T样条和T网格上的样条
|
摘要
自由曲线曲面技术是计算机辅助几何设计(CAGD)的核心,而非均匀有理B样条(NURBS)方法作为自由曲线曲面的造型方法,由于其统一的数学模型而成为计算机辅助设计和制造(CAD/CAM)的一个标准。长期以来,几何外形的数学模型化和数据平滑化促使了样条方法的发展;反过来,样条理论和方法的建立,为自由曲线曲面的发展提供了重要的依据和工具。但是我们知道NURBS曲面有着很多缺点,比如说它的控制点必须拓扑的位于一个矩形网格上,从而很多的控制点更多的是为了满足拓扑的限制而不是几何上的信息。为了克服这个缺点,Sederberg[Sederberg2003,Sederberg 2004]提出了T样条。T样条允许控制网格中出现T型控制点,同时它是从NURBS推广而来,从而它可以消除大多数的多余的控制点。
NURBS曲面的参数表示和可以精确表示圆锥曲面是它的一个很大的优点,但是它的张量积的拓扑限制是它的一个很大的缺点。为了克服这些限制创建复杂拓扑的物体,裁剪是NURBS曲面在造型上的一个很重要的操作。但是裁剪只是一种忽略裁剪区域的求值的方法,所以如果我们将两个经过裁剪的曲面放在一起,那么在它们在裁剪区域上是无法避免缝隙的。对很多的应用而言,这是一个很大的问题。这个问题并不新,但是它困扰了CAD界20多年。在本文的第二章,我和Sederberg教授用T样条解决了这个问题。
由于在大多数的情况下,T样条都是有理的,邓建松等人[邓建松2005]提出T网格上的样条,并在多项式的次数大于光滑阶的两倍的情况下研究了它的维数。在本文中,在同样的限制下,本文首先在层次T网格上构造了它的一组基函数,并讨论了节点插入、删除、插值和拟合算法。接着又研究了它的绘制、缝合算法,从而可以拟合任意拓扑的模型。我还给出了与NURBS的转化、简化等方面的一些应用。结果表明T样条不仅有着NUBRS的那些好的性质,还有着良好的自适应性。相比T样条而言,更加的简单和快速。接着本文研究了任意T网格上的样条的基函数的构造,并给出了一个通过删除多余的边来简化B样条曲面的算法。
最后本文将前面的内容推广到三维空间的T网格上的样条,从而为隐式T网格上的样条曲面和曲面自由变形提供了理论上的基础。
The technique for free-form curves and surfaces is the core of computer-aided geometry design(CAGD).As a kind of free-form curves and surfaces sculpting method, Non-uniform rational B-spline(NURBS)has become a standard in computer-aided design/manufacture(CAD/CAM)for its' uniform mathematical model.Since long ago,the mathematic models of geometrical shape and data smoothing promote the development of spline.And on the contrary,the building of spline theory and method provides the important foundations and tools for free-form curves and surface design.But as we known,NURBS has many disadvantages such as,its' control points must lie topologically in a rectangular grid.This means that typically,a large number of NURBS control points serve no purpose other than to satisfy topological constraints.In order to overcome these restrictions,Sederberg[Sederberg 2003,Sederberg 2004]inverted T-spline which allow T-junctions in the control nets. T-splines are a generalization of NURBS surfaces that are capable of significantly reducing the number of superfluous control points.
NURBS surfaces' parametric representation and the exact expression of conical sections is a big advantage,but the strict in tensor product topology is a very big disadvantages.In order to overcome these restrictions to create complex objects, trimming is an important operation in geometric modeling.However,trimming operator is a way to elevate with eliminating some trimmed region.So if we put two trimmed NURBS surfaces back together,it can't avoid the gap between these two surfaces which is a very big problem for some applications.This problem is not new,but it has puzzled CAD for over twenty years.In chapter 2,I provide a method with Dr.Sederberg to solve this problem with T-spline.
As T-spline is rational in many cases,professor Deng etc.invented splines over T-meshes in[邓建松2005]and provided the dimension formulae when half of the degree of the polynomial in each cell is more than the order of the smoothness between the adjacent cells.Under the same restricts,this paper construct a set of basis functions over a hierarchical t-mesh and provides the knot insertion,knot removal,interpolation and fitting algorithm.Then I provide some more applications such as tessellation,stitching with spline over T-mesh which provides the possibility to construct arbitrary topology spline surface over T-mesh.I also give the method to transform between the spline surface over T-mesh and NURBS.All the results indicate the spline over T-mesh not only has the merit as the NURBS but also has good adaptiveness and it is much simpler and faster than the T-spline.And then I give an algorithm for construction basis functions over arbitrary T-meshes.I also provide a B-spline surface simplification algorithm with the superfluous edges removal.
In the end,the paper extends the theory to splines over 3D T-meshes which are the theoretical foundations for implicit splines over T-meshes and free-form deformation.
