组合参数曲线磨光及自由变形的动态研究
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摘要
本文论述了二个方面的内容：一是组合参数曲线的变形。二是组合参数曲线的磨光。对于贝齐尔曲线，由于它采用一组独特的基函数，所以它具有良好的优良性质。在实践中，它表现了强大的生命力，用途广泛。
另外，我们经常会遇到一个普遍问题，就是难以用单一的贝齐尔曲线段描述复杂的形状。贝齐尔方法对形状的定义是整体方案，欲对其作局部修改，必然会影响到整体，即贝齐尔方法是有整体控制性质，但却缺乏局部控制性质。通过任意分割和任意升阶得出所要变形的一段，以及增加控制点，增加对曲线进行控制的潜在灵活性。这篇论文讨论贝齐尔曲线局部自由变形的问题。结果是使变形的曲线可以不断进行变形，得到所需的图形。然而，无论是基于伸缩因子的自由变形还是基于任意分割参数曲线的自由变形都会使参数曲线的光滑度降低。这样，本文又讨论了参数曲线的光滑变形问题，解决的方法就是通过参数曲线的磨光法实现参数曲线的光滑度。
综上所述：本论文的目的就是讨论组合参数曲线的自由光滑拼接。
The thesis discusses two respects of contents: One is deformation of combination of parametric curve.The other is smoothing of combination of parametric curve.Bezier curve has good character ,because it uses a sequnce of special basic function . Bezier curve has strong ability in practice and is widely used .
In addition ,we often encounter a general problem-it is very difficult to describe a complex shape with a single Bezier curve precise .Method of Bezier definites shape of curve ,which is global project .If you locally modify curve ,it is necessary to affact the global .It is that method of Bezier has character of global control,but it has not charater of local control .By way of arbitrary partition .arbitrary degree elevation ,that may increases control point .increases potential active character of controling curve.Therefore, the thesis discusses mainly the problem of local free-form deformation ,which may make Bezier curve deform continuely and gain wanted figures.However, no matter what free-form deformation based on extension factor for parametric curve or arbitrary partition will reduce smoothness of parametric curve. Therefore.the thesis discusses the problem of smooth deformation parametric.Solving method is to realize smooth merging of parametric curve by smoothing parametric curve.
To sum up ,the purpose of the thesis is that it discusses freely-smooth merging of combination.
另外，我们经常会遇到一个普遍问题，就是难以用单一的贝齐尔曲线段描述复杂的形状。贝齐尔方法对形状的定义是整体方案，欲对其作局部修改，必然会影响到整体，即贝齐尔方法是有整体控制性质，但却缺乏局部控制性质。通过任意分割和任意升阶得出所要变形的一段，以及增加控制点，增加对曲线进行控制的潜在灵活性。这篇论文讨论贝齐尔曲线局部自由变形的问题。结果是使变形的曲线可以不断进行变形，得到所需的图形。然而，无论是基于伸缩因子的自由变形还是基于任意分割参数曲线的自由变形都会使参数曲线的光滑度降低。这样，本文又讨论了参数曲线的光滑变形问题，解决的方法就是通过参数曲线的磨光法实现参数曲线的光滑度。
综上所述：本论文的目的就是讨论组合参数曲线的自由光滑拼接。
The thesis discusses two respects of contents: One is deformation of combination of parametric curve.The other is smoothing of combination of parametric curve.Bezier curve has good character ,because it uses a sequnce of special basic function . Bezier curve has strong ability in practice and is widely used .
In addition ,we often encounter a general problem-it is very difficult to describe a complex shape with a single Bezier curve precise .Method of Bezier definites shape of curve ,which is global project .If you locally modify curve ,it is necessary to affact the global .It is that method of Bezier has character of global control,but it has not charater of local control .By way of arbitrary partition .arbitrary degree elevation ,that may increases control point .increases potential active character of controling curve.Therefore, the thesis discusses mainly the problem of local free-form deformation ,which may make Bezier curve deform continuely and gain wanted figures.However, no matter what free-form deformation based on extension factor for parametric curve or arbitrary partition will reduce smoothness of parametric curve. Therefore.the thesis discusses the problem of smooth deformation parametric.Solving method is to realize smooth merging of parametric curve by smoothing parametric curve.
