CAGD中对偶基与几何逼近问题的应用研究
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摘要
计算机辅助几何设计,简称CAGD(Computer Aided Geometric Design),是随着航空、造船、机械设计和制造等现代工业的蓬勃发展与计算机的出现而发生与发展起来的一门新兴的学科。其中自由曲线、曲面造型是其重要内容。近些年来,新的曲线不断被提出,如Wang-Ball曲线,Said-Ball曲线,SBGB曲线,WSGB曲线和WBGB曲线等。这些曲线与CAGD中使用最为广泛的Bezier曲线相比大多也具有端点切触性,凸包性,保形性等。为了实现不同造型系统或图形系统的数据交换,也为了获得不同曲线再同一系统中混合使用,以及进行拼接、求导、绘制等几何操作的一致性,有必要研究不同曲线之间的转换,对偶基就是实现曲线在各种不同的基下相互转换时所普遍采用的工具。曲线/曲面的逼近与表示是CAGD中的两大基本理论问题,其中,降阶逼近,等距逼近,有理曲线/曲面的多项式逼近由于直接关系到几何设计系统的效率,精度,质量和功能已经成为当前的研究热点。有鉴于此,本文针对对偶基和几何逼近若干问题展开了较为深入的研究,主要的研究工作及成果如下:
·在对偶基的理论和应用方面
(1)构造了NS幂基和WBGB基的对偶基,并利用对偶基给出了NS幂基和WBGB基下的Marsden恒等式,实现了Bezier曲线到NS幂基曲线、WBGB曲线的转换,为充分利用B6zier曲线和NS幂基曲线,WBGB曲线各自的优点提供了理论基础。
(2)通过引入一组参数K,L,本文进一步研究了广义Ball曲线的统一表示,给出了Bezier-Said-Wang型广义Ball曲线(BSWGB曲线),这族曲线将Said-Ball曲线,Wang-Ball曲线,WSGB曲线,SBGB曲线和WBGB曲线统一的表示出来,使得上述广义Ball曲线成为BSWGB曲线族的特例。本文还使用对偶基作为工具对BSWGB基做了进一步的研究,推导出它们的对偶基公式,使得文献([奚9]],[OG97],[江04],[Wu04],[蒋04a],[蒋04b],[JWT06],[ZWT09b])的对偶基成为本文的特例。同时也解决了实际中的两个问题:1)推导出一般幂基的BSWGB基表示,也即相应的Marsden恒等式;2)推导出Bemstein基到BSWGB基的转换公式。
(3)通过引入两向量函数内积矩阵的概念和运算,给出了SBGB基的带权对偶基函数的显式表达式,并得到了它的满足边界约束条件的带权对偶基函数。本文还给出了SBGB基的积分形式的对偶泛函,提出了一种用SBGB基表示的、满足一定插值条件的多项式作平方可积函数最小二乘逼近的直接解法。本文的结果包含了带权的Bernstein基,Said-Ball基和一些中间基函数的对偶基,并使得文献([J(u|¨)t98],[RA07],[RA08])的结果成为本文的特例。这些结果对于研究SBGB基的理论和推广它的应用将起一定的作用。利用上述结果,可以类似讨论带权的WBGB基,WSGB基和BSWGB基的对偶基函数。作为对带权对偶基函数的应用,本文还针对平面Bezier曲线的等距曲线,给出了相应的逼近算法。
●在S幂基的应用方面
S幂基函数拥有着良好的数值性质,它与Bernstein基的转换矩阵是非病态矩阵。S幂基不仅保留了幂基函数形式简单、易于计算的优点(满足Horner嵌套算法),并且采用该幂基的多项式曲线的系数矢量具有明显的几何意义,可以作为形状操作工具。最重要的是这种幂基曲线对曲线的两个端点都能做到保端点高阶连续,特别有利于曲线/曲面的分段表示。Sanchez-Reyes对一元S幂基做了非常完备的讨论,并简要的介绍了二元S幂基。在此基础上,本文详细的讨论了二元S幂基的除法运算和求平方根运算。作为对二元S幂基的应用,本文还给出了张量积Bezier曲面的降阶,有理Bezier曲面的多项式逼近,Bezier曲面的等距逼近的二元S幂基算法。算法表明:使用二元S幂基多项式作为工具的逼近算法复杂度低,仅仅涉及到多项式的加法,减法和乘法,并在曲面的四个角点保高阶插值,无须添加额外的约束条件。
·在等距曲线/曲面的逼近方面
(1)采用带重节点的“两点式“Newton插值算法,得到了等距曲线的多项式逼近算法和有理逼近算法,算法在曲线的两个端点保端点高阶插值,特别适合曲线的分段表示。适当升高多项式的阶数并结合离散算法,我们可以轻松的提高逼近精度,同时在离散点处保高阶插值。
(2)采用修正的Thiele型插值算法,得到了等距曲线的有理逼近算法,并进一步考虑了保端点高阶插值的有理逼近算法,算法中每个系数的求解只涉及到乘法,除法运算,算法复杂度为D(n~2),较Li算法(复杂度为D(n~3))有了较大的提高。此外采用修正的二元Newton-Thiele型混合插值算法,得到了等距曲面的有理逼近算法,算法中每个系数的求解只涉及到乘法,除法运算。
Computer Aided Geometric Design(CAGD) is an emerging discipline along with the appearance of computers and the flourish of modern industrial such as aviation,shipbuilding,design and manufacturing of machine.Approximation and representation of curves and surfaces are the two fundamental theoretical themes in CAGD.In recent years,new curves have constantly been raised such as Wang-Ball curve,Said-Ball curve,SBGB curve,WSGB curve and WBGB curve.Compared with the most widely used Bezier curves,these curves also inherit the properties such as endpoints' property,convex hull property and shape preservation.In order to realize data exchange among different modeling systems and graphics systems,also in order to achieve mixed use of different curves in the same system and the consistency of different kinds of geometric operations such as splicing,derivative and drawing,we need to study conversions between different curves.The dual bases are widely applied to mutual transformation between different bases for the same curve. Therein degree reduction approximation,offset approximation and polynomial approximation of rational curves/surfaces have become the hotspots of investigation,since they directly relate to the efficiency,precision,quality and function of geometric design systems.In view of these facts,the dissertation gives a more in-depth discussion about the theories and applications of dual bases and geometric approximations.The main results in this dissertation are outlined as follows:
●In respect of theories and applications of dual basis