NURBS曲面的参数表示和可以精确表示圆锥曲面是它的一个很大的优点,但是它的张量积的拓扑限制是它的一个很大的缺点。为了克服这些限制创建复杂拓扑的物体,裁剪是NURBS曲面在造型上的一个很重要的操作。但是裁剪只是一种忽略裁剪区域的求值的方法,所以如果我们将两个经过裁剪的曲面放在一起,那么在它们在裁剪区域上是无法避免缝隙的。对很多的应用而言,这是一个很大的问题。这个问题并不新,但是它困扰了CAD界20多年。在本文的第二章,我和Sederberg教授用T样条解决了这个问题。
由于在大多数的情况下,T样条都是有理的,邓建松等人[邓建松2005]提出T网格上的样条,并在多项式的次数大于光滑阶的两倍的情况下研究了它的维数。在本文中,在同样的限制下,本文首先在层次T网格上构造了它的一组基函数,并讨论了节点插入、删除、插值和拟合算法。接着又研究了它的绘制、缝合算法,从而可以拟合任意拓扑的模型。我还给出了与NURBS的转化、简化等方面的一些应用。结果表明T样条不仅有着NUBRS的那些好的性质,还有着良好的自适应性。相比T样条而言,更加的简单和快速。接着本文研究了任意T网格上的样条的基函数的构造,并给出了一个通过删除多余的边来简化B样条曲面的算法。
最后本文将前面的内容推广到三维空间的T网格上的样条,从而为隐式T网格上的样条曲面和曲面自由变形提供了理论上的基础。
The technique for free-form curves and surfaces is the core of computer-aided geometry design(CAGD).As a kind of free-form curves and surfaces sculpting method, Non-uniform rational B-spline(NURBS)has become a standard in computer-aided design/manufacture(CAD/CAM)for its' uniform mathematical model.Since long ago,the mathematic models of geometrical shape and data smoothing promote the development of spline.And on the contrary,the building of spline theory and method provides the important foundations and tools for free-form curves and surface design.But as we known,NURBS has many disadvantages such as,its' control points must lie topologically in a rectangular grid.This means that typically,a large number of NURBS control points serve no purpose other than to satisfy topological constraints.In order to overcome these restrictions,Sederberg[Sederberg 2003,Sederberg 2004]inverted T-spline which allow T-junctions in the control nets. T-splines are a generalization of NURBS surfaces that are capable of significantly reducing the number of superfluous control points.
NURBS surfaces' parametric representation and the exact expression of conical sections is a big advantage,but the strict in tensor product topology is a very big disadvantages.In order to overcome these restrictions to create complex objects, trimming is an important operation in geometric modeling.However,trimming operator is a way to elevate with eliminating some trimmed region.So if we put two trimmed NURBS surfaces back together,it can't avoid the gap between these two surfaces which is a very big problem for some applications.This problem is not new,but it has puzzled CAD for over twenty years.In chapter 2,I provide a method with Dr.Sederberg to solve this problem with T-spline.
As T-spline is rational in many cases,professor Deng etc.invented splines over T-meshes in[邓建松2005]and provided the dimension formulae when half of the degree of the polynomial in each cell is more than the order of the smoothness between the adjacent cells.Under the same restricts,this paper construct a set of basis functions over a hierarchical t-mesh and provides the knot insertion,knot removal,interpolation and fitting algorithm.Then I provide some more applications such as tessellation,stitching with spline over T-mesh which provides the possibility to construct arbitrary topology spline surface over T-mesh.I also give the method to transform between the spline surface over T-mesh and NURBS.All the results indicate the spline over T-mesh not only has the merit as the NURBS but also has good adaptiveness and it is much simpler and faster than the T-spline.And then I give an algorithm for construction basis functions over arbitrary T-meshes.I also provide a B-spline surface simplification algorithm with the superfluous edges removal.
In the end,the paper extends the theory to splines over 3D T-meshes which are the theoretical foundations for implicit splines over T-meshes and free-form deformation.
引文
[Alfeld 1987]P.Alfeld and L.L.Schumaker,The dimsions of bivariate splint spaces of smoothness r for degree d≥4r+1,Constructive Approximation,3,189-197,1987.
[Bartels & Foley 1995]R.H.Barrels and D.R.Foley."surface fitting with hierarchical splines" ACM Trans.Graphics,Vol.14,134-161,1995.
[Bezier 1972]P.E.Bezier.Numerical control-mathematics and applications(Translated by A.R.Forrest).John Wiley and Sons,1972.
[Bezier 1977]P.E.Bezier.Essai de definition numerique des courbes et des surfaces experimentales.Thesis,University of Pierre et Marie Curie,Paris,1977.
[Bezier 1986]P.E.Bezier.The mathematical basis of the UNISURF CAD system.Butterworth,London,1986.
[Biermann 2001]H.Biemann,D.Kristjansson and D.zorin."Approximate Boolcan operations on free-form solids",In Proceedings of SIGGRAPH 2001,ACM,185-194.2001.
[Boehm 1980]W.Boehm."Inserting new knots into B-spline curves".CAD,Vol.12,No.4.199-201,1980.
[Chaikin 1974]G.M.Chaikin.An algorithm for high speed curve generation.Computer Graphics and Image Processing,Vol.3,346-349,1974.
[Cheney et al.1985]E.W.Cheney and W.A.Light."Approximation Theory in Tensor Product Space".Lecture Notes in Mathematices,Springer,Berlin,1985.