To sum up ,the purpose of the thesis is that it discusses freely-smooth merging of combination.
引文
[1] 施法中，计算机辅助几何设计与非均匀有理B样条[M]，高等教育出版社。2001．3．
[2] Farin G. Curve and surfaces for computer aided geometric design: Apractical guide; Academic Press, 1988.
[3] A H Barr, Global&local deformation of solid primitives[J]. Computer Graphics, 1984,18(3); 21—30
[4] Coquillarts. Eefended free—form deformation: Asculpting tool for 3D geometricmodeling[J]. Computer Graphics, 1990, 24(4): 187—196.
[5] L Piegl. Modifying the shape of rational B—spline, Part: Cures[J], Computer Aided Design, 1989, 21(8): 509—518.
[6] C k Au, M MF Yuen. Unified approach to NURBS Curves shape modification[J]., Computer—Aided Design,1995,27(2): 85—93.
[7] 李岳生，齐东旭。样条函数方法。科学出版社[M]。1979，148—149。
[8] Wilfried T, Hartmut P.Arbitrarily high degree elevation of Bèzier representation. CAGD. 13(1996), 387—398.
[9] 王小平等，基于伸缩因子的参数曲线自由变形[J]。计算机辅助设计与图形学学报。2002。1。P66-69。
[10] Feguson, J, Multivariable Curve Interpolation. Journal of Association for Computing Machinery. 1964,11(2); P221-228.
[11] Boehm. W. Visual continuity, CAD, 20.6.1988.
[12] Manning J R.Continuity Conditions for Spline Curves, The Computer J. 17,2.
[13] Nielson G M. Some piecewise polynomial alternatives to splines under tension, In[6].
[14] Barsky B A.The Beta Spline: A local representation based on shape parameters and fundamental geometric measure, PH. D. thesis, Dept of Computer Science, Univ of Utah, Salt Lake City, Utah, 1981.
[15] Boehm. W. on the definition of geometric continuity, CAD,20,7,1988.
[16] Farin G. smooth interpolation to scattered 3D deta, In[6]
[17] Beohm W. smooth curves and surfaces,In[62].
[18] Goodman.TNT.properties of β-splines of Approx theory,44,1985.
[19] Goodman.TNT.Unsworth K.Generation of β-splines curves using a recurrence relation, In Fundamental algorithms for computer
graphics,springer-verlag,Berlin, 1985.
[20] Dannenberg L,Nowacki H.Approximate comersion of surface representations with polynominal bases. Computer-Aided Geometric Design 1985; 2: P123-31.
[21] Hoschek J. Approximate conversion of spline curves.CAGD 1988; 20: P398-405.
[22] Watkin M A. Worsey, A J.degree reduction of Bezier curves. CAGD19988; 20: 398-405
[23] Lacance M A. chebyshev economization for parametric surfaces CAGD 1988; 5: 192-208
[24] Eck M. least square degree reduction of Bezier curves CAGD 1995; 27: 845-851
[26] Begacki P, Weinstein S E, Xu Y. degree reduction of Bezier curves by uniform approximation with endpoint interpolation CAGD 1995; 27: 651-661.
[27] Hu Shi-Min, Jin Tong-Guang, approximate degree reduction of Bezier curves. Proceedings of Symposium on Computational Geometriy, Held in Hangzhou, China, 1992, P110-126.
[28] Hu Shi-Min, wangGuo-zhao, Jin Tong-Guang. properties of two types of generalized Ball curves. CAD 1996, 28: 218-222.
[29] Shi-Min Hu, Ron-Feng Tong, Tao Ju e,t.approximate merging of a pair of Bezier curves CAD 2001, 33: 125-136.
[30] 王国瑾，汪国昭，郑建民。计算机辅助几何设计[M]。高等教育出版社。2001。
[31] 苏步青，刘鼎元。计算几何。上海科学技术出版社。1981。
[32] 关履泰，罗笑南，毛明志。计算机辅助几何设计。高等教育出版社。1999。
[33] 孙家广等编著．计算机图形学(第三版)[M]．清华大学出版社．2000．4．
[34] 孙家昶．样条函数和计算几何[M]．科学出版社．1981．
[35] 詹棠森等．基于任意分割Bezier曲线局部变形磨光．江西师大学报．2003(5)
[36] 合肥工业大学计算科学教研室．数值计算方法．2002．(内部讲义)
[2] Farin G. Curve and surfaces for computer aided geometric design: Apractical guide; Academic Press, 1988.
