几何造型中曲面的裁剪、形状调整以及界的研究
摘要
在计算机辅助几何设计中，样条曲面是最重要的数学方法和研究内容，本文针对运用样条曲面造型时三个方面的问题，作了一定的研究．
1．样条曲面的裁剪问题．曲面的裁剪是指如何在已知曲面上提取特定的子区域，从而能对该特定子区域和余下的曲面部分进行更进一步的操作．样条曲面的裁剪在曲面的细分、拼接等方面有着重要的应用．本文借助开花这一研究样条的有力工具，首先对基于直线的裁剪问题给出了一种统一模式的解决方法．其次，将基于直线的裁剪推广到基于多项式曲线的裁剪，使得裁剪的方法更灵活，裁剪的区域更自由．方法不仅对多项式形式的曲面适用，而且对有理形式的曲面也适用．
2．针对经典的B(?)zier曲线曲面在控制顶点给定，曲线曲面形状就唯一固定，缺乏进一步调整曲线曲面形状的能力这一情况，本文给出了一种带多个形状参数的广义B(?)zier曲线曲面．广义B(?)zier曲线曲面保留了经典B(?)zier曲线曲面的优良性质，并可以通过形状参数的取值变化来进一步调整曲线曲面的形状．形状参数具有直观明确的几何意义，调整曲线曲面形状的方式可以预先估计和精确计算．并且对广义B(?)zier曲线曲面如何拼接成样条曲线曲面作了分析．
3．关于样条曲面的界的问题．样条曲线曲面的界指的是相对于某一确定的位置，通过求出曲线曲面与该确定位置的距离，从而确定曲线曲面在拓扑上所处于的一个尽可能小而且精确的范围．关于B(?)zier和B-样条曲线曲面的界已经有了一定的成果，本文针对在三边曲面造型中影响极大的三向四次箱样条曲面，通过控制顶点的方向差分，求出了曲面片与控制网格的中心三角平面片的距离，从而刻画了三向四次箱样条曲面的界．
Spline surfaces are the main methods and core contents in CAGD. The following three problems arising from using spline surfaces in geometric modeling are researched in this dissertation.
1. The trimming of spline surface. Trimming of surface means how to refine a subdomain of a given surface for further processing of this subdomain and the left parts. Many problems such as subdivision and connection of surfaces can be regarded as the special cases of trimming. By means of blossoms, one united method is given for the problem of trimming with lines firstly. Then this method is generalized to trimming with polynomial curves, which makes more choices for the shape of trimming subdomain. The method is working for both polynomial surfaces and rational cases.
2. When the control points are given, the shape of classical Bezier curve or surface is uniquely decided. For remedying this cases, the generalized Bezier curve and surface with multi-shape-parameters are given, which not only hold the excellent properties of classical Bezier curve or surface, but also can changing the shape of the curve or surface with the shape parameters. The geometric significance of shape parameters is clearly, the modifying style can be foreseen and evaluated accurately. And the connecting of generalized Bezier curves and surfaces are also discussed.
3. The bound of spline surface means how to evaluate the distance from the surface to a fixed position for deciding a small and accurate topologic bound of this surface. The bounds of Bezier and B-Spline surfaces have some researched achievements. This thesis gives a bound of three direction quartic box spline surface which is well used in geometric modeling with triangular surfaces. The distance from the surface to the central triangular plane patch over control net is described with the direction differences of control points.
1．样条曲面的裁剪问题．曲面的裁剪是指如何在已知曲面上提取特定的子区域，从而能对该特定子区域和余下的曲面部分进行更进一步的操作．样条曲面的裁剪在曲面的细分、拼接等方面有着重要的应用．本文借助开花这一研究样条的有力工具，首先对基于直线的裁剪问题给出了一种统一模式的解决方法．其次，将基于直线的裁剪推广到基于多项式曲线的裁剪，使得裁剪的方法更灵活，裁剪的区域更自由．方法不仅对多项式形式的曲面适用，而且对有理形式的曲面也适用．
2．针对经典的B(?)zier曲线曲面在控制顶点给定，曲线曲面形状就唯一固定，缺乏进一步调整曲线曲面形状的能力这一情况，本文给出了一种带多个形状参数的广义B(?)zier曲线曲面．广义B(?)zier曲线曲面保留了经典B(?)zier曲线曲面的优良性质，并可以通过形状参数的取值变化来进一步调整曲线曲面的形状．形状参数具有直观明确的几何意义，调整曲线曲面形状的方式可以预先估计和精确计算．并且对广义B(?)zier曲线曲面如何拼接成样条曲线曲面作了分析．
3．关于样条曲面的界的问题．样条曲线曲面的界指的是相对于某一确定的位置，通过求出曲线曲面与该确定位置的距离，从而确定曲线曲面在拓扑上所处于的一个尽可能小而且精确的范围．关于B(?)zier和B-样条曲线曲面的界已经有了一定的成果，本文针对在三边曲面造型中影响极大的三向四次箱样条曲面，通过控制顶点的方向差分，求出了曲面片与控制网格的中心三角平面片的距离，从而刻画了三向四次箱样条曲面的界．
Spline surfaces are the main methods and core contents in CAGD. The following three problems arising from using spline surfaces in geometric modeling are researched in this dissertation.
1. The trimming of spline surface. Trimming of surface means how to refine a subdomain of a given surface for further processing of this subdomain and the left parts. Many problems such as subdivision and connection of surfaces can be regarded as the special cases of trimming. By means of blossoms, one united method is given for the problem of trimming with lines firstly. Then this method is generalized to trimming with polynomial curves, which makes more choices for the shape of trimming subdomain. The method is working for both polynomial surfaces and rational cases.