[Chui 1983a]C.K.Chui and R.H.Wang.Multivariate spline spaces.J.Math.Anal.Appl.,Vol.94,197-221,1983.
[Chui 1983b]C.K.Chui and R.H.Wang.On smooth multivariate spline functions.Math.Comp.,Vol.41,No.163,131-142,1983.
[Cohen et al.1980]E.Cohen,T.Lyche and R.F.Riesenfeld."Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphies".Computer Graphics and Image Processing,Vol.14,87-111,1980.
[Cohen et al.1985]E.Cohen,T.Lyche and L.Schumaker."Algorithms for degree-raising of splines".Trans.on Graphics,Vol.4,171-181,1985.
[Coons 1964]S.A.Coons."Surfaces for computer-aided design of space figures".MAC-M-255,M.I.T.,January 1964.
[Coons 1967]S.A.Coons."Surfaces for computer-aided design of space forms".MAC-TR-41,M.I.T.,June 1967.
[Cox 1972]M.G.Cox."The numerical evaluation of B-splines".J.Inst.Math.App.,Vol.10,134-149,1972.
[Cox et al.1997]D.Cox,J.Little and D.O'Shea.Ideas,Varieties and Algorithms - An Introduction to Computational Algebraic Geometry.Springer-Verlag,New York,2nd edition,1997.
[Cox et al.1998]D.Cox,J.Little,and D.O'shen.Using Algebraic Geometry.GTM85,Springer-Verlag,New York,Inc.,1998.
[Cox et al.1998]D.Cox,T.Sederberg and F.Chen."The moving line ideal basis of planar rational curves".CAGD,Vol.15,803-827,1998.
[Dahmen 1980]W.Dahmen.On multivariate B-splines.SIAM J.Numer.Anal.,Vol.17,179-191,1980.
[Dahmen 1983]W.Dahmen and C.A.Micchelli.In Recent progress in multivariate splines,interpolating cardinal splines as their degree rends to infinity,Isreal J.Ward(eds.),Academic Press,27-121,1983.
[Dahmen 1984]W.Dahmen and C.Micchelli.Subdivision algorithms for the generation of box splint surfaces.Computer Aided Geometric Design,Vol.1,No.2,115-129,1984.
[Dahmen 1989]W.Dahmen."Smooth Piecewise Quadric Surfaces".in Mathematical Aspects of CAGD,T.Lyche & L.L.Scumaker eds.,Acadeic Press,181-194,1989.
[Dahmen 1993]W.Dahmen and T.M.Thamm-Schaar."Cubicoids:modeling and visualization".CAGD,Vol.10,89-108,1993.
[de Boor 1972]C.de Boor."On calculating with B-splints".J.Approx.Theory,Vol.6.No.1,50-62,1972.
[de Boor 1978]C.de Boor.A pratical guide to splines.Springer-Verlag,1978.
[de Boor 1973]C.de Boor."Efficient computer manipulation of tensor products".ACM Trans.Math.Software,Vol.5,173-182,1979.
[de Boor 1982]C.de Boor.Topics in multivariate approximation theory.In Topics in Numerical Analysis,P.R.Turner(eds.),Lecture Notes Mathematics,Springer-Verlag.Vol.965,39-78,1982.
[de Boor 1983]C.de Boor and R.DeVore.Approximation by smooth multivariate splines.Trans.Amer.Math.Soc.,Vol.276,775-788,1983.
[de Boor 1987]C.de Boor.B form basis.In Geometric Modelling,G.Farin(eds.),SIAM,Philadephia,131-148,1987.
[de Casteljau 1959]P.de Casteljau."Outillages methodes calcul".Technical Report.,A.Citrie,Paris,1959.
[de Casteljau 1963]P.de Casteljau."Courbes et surfaces a poles".Technical Report,A.Citrie,Paris,1963.
[邓建松 2005]Jiansong Deng,Falai Chen,Yuyu Feng."Dimensions of spline spaces over T-meshes".Journal of Computational and Applied Mathematics,Vol.194,No.2,267-283,2006.
[Eck et al 1995]Eck,M.,DeRose,T.,Duchamp,T.,Hoppe,H.,Lounsbery,M.,and Stuetzle,W."Multiresolution analysis of arbitrary meshes".SIGGRAPH '95:Proceedings of the 22nd annual conference on Computer graphics and interactive techniques,173-182,1995.
[Farouki R.T.et al 2004]R.T.Farouki,C.Y.Han,and T.W.Sederberg."Topologically consistent trimmed surface approximations based on triangular patches".Computer Aided Geometric Design,Vol 21,459-478,2004.
[Farin 1979]G.E.Farin.Subsplines uber dreiecken.Diss.TU Braunschweig,1979.
[Farin 1980]G.Farin.Bezier polynomials over triangles and the construction of piecewise C~r polynomials.TR/91,Dept.of Math.,Brunel Univ.,Uxbridge,Middlesex,U.K.,1980.
[Farin 1986]G.Farin.Triangular Bernstein-Bezier patches.Computer Aided Geometric Design,Vol.3,83-127,1986.
[Farin 2002]G.Farin Curves and Surfaces for Computer-Aided Geometric Design:A Practical Guided.Academic Press,San Diego,CA,2002.
