computer Aided Geometric Design,2012,29:41-50.
[98]MALL AT S. A theory for nmltiresolution signal decomposition:the wavelet represen-tation[J]. IEEE Trans Pat Anal Intel,1989,11(7):674-693.
[99]DE BOOR C. A practical guide to splines[M].Springer-Verlag, New York,2001.
[100]BARRERA D, FORTES M A, GONZALEZ P, PASADAS M.Minimal energy Cr sur-faces on uniform Powell-Sabin type meshes for noisy data[J]. Journal of Computational and Applied Mathematics,2008,218(2):592-602.
[101]FORTES M A, GONZALEZ P, MONCAYO M, PASADAS M. Multiresolution anal-ysis for minimal energy Cr surfaces on Powell-Sabin type meshes[C]//Proceeding MMCS'08 Proceedings of the 7th international conference on Mathematical Methods for Curves and Surfaces, Pages 209-223, Springer-Verlag Berlin, Heidelberg (?)2010 ISBN:3-642-11619-1978-3-642-11619-3.
[102]LV X-G, HUANG T-Z, LE J. A note on computing the inverse and the determinant of a pentadiagonal Toeplitz matrix[J]. Applied Mathematics and Computation,2008, 206:327-331.
[103]KILIC E, El-Mikkawy M. A computational algorithm for special nth order pentadiag-onal Toeplitz determinants [J]. Applied Mathematics and Computation,2008,199(2): 820-822.
[104]MOLINARI L G. Determinants of Block Tridiagonal Mtrices[J]. Linear Algebra and its Applications,2008,429:2221-2226.
[105]MEURANT G. A review on the inverse of symmetric tridiagonal and block tridiagonal matrices[J]. SIAM Journal on Matrix Analysis and Applications,1992,13(3):707-728.
[106]SWEET R A. A recursive relation for the determinant of a pentadiagonal matrix[J]. Communications of the ACM,1969,12:330-332.
[107]SOGABE T. A fast numerical algorithm for the determinant of a pentadiagonal ma-trix[J]. Applied Mathematics and Computation,2008,196(2):835-841.
[108]SOLARY M S. Finding determinant for some block band matrices by reblocking[J]. Differential Equations and Control Processes,2012,3:37-48.
[109]COOS S A. Surfaces for computer aided design, Technical Report[R]. MIT Press, Available as AD 663 504 from the National Technical Information Service, Springfield, VA 22161.1964.
[110]HAO Y X, LI C J, WANG R H. An approximation method based on MRA for the quasi-Plateau problem[J]. BIT Numerical Mathematics,2013,53:411-442.
[111]DYN N, LEVIN D. Subdivision schemes in geometric modelling[J]. Acta Numerica, 2002,11:73-144.
[112]DYN N, LEVIN D, GREGORY J A. A 4-point interpolatory subdivision scheme for curve design[J]. Computer Aided Geometric Design,1987,4:257-268.
[113]HASSAN M F, IVRISSIMITZIS I P, DODGDSON N A, SABIN M A. An interpolat-ing 4-point C2 ternary stationary subdivision scheme[J]. Computer Aided Geometric Design,2002,19:1-8.
[114]SHAHID S SIDDIQI, NADEEM AHMAD. A new 3-point C2 approximating subdivi-sion scheme[J]. Applied Mathematics Letters,2007,20:707-711.
[115]SHAHID S SIDDIQI, KASHIF REHAN. A stationary binary C1 curve subdivision scheme[J]. European Journal of Scientific Research,2009,4:553-561.
[116]BECCARI C, CASCIOLA G, ROMANI L. Shape controlled interpolator ternary subdivision[J]. Applied Mathematics and Computation,2009,215:916-927.
[117]DYN N, ISKE A, QUAK E, FLOATER M S. Tutorials on multiresolution in geometric modelling[M]. Summer school lecture notes series:, Mathematics and Visualization, Springer, ISBN:3-540-43639-1,2002.
[118]DYN N, HORMANN K, SABIN M A, SHEN Z. Polynomial reproduction by symmet-ric subdivision schemes[J]. Journal of Approximation Theory,2008,155:28-42.