2. When the control points are given, the shape of classical Bezier curve or surface is uniquely decided. For remedying this cases, the generalized Bezier curve and surface with multi-shape-parameters are given, which not only hold the excellent properties of classical Bezier curve or surface, but also can changing the shape of the curve or surface with the shape parameters. The geometric significance of shape parameters is clearly, the modifying style can be foreseen and evaluated accurately. And the connecting of generalized Bezier curves and surfaces are also discussed.
3. The bound of spline surface means how to evaluate the distance from the surface to a fixed position for deciding a small and accurate topologic bound of this surface. The bounds of Bezier and B-Spline surfaces have some researched achievements. This thesis gives a bound of three direction quartic box spline surface which is well used in geometric modeling with triangular surfaces. The distance from the surface to the central triangular plane patch over control net is described with the direction differences of control points.
引文
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[3]Schumaker L.L..Spline Functions:Basic Theory[M].New York:John Wiley&Sons,1981.
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[7]孙家昶．样条函数和计算几何[M]．北京：科学出版社，1982．
[8]施法中．计算机辅助几何设计与非均匀有理B样条(CAGD&NURBS)[M]．北京：高等教育出版社，2001．
[9]朱心雄，等．自由曲线曲面造型技术[M]．北京：科学出版社，2000．
[10]王仁宏，等．计算几何教程[M]．北京：科学出版社，2008．
[11]Forrest A.R..Interactive interpolation and approximation by B(?)zier polynomials[J].Computer Journal,1972,15(1):71-79.
[12]Gordan W.J.,Riesenfeld R.F..B-Spline curves and surfaces.In Barnhill R.E.and Riesenfeld R.F eds..Computer Aided Geometric Design[M].New York:Academic Press,95-126,1974.
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[14]Boehm W.,Farin G.,Kahmann,J..A survey of curve and surface methods in CAGD[J].Computer Aided Geometric Design,1984,1(1):1-60.
[15]de Boor C.,DeVore R..Approximation by smooth multivariate splines[J],Trans.Amer.Math.Soc,1983(276):775-788.
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[18]Goldman R.N..Pyramid Algorithms:A dynamic Programming Approach to Curves and Surfaces for Geometric Modeling[M].San Francisco:Morgan Kaufmann,2002.(戈德曼．金字塔算法-曲线曲面几何模型的动态编程处理[M]．吴宗敏等，译．北京：电子工业出版社，2004)
[19]Ramshaw L..Blossoming:A connect-the-dots approach to splines[R].Systems Research Center Reports 19,Palo Alto,Califorinia,1987.
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[21]Farin G..Curves and Surfaces in Computer Aided Geometric Design[M].5rd ed,San Francisco:Morgan Kaufmann,2001.(法林．CAGD曲线与曲面[M]．李双喜，译.北京：科学出版社， 2006．)
[22]Gallier J..Curves and Surfaces in Geometric Modeling:Theorey and Algorithms[M].San Francisco:Morgan Kaufmann,2000.
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[24]Goldman R.N.,Filip D..Conversion from B(?)zier rectangles to B(?)zier triangles[J].Computer-Aided Design,1987(19):25-27.
[25]Hu S.M..Conversion of a triangular B(?)zier patch into three rectangular B(?)zier patches[J].Computer Aided Geometric Design,1996(13):219-226.
[26]Hu S.M..Conversion between triangular and rectangular B(?)zier patches[J].Computer Aided Geometric Design,2001(18):667-671.
[27]Hu S.M.,Wang G.Z.,Jin,T.G..Generalized subdivision of B(?)zier surfaces[J].Graphical Models and Image Processing,1996(58):218-222.
[28]Feng J.,Peng Q..Functional compositions via shifting operators for B(?)zier patches and their applications[J].Journa of Software,1999(10):i316-1321.
[29]Goldman R.N.On the algebraic and geometric foundations of computer graphics[J].ACM Transactions on Graphics,2002(21):52-86.
[30]胡事民，孙家广．B(?)zier曲面的广义离散及应用[J]．计算机学报，1999，22(3)：290-295．
[31]胡事民，孙家广．基于广义离散分解trimmed曲面[J]．计算机学报，1999，22(3)：297-301．
[32]Lasser D..Tensor product B(?)zier surfaces on triangle B(?)zier surfaces.Computer Aided Geometric Design,2002(19),625-643.
[33]Juhasz I.,Hoffmann M..Modifying a knot of B-spline curves[J].Computer Aided Geometric Design,2003(20):243-245.
[34]Juhasz I.,Hoffmann M..Constrained shape modification of cubic B-spline curves by means of knots[J].Computer-Aided Design,2004(36):437-445.
[35]Piegl L..Modifying the shape of rational B-splines[J].Computer-Aided Design,1989(21):509-518.
[36]Chen Q.,Wang G.Z..A class of B~zier-like curves[J].Computer Aided Geometric Design,2003(20):29-39.
[37]Wang W.T.,Wang,G.Z..Bezier curve with shape parameter[J].J.of Zhejiang Univ.Sci.,2005(6):497-501.
[38]Han X.L..Quadratic trigonometric polynomial curves with a shape parameter[J].Computer Aided Geometric Design,2002(19):503-512.
[39]Han X.L..Cubic trigonometric polynomial curves with a shape parameter[J].Computer Aided Geometric Design,2004(21):535-548.
[40]Zhang J..C-curves:An extension of cubic curves[J].Computer Aided Geometric Design,1996(13):199-217.
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