[Farin 2002]G.Farin."A history of curves and surfaces in CAGD".In Handbook of Computer Aided Geometric Design,G.Frain,J.Hoschek and M.S.Kim(eds.),1-14,Elsevier Science B.V.,2002.
[Faux & Pratt 1979]I.Faux and M.Pratt.Computational Geometry for Design and Manufacture.Ellis Horwood,1979.
[Ferguson 1963]J.C.Ferguson."Multivariable curve interpolation".Report No.D2-22504,The Boeing Co.,Seattle,Washington,1963.
[Ferguson 1964]J.C.Ferguson."Multivariable curve interpolation".J.ACM,Vol.11,221-228,1964.
[Floator 2002]M.S.Floator and K.Hormann."Parameterization of Triangulations and Unorganized Points".in Tutorials on Multiresolution in Geometric Modelling(A.Iske,E.Quak and M.S.Floater,eds.),Springer-Verlag,287-316,2002.
[Floator 2004]M.S.Floator and K.Hormann."Surface Parameterization:a Tutorial and Survey".in Advances in Multiresolution for Geometric Modelling(N.A.Dodgson and M.S.Floater and M.A.Sabin,eds.),Springer-Verlag,Heidelberg,157-186,2004.
[Forrest 1972]A.R.Forrest."Interactive interpolation and approximation by Bezier polynomials".Computer J.,Vol.15,No.1,71-79,1972.
[Forsey 1988]David Forsey and Richard H.Bartels."Hierarchical B-spline refinement".Computer Graphics,Vol.22,No.4,205-212,1988.
[Franke 80]R.Pranke,G.Nielson.Smooth interpolation of large sets of scattered data.International Journal of Numberical Methods in Engineering,vol.15,1980,1691-1704.
[Garrity & Warren 1991]T.Garrity and J.Warren."Geometric continuity".CAD,Vol.8,51-56,1991.
[Gordon & Riesenfeld 1974a]W.J.Gordon and R.F.Riesenfcld."Bernstein-Bezier methods for the computer-aided design of free form curves and surfaces".J.ACM,Vol.21,No.2,293-310,1974.
[Gordon & Riesenfeld 1974b]W.J.Gordon and R.F.Riesenfeld."B-spline curves and surfaces".In Computer Aided Geometric Design,R.E.Barnhill and R.F.Riesenfeld(eds.),95-126,Academic Press,1974.
[Greiner & Hormann 1997]G.Greiner and K.Hormann."Interpolation and approxima tion scattered 3D data with hierarchical tensor product B-spline".In Surface Fitting and Multiresolution Methods,A.Le Mehaute,C.Rabut,and L.L.Schumaker(eds.),163-177,Vanderbilt University Press,1997.
[郭竹瑞1990]贾荣庆.多元样条研究中的B网方法.数学进展,Vol.19,189-198,1990.
[Hayes 1974]J.C.Hayes and J.Halliday."The least squares fitting of cubic spline surfaces to general data sets".J.Math.Appl.,Vol.14,89-113,1974.
[Hoffmann 1989]C.Hoffmann.Geometric and Solid Modeling:An Introduction.San Francisco,Morgan Kaufmann,1989.
[H.Ipson 2005]H.Ipson."T-spline Merging".Master's thesis,Brigham Young University.2005.
[贾荣庆1987a]贾荣庆.箱样条研究的新进展.高校应用数学学报,Vol.2,330-343.1987.
[贾荣庆1987b]贾荣庆.多元样条的B网表示.科学通报,Vol.11,804-807,1987.
[Katz 1988]S.Katz,T.W.Sederberg."Genus of the intersection curve of two rational surface patches".In Computer Aided Geometric Design,Vol 5,253-258,1988.
[Krishnamurthy 1996]V.Krishnamurthy,and M.Levoy."Fitting smooth surfaces to dense polygon meshes".In SIGGRAPH '96:Proceedings of the 23rd annual conference on Computer graphics and interactive techniques,ACM Press,New York,NY,USA.313-324,1996.
[Lai 1996]M.J.Lai."Scattered data interpolation and approximation using bivariate C~1piecewise cubic polynomials".CAGD,Vol.13,81-88,1996.
[Lai 2007]S.Lai,F.Cheng."Robust and error controllable boolean operations on freeform solids represented by catmullclark subdivision surfaces".Computer Aided Design and Applications,Vol.4,2007.
[Litke etal.2001]N.Litke,A.Levin and P.Schroder."Trimming for subdivision surfaces".CAGD,Vol.18,463-481,2001.
[Loop 2004]C.Loop."Second order smoothness over extraordinary vertices".In Eurographics /ACM SIGGRAPH Symposium on Geometry Processing,165-174,2004.
[Peter 2000]Jorg Peter."Patching Catmull-Clark Meshes",Siggraph 2000,2000.
[Prautzsch 1984a]H.Prautzsch."Degree elevation of B-spline curves".CAGD,Vol.1,No.2,193-198,1984.
[Riesenfeld 1975]R.F.Riesenfeld.On Chaikin's algorithm.IEEE Computer Graphics and Applications,Vol.4,304-310,1975.