[119]DYN N, KUIJT F, LEVIN D, VAN DAMME R. Convexity preservation of the four-point interpolatory subdivision scheme[J]. Computer Aided Geometric Design,1999, 16:789-792.
[120]SHAHID S SIDDIQI, NADEEM AHMAD. A C6 approximating subdivision scheme[J]. Applied Mathematics Letters,2008,21:722-728.
[121]KUIJT F, VAN DAMME R. Convexity preserving interpolatory subdivision schemes[J]. Constructive Approximation,1998,14(4):609-630.
[122]CAI ZHIJIE. Convexity preservation of the interpolating four-point C2 ternary sta-tionary subdivision scheme[J]. Computer Aided Geometric Design,2009,26:560-565.
[123]DYN N, FLOATER M S, HORMANN. A C2 four-point subdivision scheme with fourth order accuracy and its extensions[M]. In:M.Daehlen, K.M(?)rken, and L.L.schumaker, editors, Mathematical Methods for Curves and Surfaces. Troms(?) 2004. Nashboro Press, Brentwood, TN, (2005) 145-156.
[124]LAI M J. Fortran subroutines for B-nets of Box splines on three and four directional meshes[J]. Numerical Algorithms,1992,2:33-38.
[2]BEZIER P. Mathematical and practical possibilities of UNISURF[M]. In:Barnhill, R and Riesenfeld, R.F., eds., Computer Aided Geometric Design, New York:Academic Press,1974,127-152.
[3]BEZIER P. The mathematical basis of the UNISURF CAD systems[M]. Butterworths, England,1986.
[4]PIEGL L. Modifying the shape of rational B-splines, part 1:curves [J]. Computer-Aided Design,1989,21:509-518.
[5]PIEGL L. Modifying the shape of rational B-splines, part 2:surfaces[J]. Computer-Aided Design,1989,21:538-546.
[6]PIEGL L. On NURBS:a survey [J]. IEEE Computer Graphics and Application.1991, 10:55-71.
[7]PIEGL L, TILLER W. Curve and surface constructions using rational B-splines [J]. Computer-Aided Design,1987,19:485-498.
[8]FARIN G. NURBS for Curves and Surface Design[M]. SIAM, Philadelphia, PA,1991.
[9]BAJAJ C L, CHEN J, XU G. Modelling with cubic A-patches[J]. ACM Transactions on Graphics,1995,14(2):103-133.
[10]LI J G, HOSCHEK J, HARTMANN E. Gnn-1-function splines for interpolation and approximation of curves, surfaces and solids[J]. Computer Aided Geometric Design, 1990,7(1/4):209-220.
[11]CHEN C, CHEN F, FENG Y Y. Blending quadric surfaces with piecewise algebraic surfaces[J]. Graphical Models,2001,63(4):212-227.
[12]ZHU C G, WANG R H, SHI X, et al. Functional splines with different degrees of smoothness and their applications [J]. Computer-Aided Design,2008,40(5):616-624.
[13]PALUSZNY M, PATTERSON R R. A family of tangent continuous cubic algebraic splines[J]. ACM Transactions on Graphics,1993,12(3):209-232.
[14]HARTMANN E. Blending of implicit surfaces with functional splines[J]. Computer-Aided Design,1990,22(10):500-506.
[15]WU T R, ZHOU Y S. On blending of severalquadratic algebraic surfaces[J]. Computer Aided Geometric Design,2000,17(8):759-766.
[16]张三元.隐式曲线、曲面的几何不变量及几何连续性[J].计算机学报,1999,22(7)：774-776.
[17]FENG Y, CHEN F, DENG J, et al. Construction piecewise algebraic blending sur-faces[J]. Geometric Computation, Lecture Notes Ser. Comput.,11, River Edge, NJ: World Sci. Publishing,2004:34-64.
[18]HARTMANN E. Implicit Gn-blending of vertices[J]. Computer-Aided Design,2001, 18(3):267-285.
[19]ZHANG S, BAO H, WEI B. Cubic algebraic curves based on geometric constraints[J]. Computer Aided Geometric Design,2001,4(18):299-307.
[20]FARIN G. Curves and surfaces for computer aided geometric design:A practical guide[M]. Fifth edition, Morgan Kaufmann, San Francisco,2002.
[21]PIEGL L, TILLER W. The NURBS book[M]. Section edition, Springer-Verlag,1997.