(1) Dual bases of NS-power basis and WBGB basis are constructed.By means of dual bases of NS-power basis and WBGB basis,we also obtained the Marsden identity and realized the conversions from Bezier curve to NS power curve and WBGB curve.These results are very useful for the applications of NS curve and WBGB curve and their popularizations in Computer Aided Geometric Design.
(2) By introducing a set of parameters K,L,we further investigated the unifying representation of generalized Ball basis and presented Bezier-Said-Wang type generalized Ball curve (BSWGB curve).BSWGB basis unifies Said-Ball basis,Wang-Ball basis,SBGB basis,WSGB basis and WBGB basis and each of the latter five bases is the special case of the new one.We also used dual basis as a tool to further investigate BSWGB basis and deduce the dual basis of it. Accordingly,we discussed the dual bases of some paper as special cases.We also solve the two problems in practice:1) deducing the BSWGB basis representation of the power basis,namely, Marsden identity,2) deducing the conversion matrices from Bernstein basis to BSWGB basis.
(3) By introducing the definition and operations of the inner-product matrix of two vector functions,explicit formulas for the dual basis functions of Said-Bezier type generalized Ball basis (SBGB) with respect to the Jacobi weight function are given.The dual basis functions satisfying boundary constraints are also considered.We also gave integral dual functional of SBGB basis and presented a direct solution of the least squares approximation polynomials satisfying interpolating conditions of functions by means of SBGB basis.As a result,the dissertation includes the weighted dual basis functions of Bernstein basis,Said-Ball basis and some intermediate bases and discusses the dual bases of paper([J(u|¨)t98],[RA07],[RA08]) as special cases.These results are very useful for the study of SBGB basis and wide range of application of SBGB basis.Using the above results, we can discuss the weighted dual basis functions of WBGB,WSGB,and BSWGB basis similarly. As its applications,the dissertation also gave the approximation algorithms of Bezier curves' offsets.
●In respect of the applications of S-power bases
S-power basis possesses good numerical condition,the conversion matrices between the Bernstein basis and the S-power basis are not ill-conditioned.S-power basis preserves the advantages of the power form such as simple form,easy to calculate(admits a Horner's evaluation scheme) and the coefficients of it obviously admit geometrical interpretation and can be used as shape control tools.The most important thing is S-power curve can keep high continuities on two endpoints automatically and especially suited for the piecewise representations of curves and surfaces.Sanchez-Reyes gave detailed studies to unitary S-power basis and gave a brief introduction to bivariate one.Based on these,we gave detailed discussions of division operations and extraction function of bivariate S-power basis.As the applications to bivariate S-power basis,we studied degree reduction of tensor product Bezier surfaces,polynomial approximation of rational Bezier surfaces and offset approximation of Bezier surfaces.The algorithms show that the complexity is low by using bivariate S-power basis,which only involves in additions,subtractions and multiplications,and the approximation surface automatically interpolates to the four corner points of the original one.
●In respect of the applications of S-power bases
(1) We obtained rational and polynomial approximation algorithms of offset curves by using "two points" Newton interpolation with confluents.The algorithms could keep high continuities on two endpoints and especially suited for the piecewise representation of curves.Increasing the orders of the polynomials and combining with subdivision algorithm,we can improve approximation precision easily and the curve could keep high continuities on subdivision points.