[Sabin 1976]M.A.Sabin.The use of piecewise form for the numerical representation of shape.Ph.D.Thesis,MTA Budapest,Hungary,1976.
[Schoenberg 1946]I.J.Schhoenberg."Contributions to the problem of approximation of equidistant data by analytic functions".Quart.Appl.Math.,Vol.4,No.1,44-99,1946.
[Schumaker 1981]L.L.Schumaker.Spline functions:Basic theory,John Wiley & Sons,Inc.,1981.
[Sederberg 1984]T.W.Sederberg."Piecewise Algebraic Curves".CAGD,Vol.1,241-255,1984.
[Sederberg 1984]T.W.Sederberg,D.Anderson,and R.Goldman."Implicit representation of parametric curves and surfaces".Computer Vision,Graphics and Image Processing,Vol.28,72-84,1984.
[Sederberg 1998]Thomas W.Sederberg,Jianmin Zheng,David Sewell,and Malcolm Sabin.Non-uniform recursive subdivision surfaces.Computer Graphics (SIGGRAPH'98 Proceedings),387-394,1998.
[Sederberg 2003]T.W.Sederberg,Jianmin Zheng and Xiaowen Song.Knot intervals and multi-degree splines.Computer Aided Geometric Design,Vol.20,No.7,455-468.2003.
[Sederberg 2003]Thomas W.Sederberg,Jianmin Zheng,Almaz Bakenov,and Ahmad Nasri.T-splines and T-NURCCs.ACM Transactions on Graphics,Vol.22,No.3,161-172,2003.
[Sederberg 2004]Thomas W.Sederberg,David L.Cardon,G.Thomas Finnigan,Nicholas S.North,Jianmin Zheng,and Tom Lyche.T-spline simplification and local refinement.ACM Transactions on Graphics,Vol.23,No.3,276-283,2003.
[Shewchuk 2002]Shewchuk,J.R."Delaunay refinement algorithms for triangular mesh generation".Computational Geometry:Theory and Applications 22,1-3,21-74,2002.
[施法中1994]施法中.计算机辅助几何设计与非均匀有理B样条.北京航天航空大学出版社,1994.
[Song 2004]X.Song,T.W.Sederberg,J.Zheng,R.T.Farouki,and J.Hass."Linear perturbation methods for topologically consistent representations of free-form surface intersections".Computer AIded Geometric Design,Vol 21,303-319,2004.
[Tur99]G.Truk and J.F.O'Brien.Shape transformation using variational implicit functions.In:Siggraph'99 Conference Proceedings,pages 335-342,1999.
[Tur02]G.Turk,J.F.O'Brien.Modelling with implicit surfaces that interpolate.ACM Transaction on Graphics,vol.21,855-873,2002.
[Wang W.2006]W.Wang,H.Pottmann,and Y.Liu."Fitting B-spline curves to point clouds by curvature-based squared distance minimization".ACM Transactions on Graphics,Vol25,214-238,2006.
[Wang 2005a]Y Wang,J Zheng,H.Soon Seah."Conversion between T-Splines and Hierarchical B-Splines".Proceedings of the eighth IASTED International Conference Computer Graphics and Imaging.8-13,2005.
[Weller 1995]Frank Weller and Hans Hagen.Tensor-product spline spaces with knot segments.in Mathematical Methods for Curves and Surfaces,M.Daehlen,T.Lyche,and L.L.Schumaker(eds.),Vanderbilt Unversity Press,563-572,1995.
[Wu & Wang 1994]W.T.Wu and D.K.Wang."On the algebraic surface - fitting problem in CAGD" Mathematics in Practice and Theory,No.3,26-31,1994.
[J.Zhang 1998]J.Zhang,Z.Tang,and J.Li."Adaptive hierarchical B-spline surface approxination of large-scale scattered data".Pacific Graphics.8-16,1998.
[Zheng 2005]J Zheng,Y Wang,H.Soon Seah."Adaptive T-Spline Surface Fitting to ZMap Models".Proceedings of the 3rd international conference on Computer graphics and interactive techniques in Australasia and South East Asia 405-411,2005.
[Li 2006]Wan-Chiu Li,Nicholas Ray,Bruno Levy."Automatic and interactive mesh to T-spline conversion".Proceedings of the fourth Eurographics symposium on Geometry processing.191-200,2006.
[周建伟1986]周建伟.三角域上Bernstein多项式的正性与凸性.计算数学,Vol.8,185-190,1986.
[朱心雄2000]朱心雄.自由曲线曲面造型技术.北京:科学出版社,2000.
[Bartels & Foley 1995]R.H.Barrels and D.R.Foley."surface fitting with hierarchical splines" ACM Trans.Graphics,Vol.14,134-161,1995.
[Bezier 1972]P.E.Bezier.Numerical control-mathematics and applications(Translated by A.R.Forrest).John Wiley and Sons,1972.
[Bezier 1977]P.E.Bezier.Essai de definition numerique des courbes et des surfaces experimentales.Thesis,University of Pierre et Marie Curie,Paris,1977.
[Bezier 1986]P.E.Bezier.The mathematical basis of the UNISURF CAD system.Butterworth,London,1986.