[22]ZHANG J W. C-curves:An extension of cubic curves[J]. Computer Aided Geometric-Design,1996,13(3):199-217.
[23]ZHANG J W, KRAUSE F-L. Extending cubic uniform B-splinos by unified trigono-metric and hyperbolic basis[J]. Graphical Models,2005,67(2):100-119.
[24]CHEN Q, WANG G. A class of Bezier-like curves[J]. Computer Aided Geometric De-sign,2003,20:29-39.
[25]LU Y G, WANG G Z, YANG X N. Uniform hyperbolic polynomial B-spline curves[J]. Computer Aided Geometric Design,2002,19(6):379-393.
[26]WANG G Z, FANG M E. Unified and extened form of three types of splines[J]. Journal of Computational and Applied Mathematics,2008,216(2):418-508.
[27]WANG G Z, LI Y J. Optimal properties of the uniform algebraic trigonometric B-splines[J]. Computer Aided Geometric Design,2006,23(2):226-238.
[28]WANG G Z, YANG Q M. Planar cubic hyperbolic polynomial curve and its shape classification[J]. Progress in Natural Science,2004,14(1):41-46.
[29]MAINAR E, PENA. A basis of C-Bezier splines with optimal properties [J]. Computer Aided Geometric Design,2002,19(4):291-295.
[30]王仁宏,施锡泉,罗钟铉,苏志勋.多元样条函数及其应用[M].北京：科学出版社,1994.
[31]王仁宏,李崇君,朱春钢.计算几何教程[M].北京：科学出版社,2008.
[32]王仁宏.数值逼近[M].北京：高等教育出版社,1999.
[33]李宝军.指数多项式曲线细分重构与插值细分曲面快速计算[D].大连：大连理工大学,2008.
[34]郑彩玲.偏微分方程过渡面构造与曲面控制问题的研究[D].大连：大连理工大学,2005.
[35]QIN H, TERZOPOULOS D. D-NURBS:a physics-based framework for geometric de-sign[J]. IEEE Transactions on Visualization and Computer Graphics,1996,2(1):85-96.
[36]王志国.曲线曲面形状修改和变形关键技术研究[D].南京：南京航空航天大学,2006.
[37]周晓杰.曲线曲面造型中的变形方法研究与应用[D].大连：大连理工大学,2008.
[38]PLATEAU.实验和理论流体静力学[M].1873.
[39]MONGE.画法几何学[M].1794.
[40]DO CARMO M. Differential geometry of curves and surfaces[M]. Prentice-Hall, Engle-wood Cliffs,1976.
[41]DOUGLAS J. Solution of the problem of Plateau[J]. Transactions of the AMS,1931, 33:263-321.
[42]RATO T. On Plateau's problem[J]. Annals of Mathematics,1930,31(2):457-469.
[43]王子航.利用四点插值细分法构造光顺极小曲面[D].大连：大连理工大学,2006.
[44]SEQUIN C H. CAD Tools for Aesthetic Engineering[J]. Computer-Aided Design,2005, 7:737-750.
[45]SULLIVAN J. The aesthetic value of optimal geometry[M]. In:Emmer, M. (ed.) The Visual Mind II,547-563. MIT Press, Cambridge,2005.
[46]WANG Y. Periodic surface modeling for computer aided nano design[J]. Computer-Aided Design,2007,3:179-189.
[47]JIN W, WANG G. Geometric modeling using minimal surfaces[J]. Chinese Journa of Computers,1999,22(12):1276-1280.
[48]金文标,汪国昭.一类极小曲面的几何设计[J].计算机学报,1999,23(12)：1277-1279.
[49]金文标.若干特殊曲面的造型和应用[D].杭州：浙江大学,1999.
[50]满家巨,汪国昭.正螺面与悬链面的表示和构造[J].计算机辅助设计与图形学学报,2005,17(3)：431-436.
[51]XU G, WANG G Z. AHT Bezier surfaces and NUAHT B-spline curves[J]. Journal of Computer Science and Technology,2007,22(4):597-607.
[52]JIN W B. Modeling and applications of some special surfaces[D]. Hangzhou:Zhejiang University,1999 (in Chinese with English abstract).