[3] A H Barr, Global&local deformation of solid primitives[J]. Computer Graphics, 1984,18(3); 21—30
[4] Coquillarts. Eefended free—form deformation: Asculpting tool for 3D geometricmodeling[J]. Computer Graphics, 1990, 24(4): 187—196.
[5] L Piegl. Modifying the shape of rational B—spline, Part: Cures[J], Computer Aided Design, 1989, 21(8): 509—518.
[6] C k Au, M MF Yuen. Unified approach to NURBS Curves shape modification[J]., Computer—Aided Design,1995,27(2): 85—93.
[7] 李岳生，齐东旭。样条函数方法。科学出版社[M]。1979，148—149。
[8] Wilfried T, Hartmut P.Arbitrarily high degree elevation of Bèzier representation. CAGD. 13(1996), 387—398.
[9] 王小平等，基于伸缩因子的参数曲线自由变形[J]。计算机辅助设计与图形学学报。2002。1。P66-69。
[10] Feguson, J, Multivariable Curve Interpolation. Journal of Association for Computing Machinery. 1964,11(2); P221-228.
[11] Boehm. W. Visual continuity, CAD, 20.6.1988.
[12] Manning J R.Continuity Conditions for Spline Curves, The Computer J. 17,2.
[13] Nielson G M. Some piecewise polynomial alternatives to splines under tension, In[6].
[14] Barsky B A.The Beta Spline: A local representation based on shape parameters and fundamental geometric measure, PH. D. thesis, Dept of Computer Science, Univ of Utah, Salt Lake City, Utah, 1981.
[15] Boehm. W. on the definition of geometric continuity, CAD,20,7,1988.
[16] Farin G. smooth interpolation to scattered 3D deta, In[6]
[17] Beohm W. smooth curves and surfaces,In[62].
[18] Goodman.TNT.properties of β-splines of Approx theory,44,1985.
[19] Goodman.TNT.Unsworth K.Generation of β-splines curves using a recurrence relation, In Fundamental algorithms for computer
graphics,springer-verlag,Berlin, 1985.
[20] Dannenberg L,Nowacki H.Approximate comersion of surface representations with polynominal bases. Computer-Aided Geometric Design 1985; 2: P123-31.
[21] Hoschek J. Approximate conversion of spline curves.CAGD 1988; 20: P398-405.
[22] Watkin M A. Worsey, A J.degree reduction of Bezier curves. CAGD19988; 20: 398-405
[23] Lacance M A. chebyshev economization for parametric surfaces CAGD 1988; 5: 192-208
[24] Eck M. least square degree reduction of Bezier curves CAGD 1995; 27: 845-851
[26] Begacki P, Weinstein S E, Xu Y. degree reduction of Bezier curves by uniform approximation with endpoint interpolation CAGD 1995; 27: 651-661.
[27] Hu Shi-Min, Jin Tong-Guang, approximate degree reduction of Bezier curves. Proceedings of Symposium on Computational Geometriy, Held in Hangzhou, China, 1992, P110-126.
[28] Hu Shi-Min, wangGuo-zhao, Jin Tong-Guang. properties of two types of generalized Ball curves. CAD 1996, 28: 218-222.
[29] Shi-Min Hu, Ron-Feng Tong, Tao Ju e,t.approximate merging of a pair of Bezier curves CAD 2001, 33: 125-136.
[30] 王国瑾，汪国昭，郑建民。计算机辅助几何设计[M]。高等教育出版社。2001。
[31] 苏步青，刘鼎元。计算几何。上海科学技术出版社。1981。
[32] 关履泰，罗笑南，毛明志。计算机辅助几何设计。高等教育出版社。1999。
[33] 孙家广等编著．计算机图形学(第三版)[M]．清华大学出版社．2000．4．
[34] 孙家昶．样条函数和计算几何[M]．科学出版社．1981．
[35] 詹棠森等．基于任意分割Bezier曲线局部变形磨光．江西师大学报．2003(5)
[36] 合肥工业大学计算科学教研室．数值计算方法．2002．(内部讲义)