[Biermann 2001]H.Biemann,D.Kristjansson and D.zorin."Approximate Boolcan operations on free-form solids",In Proceedings of SIGGRAPH 2001,ACM,185-194.2001.
[Boehm 1980]W.Boehm."Inserting new knots into B-spline curves".CAD,Vol.12,No.4.199-201,1980.
[Chaikin 1974]G.M.Chaikin.An algorithm for high speed curve generation.Computer Graphics and Image Processing,Vol.3,346-349,1974.
[Cheney et al.1985]E.W.Cheney and W.A.Light."Approximation Theory in Tensor Product Space".Lecture Notes in Mathematices,Springer,Berlin,1985.
[Chui 1983a]C.K.Chui and R.H.Wang.Multivariate spline spaces.J.Math.Anal.Appl.,Vol.94,197-221,1983.
[Chui 1983b]C.K.Chui and R.H.Wang.On smooth multivariate spline functions.Math.Comp.,Vol.41,No.163,131-142,1983.
[Cohen et al.1980]E.Cohen,T.Lyche and R.F.Riesenfeld."Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphies".Computer Graphics and Image Processing,Vol.14,87-111,1980.
[Cohen et al.1985]E.Cohen,T.Lyche and L.Schumaker."Algorithms for degree-raising of splines".Trans.on Graphics,Vol.4,171-181,1985.
[Coons 1964]S.A.Coons."Surfaces for computer-aided design of space figures".MAC-M-255,M.I.T.,January 1964.
[Coons 1967]S.A.Coons."Surfaces for computer-aided design of space forms".MAC-TR-41,M.I.T.,June 1967.
[Cox 1972]M.G.Cox."The numerical evaluation of B-splines".J.Inst.Math.App.,Vol.10,134-149,1972.
[Cox et al.1997]D.Cox,J.Little and D.O'Shea.Ideas,Varieties and Algorithms - An Introduction to Computational Algebraic Geometry.Springer-Verlag,New York,2nd edition,1997.
[Cox et al.1998]D.Cox,J.Little,and D.O'shen.Using Algebraic Geometry.GTM85,Springer-Verlag,New York,Inc.,1998.
[Cox et al.1998]D.Cox,T.Sederberg and F.Chen."The moving line ideal basis of planar rational curves".CAGD,Vol.15,803-827,1998.
[Dahmen 1980]W.Dahmen.On multivariate B-splines.SIAM J.Numer.Anal.,Vol.17,179-191,1980.
[Dahmen 1983]W.Dahmen and C.A.Micchelli.In Recent progress in multivariate splines,interpolating cardinal splines as their degree rends to infinity,Isreal J.Ward(eds.),Academic Press,27-121,1983.
[Dahmen 1984]W.Dahmen and C.Micchelli.Subdivision algorithms for the generation of box splint surfaces.Computer Aided Geometric Design,Vol.1,No.2,115-129,1984.
[Dahmen 1989]W.Dahmen."Smooth Piecewise Quadric Surfaces".in Mathematical Aspects of CAGD,T.Lyche & L.L.Scumaker eds.,Acadeic Press,181-194,1989.
[Dahmen 1993]W.Dahmen and T.M.Thamm-Schaar."Cubicoids:modeling and visualization".CAGD,Vol.10,89-108,1993.
[de Boor 1972]C.de Boor."On calculating with B-splints".J.Approx.Theory,Vol.6.No.1,50-62,1972.
[de Boor 1978]C.de Boor.A pratical guide to splines.Springer-Verlag,1978.
[de Boor 1973]C.de Boor."Efficient computer manipulation of tensor products".ACM Trans.Math.Software,Vol.5,173-182,1979.
[de Boor 1982]C.de Boor.Topics in multivariate approximation theory.In Topics in Numerical Analysis,P.R.Turner(eds.),Lecture Notes Mathematics,Springer-Verlag.Vol.965,39-78,1982.
[de Boor 1983]C.de Boor and R.DeVore.Approximation by smooth multivariate splines.Trans.Amer.Math.Soc.,Vol.276,775-788,1983.
[de Boor 1987]C.de Boor.B form basis.In Geometric Modelling,G.Farin(eds.),SIAM,Philadephia,131-148,1987.
[de Casteljau 1959]P.de Casteljau."Outillages methodes calcul".Technical Report.,A.Citrie,Paris,1959.
[de Casteljau 1963]P.de Casteljau."Courbes et surfaces a poles".Technical Report,A.Citrie,Paris,1963.
[邓建松 2005]Jiansong Deng,Falai Chen,Yuyu Feng."Dimensions of spline spaces over T-meshes".Journal of Computational and Applied Mathematics,Vol.194,No.2,267-283,2006.
[Eck et al 1995]Eck,M.,DeRose,T.,Duchamp,T.,Hoppe,H.,Lounsbery,M.,and Stuetzle,W."Multiresolution analysis of arbitrary meshes".SIGGRAPH '95:Proceedings of the 22nd annual conference on Computer graphics and interactive techniques,173-182,1995.
[Farouki R.T.et al 2004]R.T.Farouki,C.Y.Han,and T.W.Sederberg."Topologically consistent trimmed surface approximations based on triangular patches".Computer Aided Geometric Design,Vol 21,459-478,2004.