[53]MAN J J, WANG G Z. Polynomial minimal surface in isothermal parameter[J]. Journal of Computers,2002,25(2):197-201.
[54]XU G, WANG G Z. Quintic parametric polynomial minimal surfaces and their prop-erties[J]. Differential Geometry and its Applications,2010,28:697-704.
[55]XU G, WANG G Z. Parametric polynomial minimal surfaces of degree six with isother-mal parameter[C]//Proc. of Geometric Modeling and Processing (GMP 2008),2008, LNCS 4975,329-343.
[56]徐岗,汪国昭.计算机辅助几何设计中极小曲面造型的研究进展[J].系统科学与数学,2008,28(8)：984-992.
[57]MAN J J, WANG G Z. Approximating to nonparameterzied minimal surface with B-spline surface[J]. Journal of Software,2003,14(4):824-829.
[58]MAN J J, WANG G Z. Minimal surface modeling using finite element method[J]. Chinese Journal of Computers,2003,26(4):507-510.
[59]DZIUK G. An algorithm for evolutionary surfaces[J].Numerische Mathematik,1991, 58:603-611.
[60]CONCUS P. Numerical solution of the minimal surface equation[J]. Mathematics of Computation,1967,21:340-350.
[61]PINKALL U, POLTHIER K. Computing discrete minimal surface and their conju-gates[J]. Exp. Math,1993,2(1):15-36.
[62]MONTERDE J. Bezier surfaces of minimal area:The Dirichlet approach[J]. Computer Aided Geometric Design,2004,21:117-136.
[63]MONTERDE J. The Plateau-Bezier Problem In:Wilson, M. J., Martin, R.R. (eds.) Mathematics of Surfaces[M]. LNCS, vol.2768,262-273. Springer, Heidelberg (2003).
[64]COSIN C, MONTERDE J. Bezier Surfaces of Minimal Area[M]. In:Sloot, P.M.A., Tan, C.J.K., Dongarra, J., Hoekstra, A.G. (eds.) ICCS 2002. LNCS, vol.2330,72-81. Springer, Heidelberg,2002.
[65]FARIN G, HANSFORD D. Discrete Coons patches[J]. Computer Aided Geometric Design 1999,16:691-700.
[66]ARNAL A, LLUCH A, MONTERDE J. Triangular Bezier surfaces of minimal area[C]//Proceedings of the Int. Workshop on Computer Graphics and Geometric Mod-elling, CG GM'2003, Montreal. Leture Notes in Comput. Sci,2003, Springer-Verlag, Berlin/New York,366-375.
[67]刘计善.一类极小曲面造型设计的新方法[J].复旦学报(自然科学版),2007,46(2)：192-197.
[68]吴花精灵,王国瑾.推广Dirichlet方法用于B样条极小曲面设计[J].软件学报,2011,22(12)：3015-3022.
[69]吴花精灵,王国瑾.插值边界曲线的NURBS近似极小曲面设计[J].软件学报,2010,21：67-74.
[70]覃廉,关履泰.基于NURBS的极小曲面造型[J].高等学校计算数学学报.2005,16(S1)：175-181.
[71]CHEN X. Design of rational Bezier minimal surface:an extension of Dirichlet ap-proach [D].杭州：浙江大学,2006.
[72]COX J, CHARLESWORTH W, ANDERSON D. Domain composition methods for as-sociating geometric modelling with finite element modeling[C]//Proceeding of the ACM SIGGRAPH, Symposium on Solid Modeling Foundation and CAD/CAM application, Austin, TX,1991,5-7.
[73]CELNIKE G, GOSSARD D. Deformable curve and surface finite elements for free-form shape design[J]. Proc. SIGGRAPH,1991,257-266.
[74]季洁,满家巨.任意边界下的极小曲面造型问题[J].计算机与现代化.2007,4：7-9.
[75]LI Y, WANG G. Two kinds of B-basis of the algebraic hyperbolic space[J]. Journal of Zhejiang University,2005,6A(7):750-759.
[76]RIESENFELD R F. On Chaikin's algorithm[J]. Computer Graphics and Image Pro-cessing,1975,4:153-174.
[77]FORREST A R. Curves and surfaces for computer sided design[D]. Cambridge:Uni-versity of Cambridge,1968.