(2) We obtained rational approximation algorithms of offset curves by using modified Thiele type interpolation.We also further considered rational approximation algorithms satisfying endpoints constraints.Our algorithms only involved multiplication,division and their complexity was O(n~2).It had a better improvement,compared with Li's algorithm(its complexity:O(n~3)). We also obtained rational approximation algorithms of offset surfaces by using modified bivariate Newton-Thiele blending interpolation algorithms and the algorithms only involved multiplication and division.
·在对偶基的理论和应用方面
(1)构造了NS幂基和WBGB基的对偶基,并利用对偶基给出了NS幂基和WBGB基下的Marsden恒等式,实现了Bezier曲线到NS幂基曲线、WBGB曲线的转换,为充分利用B6zier曲线和NS幂基曲线,WBGB曲线各自的优点提供了理论基础。
(2)通过引入一组参数K,L,本文进一步研究了广义Ball曲线的统一表示,给出了Bezier-Said-Wang型广义Ball曲线(BSWGB曲线),这族曲线将Said-Ball曲线,Wang-Ball曲线,WSGB曲线,SBGB曲线和WBGB曲线统一的表示出来,使得上述广义Ball曲线成为BSWGB曲线族的特例。本文还使用对偶基作为工具对BSWGB基做了进一步的研究,推导出它们的对偶基公式,使得文献([奚9]],[OG97],[江04],[Wu04],[蒋04a],[蒋04b],[JWT06],[ZWT09b])的对偶基成为本文的特例。同时也解决了实际中的两个问题:1)推导出一般幂基的BSWGB基表示,也即相应的Marsden恒等式;2)推导出Bemstein基到BSWGB基的转换公式。
(3)通过引入两向量函数内积矩阵的概念和运算,给出了SBGB基的带权对偶基函数的显式表达式,并得到了它的满足边界约束条件的带权对偶基函数。本文还给出了SBGB基的积分形式的对偶泛函,提出了一种用SBGB基表示的、满足一定插值条件的多项式作平方可积函数最小二乘逼近的直接解法。本文的结果包含了带权的Bernstein基,Said-Ball基和一些中间基函数的对偶基,并使得文献([J(u|¨)t98],[RA07],[RA08])的结果成为本文的特例。这些结果对于研究SBGB基的理论和推广它的应用将起一定的作用。利用上述结果,可以类似讨论带权的WBGB基,WSGB基和BSWGB基的对偶基函数。作为对带权对偶基函数的应用,本文还针对平面Bezier曲线的等距曲线,给出了相应的逼近算法。
●在S幂基的应用方面
S幂基函数拥有着良好的数值性质,它与Bernstein基的转换矩阵是非病态矩阵。S幂基不仅保留了幂基函数形式简单、易于计算的优点(满足Horner嵌套算法),并且采用该幂基的多项式曲线的系数矢量具有明显的几何意义,可以作为形状操作工具。最重要的是这种幂基曲线对曲线的两个端点都能做到保端点高阶连续,特别有利于曲线/曲面的分段表示。Sanchez-Reyes对一元S幂基做了非常完备的讨论,并简要的介绍了二元S幂基。在此基础上,本文详细的讨论了二元S幂基的除法运算和求平方根运算。作为对二元S幂基的应用,本文还给出了张量积Bezier曲面的降阶,有理Bezier曲面的多项式逼近,Bezier曲面的等距逼近的二元S幂基算法。算法表明:使用二元S幂基多项式作为工具的逼近算法复杂度低,仅仅涉及到多项式的加法,减法和乘法,并在曲面的四个角点保高阶插值,无须添加额外的约束条件。
·在等距曲线/曲面的逼近方面
(1)采用带重节点的“两点式“Newton插值算法,得到了等距曲线的多项式逼近算法和有理逼近算法,算法在曲线的两个端点保端点高阶插值,特别适合曲线的分段表示。适当升高多项式的阶数并结合离散算法,我们可以轻松的提高逼近精度,同时在离散点处保高阶插值。
(2)采用修正的Thiele型插值算法,得到了等距曲线的有理逼近算法,并进一步考虑了保端点高阶插值的有理逼近算法,算法中每个系数的求解只涉及到乘法,除法运算,算法复杂度为D(n~2),较Li算法(复杂度为D(n~3))有了较大的提高。此外采用修正的二元Newton-Thiele型混合插值算法,得到了等距曲面的有理逼近算法,算法中每个系数的求解只涉及到乘法,除法运算。
Computer Aided Geometric Design(CAGD) is an emerging discipline along with the appearance of computers and the flourish of modern industrial such as aviation,shipbuilding,design and manufacturing of machine.Approximation and representation of curves and surfaces are the two fundamental theoretical themes in CAGD.In recent years,new curves have constantly been raised such as Wang-Ball curve,Said-Ball curve,SBGB curve,WSGB curve and WBGB curve.Compared with the most widely used Bezier curves,these curves also inherit the properties such as endpoints' property,convex hull property and shape preservation.In order to realize data exchange among different modeling systems and graphics systems,also in order to achieve mixed use of different curves in the same system and the consistency of different kinds of geometric operations such as splicing,derivative and drawing,we need to study conversions between different curves.The dual bases are widely applied to mutual transformation between different bases for the same curve. Therein degree reduction approximation,offset approximation and polynomial approximation of rational curves/surfaces have become the hotspots of investigation,since they directly relate to the efficiency,precision,quality and function of geometric design systems.In view of these facts,the dissertation gives a more in-depth discussion about the theories and applications of dual bases and geometric approximations.The main results in this dissertation are outlined as follows:
●In respect of theories and applications of dual basis
(1) Dual bases of NS-power basis and WBGB basis are constructed.By means of dual bases of NS-power basis and WBGB basis,we also obtained the Marsden identity and realized the conversions from Bezier curve to NS power curve and WBGB curve.These results are very useful for the applications of NS curve and WBGB curve and their popularizations in Computer Aided Geometric Design.
(2) By introducing a set of parameters K,L,we further investigated the unifying representation of generalized Ball basis and presented Bezier-Said-Wang type generalized Ball curve (BSWGB curve).BSWGB basis unifies Said-Ball basis,Wang-Ball basis,SBGB basis,WSGB basis and WBGB basis and each of the latter five bases is the special case of the new one.We also used dual basis as a tool to further investigate BSWGB basis and deduce the dual basis of it. Accordingly,we discussed the dual bases of some paper as special cases.We also solve the two problems in practice:1) deducing the BSWGB basis representation of the power basis,namely, Marsden identity,2) deducing the conversion matrices from Bernstein basis to BSWGB basis.
(3) By introducing the definition and operations of the inner-product matrix of two vector functions,explicit formulas for the dual basis functions of Said-Bezier type generalized Ball basis (SBGB) with respect to the Jacobi weight function are given.The dual basis functions satisfying boundary constraints are also considered.We also gave integral dual functional of SBGB basis and presented a direct solution of the least squares approximation polynomials satisfying interpolating conditions of functions by means of SBGB basis.As a result,the dissertation includes the weighted dual basis functions of Bernstein basis,Said-Ball basis and some intermediate bases and discusses the dual bases of paper([J(u|¨)t98],[RA07],[RA08]) as special cases.These results are very useful for the study of SBGB basis and wide range of application of SBGB basis.Using the above results, we can discuss the weighted dual basis functions of WBGB,WSGB,and BSWGB basis similarly. As its applications,the dissertation also gave the approximation algorithms of Bezier curves' offsets.
●In respect of the applications of S-power bases
S-power basis possesses good numerical condition,the conversion matrices between the Bernstein basis and the S-power basis are not ill-conditioned.S-power basis preserves the advantages of the power form such as simple form,easy to calculate(admits a Horner's evaluation scheme) and the coefficients of it obviously admit geometrical interpretation and can be used as shape control tools.The most important thing is S-power curve can keep high continuities on two endpoints automatically and especially suited for the piecewise representations of curves and surfaces.Sanchez-Reyes gave detailed studies to unitary S-power basis and gave a brief introduction to bivariate one.Based on these,we gave detailed discussions of division operations and extraction function of bivariate S-power basis.As the applications to bivariate S-power basis,we studied degree reduction of tensor product Bezier surfaces,polynomial approximation of rational Bezier surfaces and offset approximation of Bezier surfaces.The algorithms show that the complexity is low by using bivariate S-power basis,which only involves in additions,subtractions and multiplications,and the approximation surface automatically interpolates to the four corner points of the original one.
●In respect of the applications of S-power bases
(1) We obtained rational and polynomial approximation algorithms of offset curves by using "two points" Newton interpolation with confluents.The algorithms could keep high continuities on two endpoints and especially suited for the piecewise representation of curves.Increasing the orders of the polynomials and combining with subdivision algorithm,we can improve approximation precision easily and the curve could keep high continuities on subdivision points.
(2) We obtained rational approximation algorithms of offset curves by using modified Thiele type interpolation.We also further considered rational approximation algorithms satisfying endpoints constraints.Our algorithms only involved multiplication,division and their complexity was O(n~2).It had a better improvement,compared with Li's algorithm(its complexity:O(n~3)). We also obtained rational approximation algorithms of offset surfaces by using modified bivariate Newton-Thiele blending interpolation algorithms and the algorithms only involved multiplication and division.
引文
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