[Farin 1979]G.E.Farin.Subsplines uber dreiecken.Diss.TU Braunschweig,1979.
[Farin 1980]G.Farin.Bezier polynomials over triangles and the construction of piecewise C~r polynomials.TR/91,Dept.of Math.,Brunel Univ.,Uxbridge,Middlesex,U.K.,1980.
[Farin 1986]G.Farin.Triangular Bernstein-Bezier patches.Computer Aided Geometric Design,Vol.3,83-127,1986.
[Farin 2002]G.Farin Curves and Surfaces for Computer-Aided Geometric Design:A Practical Guided.Academic Press,San Diego,CA,2002.
[Farin 2002]G.Farin."A history of curves and surfaces in CAGD".In Handbook of Computer Aided Geometric Design,G.Frain,J.Hoschek and M.S.Kim(eds.),1-14,Elsevier Science B.V.,2002.
[Faux & Pratt 1979]I.Faux and M.Pratt.Computational Geometry for Design and Manufacture.Ellis Horwood,1979.
[Ferguson 1963]J.C.Ferguson."Multivariable curve interpolation".Report No.D2-22504,The Boeing Co.,Seattle,Washington,1963.
[Ferguson 1964]J.C.Ferguson."Multivariable curve interpolation".J.ACM,Vol.11,221-228,1964.
[Floator 2002]M.S.Floator and K.Hormann."Parameterization of Triangulations and Unorganized Points".in Tutorials on Multiresolution in Geometric Modelling(A.Iske,E.Quak and M.S.Floater,eds.),Springer-Verlag,287-316,2002.
[Floator 2004]M.S.Floator and K.Hormann."Surface Parameterization:a Tutorial and Survey".in Advances in Multiresolution for Geometric Modelling(N.A.Dodgson and M.S.Floater and M.A.Sabin,eds.),Springer-Verlag,Heidelberg,157-186,2004.
[Forrest 1972]A.R.Forrest."Interactive interpolation and approximation by Bezier polynomials".Computer J.,Vol.15,No.1,71-79,1972.
[Forsey 1988]David Forsey and Richard H.Bartels."Hierarchical B-spline refinement".Computer Graphics,Vol.22,No.4,205-212,1988.
[Franke 80]R.Pranke,G.Nielson.Smooth interpolation of large sets of scattered data.International Journal of Numberical Methods in Engineering,vol.15,1980,1691-1704.
[Garrity & Warren 1991]T.Garrity and J.Warren."Geometric continuity".CAD,Vol.8,51-56,1991.
[Gordon & Riesenfeld 1974a]W.J.Gordon and R.F.Riesenfcld."Bernstein-Bezier methods for the computer-aided design of free form curves and surfaces".J.ACM,Vol.21,No.2,293-310,1974.
[Gordon & Riesenfeld 1974b]W.J.Gordon and R.F.Riesenfeld."B-spline curves and surfaces".In Computer Aided Geometric Design,R.E.Barnhill and R.F.Riesenfeld(eds.),95-126,Academic Press,1974.
[Greiner & Hormann 1997]G.Greiner and K.Hormann."Interpolation and approxima tion scattered 3D data with hierarchical tensor product B-spline".In Surface Fitting and Multiresolution Methods,A.Le Mehaute,C.Rabut,and L.L.Schumaker(eds.),163-177,Vanderbilt University Press,1997.
[郭竹瑞1990]贾荣庆.多元样条研究中的B网方法.数学进展,Vol.19,189-198,1990.
[Hayes 1974]J.C.Hayes and J.Halliday."The least squares fitting of cubic spline surfaces to general data sets".J.Math.Appl.,Vol.14,89-113,1974.
[Hoffmann 1989]C.Hoffmann.Geometric and Solid Modeling:An Introduction.San Francisco,Morgan Kaufmann,1989.
[H.Ipson 2005]H.Ipson."T-spline Merging".Master's thesis,Brigham Young University.2005.
[贾荣庆1987a]贾荣庆.箱样条研究的新进展.高校应用数学学报,Vol.2,330-343.1987.
[贾荣庆1987b]贾荣庆.多元样条的B网表示.科学通报,Vol.11,804-807,1987.
[Katz 1988]S.Katz,T.W.Sederberg."Genus of the intersection curve of two rational surface patches".In Computer Aided Geometric Design,Vol 5,253-258,1988.
[Krishnamurthy 1996]V.Krishnamurthy,and M.Levoy."Fitting smooth surfaces to dense polygon meshes".In SIGGRAPH '96:Proceedings of the 23rd annual conference on Computer graphics and interactive techniques,ACM Press,New York,NY,USA.313-324,1996.
[Lai 1996]M.J.Lai."Scattered data interpolation and approximation using bivariate C~1piecewise cubic polynomials".CAGD,Vol.13,81-88,1996.
[Lai 2007]S.Lai,F.Cheng."Robust and error controllable boolean operations on freeform solids represented by catmullclark subdivision surfaces".Computer Aided Design and Applications,Vol.4,2007.
[Litke etal.2001]N.Litke,A.Levin and P.Schroder."Trimming for subdivision surfaces".CAGD,Vol.18,463-481,2001.