[78]CATMULL E, CLARK J. Recursively generated B-spline surfaces on arbitrary topo-logical meshes[J]. Computer-Aided Design,1978,10(6):350-355.
[79]RIESENFELD R F. An algorithm for high speed curve generation [J]. Computer Graph-ics and Image Processing,1974,4:346-349.
[80]DEROSE T. Necessary and sufficient conditions for tangent plane continuity of Bozior surfaces[J]. CAGD, Special issue curves and surfaces in CAGD'89,1990,7(4):165-179.
[81]DOO D, SABIN M. Behavior of recursive division surfaces near extraordinary points[J]. Computer-Aided Design,1978,10(6):356-360.
[82]DYN NIRA, LEVIN DAVID, GREGORY J A. A butterfly subdivision scheme for surface interpolatory with tension control [J]. ACM Transactions on Graphics,1990, 9(2):160-169.
[83]KOBBELT LEIF.(?)-subdivision[C]//In Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH,2000.
[84]LOOP C. Smooth subdivision surfaces based on triangles[D]. Utah:University of Utah, Department of Mathematics,1987.
[85]NITSCHE J C C. Lectures on minimal surfaces[M]. Cambridge University Press, Cam-bridge,1989.
[86]STRUWE M. Plateau's problem and the calculus of variations[M]. Princeton University Press,1988.
[87]OSSERMAN R. A survey of minimal surfaces[M].Dover publ.2nd ed.,New York,1986.
[88]MONTERDE J, UGAIL H. A general 4th-order PDE method to generate Bezier sur-faces from the boundary[J]. Computer Aided Geometric Design,2006,23:208-225.
[89]CENTELLA P, MONTERDE J, MORENO E, OSET R. Two C1-methods to generate Bezier surfaces from the boundary[J]. Computer Aided Geometric Design,2009,20: 152-173.
[90]PAN Q, XU G L. Construction of minimal subdivision surface with a given boundary[J]. Computer-Aided Design,2011,43:374-380.
[91]XU G, WANG G Z. Parametric polynomial minimal surfaces of arbitrary degree[R]. INRIA Research Report 00507790,2010.
[92]FRANK MORGAN. Minimal surfaces, crystals, and norms on Rn[C]//Proc.7th Annual Symp.on Computational Geom., June,1991, N. Conway, NH.
[93]TRASDAHL (?), R(?)NQUIST E M.High order numerical approximation of minimal surfaces[J]. Journal of Computational Physics,2011,230:4795-4810.
[94]POLTHIER K. Polyhedral surfaces of constant mean curvature[M]. Habilitationsschrift, Technische Universitat Berlin,2002.
[95]MONTERDE J, UGAIL H. On harmonic and biharmonic Bezier surfaces[J]. Computer Aided Geometric Design,2004,7:697-715.
[96]XU G, WANG G Z. Harmonic-type Bezicr surface over rectangular and triangualr doinain[J]. Journal of Information and Computational Science,2006,3:325-332.
[97]WU H J L, WANG G J. Extending and correcting some research results on minimal and harmonic surfaces[J]. Computer Aided Geometric Design,2012,29:41-50.
[98]MALL AT S. A theory for nmltiresolution signal decomposition:the wavelet represen-tation[J]. IEEE Trans Pat Anal Intel,1989,11(7):674-693.
[99]DE BOOR C. A practical guide to splines[M].Springer-Verlag, New York,2001.
[100]BARRERA D, FORTES M A, GONZALEZ P, PASADAS M.Minimal energy Cr sur-faces on uniform Powell-Sabin type meshes for noisy data[J]. Journal of Computational and Applied Mathematics,2008,218(2):592-602.
[101]FORTES M A, GONZALEZ P, MONCAYO M, PASADAS M. Multiresolution anal-ysis for minimal energy Cr surfaces on Powell-Sabin type meshes[C]//Proceeding MMCS'08 Proceedings of the 7th international conference on Mathematical Methods for Curves and Surfaces, Pages 209-223, Springer-Verlag Berlin, Heidelberg (?)2010 ISBN:3-642-11619-1978-3-642-11619-3.
[102]LV X-G, HUANG T-Z, LE J. A note on computing the inverse and the determinant of a pentadiagonal Toeplitz matrix[J]. Applied Mathematics and Computation,2008, 206:327-331.
[103]KILIC E, El-Mikkawy M. A computational algorithm for special nth order pentadiag-onal Toeplitz determinants [J]. Applied Mathematics and Computation,2008,199(2): 820-822.
[104]MOLINARI L G. Determinants of Block Tridiagonal Mtrices[J]. Linear Algebra and its Applications,2008,429:2221-2226.
[105]MEURANT G. A review on the inverse of symmetric tridiagonal and block tridiagonal matrices[J]. SIAM Journal on Matrix Analysis and Applications,1992,13(3):707-728.
[106]SWEET R A. A recursive relation for the determinant of a pentadiagonal matrix[J]. Communications of the ACM,1969,12:330-332.
[107]SOGABE T. A fast numerical algorithm for the determinant of a pentadiagonal ma-trix[J]. Applied Mathematics and Computation,2008,196(2):835-841.
[108]SOLARY M S. Finding determinant for some block band matrices by reblocking[J]. Differential Equations and Control Processes,2012,3:37-48.
[109]COOS S A. Surfaces for computer aided design, Technical Report[R]. MIT Press, Available as AD 663 504 from the National Technical Information Service, Springfield, VA 22161.1964.
[110]HAO Y X, LI C J, WANG R H. An approximation method based on MRA for the quasi-Plateau problem[J]. BIT Numerical Mathematics,2013,53:411-442.
[111]DYN N, LEVIN D. Subdivision schemes in geometric modelling[J]. Acta Numerica, 2002,11:73-144.
[112]DYN N, LEVIN D, GREGORY J A. A 4-point interpolatory subdivision scheme for curve design[J]. Computer Aided Geometric Design,1987,4:257-268.
[113]HASSAN M F, IVRISSIMITZIS I P, DODGDSON N A, SABIN M A. An interpolat-ing 4-point C2 ternary stationary subdivision scheme[J]. Computer Aided Geometric Design,2002,19:1-8.
[114]SHAHID S SIDDIQI, NADEEM AHMAD. A new 3-point C2 approximating subdivi-sion scheme[J]. Applied Mathematics Letters,2007,20:707-711.
[115]SHAHID S SIDDIQI, KASHIF REHAN. A stationary binary C1 curve subdivision scheme[J]. European Journal of Scientific Research,2009,4:553-561.
[116]BECCARI C, CASCIOLA G, ROMANI L. Shape controlled interpolator ternary subdivision[J]. Applied Mathematics and Computation,2009,215:916-927.
[117]DYN N, ISKE A, QUAK E, FLOATER M S. Tutorials on multiresolution in geometric modelling[M]. Summer school lecture notes series:, Mathematics and Visualization, Springer, ISBN:3-540-43639-1,2002.
[118]DYN N, HORMANN K, SABIN M A, SHEN Z. Polynomial reproduction by symmet-ric subdivision schemes[J]. Journal of Approximation Theory,2008,155:28-42.
[119]DYN N, KUIJT F, LEVIN D, VAN DAMME R. Convexity preservation of the four-point interpolatory subdivision scheme[J]. Computer Aided Geometric Design,1999, 16:789-792.
[120]SHAHID S SIDDIQI, NADEEM AHMAD. A C6 approximating subdivision scheme[J]. Applied Mathematics Letters,2008,21:722-728.
[121]KUIJT F, VAN DAMME R. Convexity preserving interpolatory subdivision schemes[J]. Constructive Approximation,1998,14(4):609-630.
[122]CAI ZHIJIE. Convexity preservation of the interpolating four-point C2 ternary sta-tionary subdivision scheme[J]. Computer Aided Geometric Design,2009,26:560-565.
[123]DYN N, FLOATER M S, HORMANN. A C2 four-point subdivision scheme with fourth order accuracy and its extensions[M]. In:M.Daehlen, K.M(?)rken, and L.L.schumaker, editors, Mathematical Methods for Curves and Surfaces. Troms(?) 2004. Nashboro Press, Brentwood, TN, (2005) 145-156.
[124]LAI M J. Fortran subroutines for B-nets of Box splines on three and four directional meshes[J]. Numerical Algorithms,1992,2:33-38.