[Loop 2004]C.Loop."Second order smoothness over extraordinary vertices".In Eurographics /ACM SIGGRAPH Symposium on Geometry Processing,165-174,2004.
[Peter 2000]Jorg Peter."Patching Catmull-Clark Meshes",Siggraph 2000,2000.
[Prautzsch 1984a]H.Prautzsch."Degree elevation of B-spline curves".CAGD,Vol.1,No.2,193-198,1984.
[Riesenfeld 1975]R.F.Riesenfeld.On Chaikin's algorithm.IEEE Computer Graphics and Applications,Vol.4,304-310,1975.
[Sabin 1976]M.A.Sabin.The use of piecewise form for the numerical representation of shape.Ph.D.Thesis,MTA Budapest,Hungary,1976.
[Schoenberg 1946]I.J.Schhoenberg."Contributions to the problem of approximation of equidistant data by analytic functions".Quart.Appl.Math.,Vol.4,No.1,44-99,1946.
[Schumaker 1981]L.L.Schumaker.Spline functions:Basic theory,John Wiley & Sons,Inc.,1981.
[Sederberg 1984]T.W.Sederberg."Piecewise Algebraic Curves".CAGD,Vol.1,241-255,1984.
[Sederberg 1984]T.W.Sederberg,D.Anderson,and R.Goldman."Implicit representation of parametric curves and surfaces".Computer Vision,Graphics and Image Processing,Vol.28,72-84,1984.
[Sederberg 1998]Thomas W.Sederberg,Jianmin Zheng,David Sewell,and Malcolm Sabin.Non-uniform recursive subdivision surfaces.Computer Graphics (SIGGRAPH'98 Proceedings),387-394,1998.
[Sederberg 2003]T.W.Sederberg,Jianmin Zheng and Xiaowen Song.Knot intervals and multi-degree splines.Computer Aided Geometric Design,Vol.20,No.7,455-468.2003.
[Sederberg 2003]Thomas W.Sederberg,Jianmin Zheng,Almaz Bakenov,and Ahmad Nasri.T-splines and T-NURCCs.ACM Transactions on Graphics,Vol.22,No.3,161-172,2003.
[Sederberg 2004]Thomas W.Sederberg,David L.Cardon,G.Thomas Finnigan,Nicholas S.North,Jianmin Zheng,and Tom Lyche.T-spline simplification and local refinement.ACM Transactions on Graphics,Vol.23,No.3,276-283,2003.
[Shewchuk 2002]Shewchuk,J.R."Delaunay refinement algorithms for triangular mesh generation".Computational Geometry:Theory and Applications 22,1-3,21-74,2002.
[施法中1994]施法中.计算机辅助几何设计与非均匀有理B样条.北京航天航空大学出版社,1994.
[Song 2004]X.Song,T.W.Sederberg,J.Zheng,R.T.Farouki,and J.Hass."Linear perturbation methods for topologically consistent representations of free-form surface intersections".Computer AIded Geometric Design,Vol 21,303-319,2004.
[Tur99]G.Truk and J.F.O'Brien.Shape transformation using variational implicit functions.In:Siggraph'99 Conference Proceedings,pages 335-342,1999.
[Tur02]G.Turk,J.F.O'Brien.Modelling with implicit surfaces that interpolate.ACM Transaction on Graphics,vol.21,855-873,2002.
[Wang W.2006]W.Wang,H.Pottmann,and Y.Liu."Fitting B-spline curves to point clouds by curvature-based squared distance minimization".ACM Transactions on Graphics,Vol25,214-238,2006.
[Wang 2005a]Y Wang,J Zheng,H.Soon Seah."Conversion between T-Splines and Hierarchical B-Splines".Proceedings of the eighth IASTED International Conference Computer Graphics and Imaging.8-13,2005.
[Weller 1995]Frank Weller and Hans Hagen.Tensor-product spline spaces with knot segments.in Mathematical Methods for Curves and Surfaces,M.Daehlen,T.Lyche,and L.L.Schumaker(eds.),Vanderbilt Unversity Press,563-572,1995.
[Wu & Wang 1994]W.T.Wu and D.K.Wang."On the algebraic surface - fitting problem in CAGD" Mathematics in Practice and Theory,No.3,26-31,1994.
[J.Zhang 1998]J.Zhang,Z.Tang,and J.Li."Adaptive hierarchical B-spline surface approxination of large-scale scattered data".Pacific Graphics.8-16,1998.
[Zheng 2005]J Zheng,Y Wang,H.Soon Seah."Adaptive T-Spline Surface Fitting to ZMap Models".Proceedings of the 3rd international conference on Computer graphics and interactive techniques in Australasia and South East Asia 405-411,2005.
[Li 2006]Wan-Chiu Li,Nicholas Ray,Bruno Levy."Automatic and interactive mesh to T-spline conversion".Proceedings of the fourth Eurographics symposium on Geometry processing.191-200,2006.
[周建伟1986]周建伟.三角域上Bernstein多项式的正性与凸性.计算数学,Vol.8,185-190,1986.
[朱心雄2000]朱心雄.自由曲线曲面造型技术.北京:科学出版社,2000.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |