CAGD中等距线及测地线相关问题的研究
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摘要
等距线和测地线是计算机辅助几何设计中具有重要研究价值的两类基本曲线.其中,等距线的近似有理表示、逼近误差估计、低次曲线逼近,以及过测地线的有理曲面、可展曲面设计和调控由于直接关系到几何设计系统、工程技术应用、工业生产加工的功能、质量、精度及效率等而成为当前的研究热点.然而它们迄今未有令人满意的解决方案.本文围绕这些问题展开深入研究,建立起一系列方便高效的几何算法,取得了以下丰富的创新性理论成果:
(1)提出一种能够精确等距圆弧的高精度、高连续阶的等距逼近新算法.基于对现有各种等距逼近算法局限性的深入剖析,提炼出按算法几何意义、逼近误差精度、逼近曲线形式、等距常用曲线等原则来评价等距逼近方法优劣的一个基本准则;并以此为目标,借助基圆重新参数化的等距逼近思想,采用独特的参数化函数,创造出一种全新的等距逼近算法:无需预先识别就能精确等距圆弧这类常用曲线,在高精度要求下比现有等距逼近算法产生更少的分段数和控制顶点数,且将逼近连续阶由一般情况的G~1提高到C~1;从而一举克服了现有算法中不能精确等距圆弧、高度依赖于点采样、全局误差难以有效控制等种种弊病,特别适合应用于几何设计系统中,在压缩数据存储量、提高计算效率、改善曲线整体光滑性效果方面有特别重要的现实意义.
(2)对基于基圆重新参数化的等距逼近算法进行了透彻的误差估计.着眼于误差产生来源,发现并阐述了基圆重新参数化等距逼近算法与基圆逼近算法之间的对偶性;揭示出前者的误差来源于近似等距方向与基曲线法向之间的偏角,并指出近似等距曲线与基曲线上对应点之间的距离向量其模长恒为已知等距距离但其方向却偏离基曲线法向,只有彻底研究偏角的几何内涵及代数形式才能精确估计等距逼近误差;基于这种发现,摒弃了常用的对应参数点误差估计方式,巧用几何信息,提出并成功地实现了用Hausdorff距离来计算近似等距曲线与精确等距曲线上对应点之间误差的新方法:从而在极大程度上完善了基圆重新参数化这一等距逼近算法的理论研究,并为该算法和其他各类算法的比较提供了理论依据.
(3)提出了一种快速实现易于造型、兼容于不同几何设计系统的低次等距逼近的新算法.通过改进传统的向量值Padé逼近方法,结合曲线细分、中点展开等技术,构造出可以达到预设精度的任意次的有理等距逼近,从而消除了现有的低次等距逼近算法中大多采用点采样技术,缺乏稳定性且误差估计不便等局限性;利用线性方程组求解,使得算法敏捷,误差估计方便,为低次等距几何逼近提供了一个崭新工具.
(4)给出了过给定测地线且兼具某些几何特征的一类参数曲面的设计和调控技术,解决了以往曲面造型中仅有一般设计原理,无法具体进行曲面表示和调整的问题,为服装鞋帽类的变型设计自动化开辟了一个新途径.与此同时,鉴于计算机辅助设计和制造系统对曲面表示和数据交换的需要,实现了过给定测地线的有理曲面设计,给出了常用的三次有理曲面的设计算法,充分考虑到设计中调整曲面形状的需要,采用变分优化技术,便于对曲面直接操作,直观便捷,且兼顾到曲面造型和光顺的需要.
(5)提出了全新的可展曲面设计思路.基于将曲面看作动曲线上的点沿局部坐标架在空间中运动所得轨迹的观点,得到了过给定曲线的一张可展曲面的充分必要条件.受鞋衣制造业的启发,特别考虑了给定曲线为所求曲面上一条测地线的情况.根据可展曲面的类型,把给定曲线作相应分类,使得算法在工程中的应用与实现能够十分方便而有效.此外,给出了可展曲面的多项式表示,使得算法效率进一步提高.
(6)讨论了广泛应用于工程、能精确表示双曲线、悬链线等超越曲线的H-Bézier曲线的奇异点情况.利用活动控制顶点技术,刻划曲线奇异点与活动控制顶点分布的对应情况,并就判别曲线的类型比较了三次H-Bézier曲线、Bézier曲线、有理Bézier曲线及C-Bézier曲线,并指出了这种曲线刻划方法的重要应用.
Offsets and geodesics are two fundamental curves with great importance in researches of computer aided geometric design(CAGD).The offset approximation in terms of rational representation,error estimation,and low-order approximation, as well as the surface design and modification passing through the given geodesic,has become one of the hotspots of investigation,due to the direct relationship with the function,quality,precision and efficiency of geometric design systems,engineering techniques,and processing industries.However,up to now there are still no satisfying solutions to the issues.Centering on those areas,this thesis carries out an in-depth study,and provides a series of convenient and efficient geometric algorithms.The abundant and innovative results are presented as follows:
(1) Propose a high precision and high-order continuity offset approximation algorithm yielding precise circular offsetting.Based on the thorough analysis of all existing offset approximation methods,we abstract several principles for algorithm evaluation,including geometric interpretation,approximation precision, approximate curve representation and the approximation effect for curves in common use.According to these principles and based on the circle reparametrization theory,we apply a unique parametrization function,and deduce an innovative algorithm for planar offset approximation.The algorithm has the advantages of implementing precise offsetting for circulars without curve identification in advance,generating a much smaller number of curve segmentations and control points in the demand of high precision approximation,and improving the continuity of the resultant curve from G~1 to C~1.So it successfully overcomes different kinds of intrinsic disadvantages of existing methods,which cannot offset circulars precisely,or highly depend on the sampling technique,or cannot achieve global error control.The algorithm has practical significance in saving data storage,raising computing efficiency and improving the whole smoothness, so is especially suitable for geometric design systems.
(2) Carry out a thorough analysis for error estimation of the offset approximation technique based on circle reparametrization.Focusing on the sourse of the approximation error,we discover and explain the duality of the two offset approximation methods based on circle reparametrization and circle approximation respectively.It reveals that the approximation error of the first method origins from the deflection of the approximate offset direction and the normal direction of the base curve,but the distance between the corresponding points of the approximate offset and the base curve is always equal to the given offset radius. In this sense,no precise offset approximation can be achieved until the thorough researches of the geometric meaning and the algebraic form of the deflection angle. We get rid of the error estimation way of measuring the distance between the corresponding points with the same parameter.By skillfully using the geometric information,Hausdorff distance estimation is proposed and successfully computed between the precise offset and the approximate offset.The discussion perfects the theoretical research on the circle reparametrization method, and provides theoretical basis for the comparison between the method and other algorithms.
(3) Develop a new algorithm for low-order offset approximation in terms of fast implementation of curve modeling,and good compatibility of different geometric design systems.By improving the traditional vector Padéapproximation method and combining the curve subdivision and expansion technique at the mid-point,the method yielding arbitrary order rational offset approximation and achieving the given precision is obtained.The method avoids the limits of the sampling technique,which leads to unstable results and cannot achieve global error control.Instead it applies the linear system for the final solution,so the computation and error estimation is convenient and fast in implementation.The method provides a new geometric tool for low-order offset approximation.
(4) Provide a technique for designing and modeling surfaces with the given curve as a geodesic.The previous research only proposed a common principle without the consideration of the surface representation and modification requirements in geometric design systems.Our method solves the problem,developing a new way for design automation in garment manufacture and shoe-making indus- try.Considering the demand of data representation and exchange in computer aided design and computer aided manufacture(CAD/CAM) system,we research the rational surface design,and provide an algorithm for the construction of the cubic rational surface.Variation optimization technique is used,which can be operated directly on the surface in an intuitive way,giving consideration of both surface modeling and smoothing.
(5) Offer a wholly bran-new idea for developable surface design.Based on the view of regarding a surface as a locus of the moving point of the given curve moving along the local frame in the space,we deduce the sufficient and necessary conditions for the surface passing through the given curve to be developable. Inspired by the practice in garment manufacture and shoe-making industry,we give special attention to the situation when the given curve is a geodesic on the surface.According to the types of developable surfaces,the given curve is characterized in order to make the algorithm more convenient and effective in engineering application.Besides,rational representation is provided,which further improves the computation efficiency.
(6) Discuss the singularity distribution of H-Bézier curves,which can precisely represent the widely used curves,hyperbola and catenary.Using the moving control point technique,the correspondence between the distribution of the moving control point and the curve singularities is characterized.Comparison in terms of the type of the discriminant curve is provided between the cubic H-Bézier curve,Bézier curve,rational Bézier curve and C-Bézier curve.We also illustrate the application of this curve characterizing method.
(1)提出一种能够精确等距圆弧的高精度、高连续阶的等距逼近新算法.基于对现有各种等距逼近算法局限性的深入剖析,提炼出按算法几何意义、逼近误差精度、逼近曲线形式、等距常用曲线等原则来评价等距逼近方法优劣的一个基本准则;并以此为目标,借助基圆重新参数化的等距逼近思想,采用独特的参数化函数,创造出一种全新的等距逼近算法:无需预先识别就能精确等距圆弧这类常用曲线,在高精度要求下比现有等距逼近算法产生更少的分段数和控制顶点数,且将逼近连续阶由一般情况的G~1提高到C~1;从而一举克服了现有算法中不能精确等距圆弧、高度依赖于点采样、全局误差难以有效控制等种种弊病,特别适合应用于几何设计系统中,在压缩数据存储量、提高计算效率、改善曲线整体光滑性效果方面有特别重要的现实意义.
(2)对基于基圆重新参数化的等距逼近算法进行了透彻的误差估计.着眼于误差产生来源,发现并阐述了基圆重新参数化等距逼近算法与基圆逼近算法之间的对偶性;揭示出前者的误差来源于近似等距方向与基曲线法向之间的偏角,并指出近似等距曲线与基曲线上对应点之间的距离向量其模长恒为已知等距距离但其方向却偏离基曲线法向,只有彻底研究偏角的几何内涵及代数形式才能精确估计等距逼近误差;基于这种发现,摒弃了常用的对应参数点误差估计方式,巧用几何信息,提出并成功地实现了用Hausdorff距离来计算近似等距曲线与精确等距曲线上对应点之间误差的新方法:从而在极大程度上完善了基圆重新参数化这一等距逼近算法的理论研究,并为该算法和其他各类算法的比较提供了理论依据.
(3)提出了一种快速实现易于造型、兼容于不同几何设计系统的低次等距逼近的新算法.通过改进传统的向量值Padé逼近方法,结合曲线细分、中点展开等技术,构造出可以达到预设精度的任意次的有理等距逼近,从而消除了现有的低次等距逼近算法中大多采用点采样技术,缺乏稳定性且误差估计不便等局限性;利用线性方程组求解,使得算法敏捷,误差估计方便,为低次等距几何逼近提供了一个崭新工具.
(4)给出了过给定测地线且兼具某些几何特征的一类参数曲面的设计和调控技术,解决了以往曲面造型中仅有一般设计原理,无法具体进行曲面表示和调整的问题,为服装鞋帽类的变型设计自动化开辟了一个新途径.与此同时,鉴于计算机辅助设计和制造系统对曲面表示和数据交换的需要,实现了过给定测地线的有理曲面设计,给出了常用的三次有理曲面的设计算法,充分考虑到设计中调整曲面形状的需要,采用变分优化技术,便于对曲面直接操作,直观便捷,且兼顾到曲面造型和光顺的需要.
(5)提出了全新的可展曲面设计思路.基于将曲面看作动曲线上的点沿局部坐标架在空间中运动所得轨迹的观点,得到了过给定曲线的一张可展曲面的充分必要条件.受鞋衣制造业的启发,特别考虑了给定曲线为所求曲面上一条测地线的情况.根据可展曲面的类型,把给定曲线作相应分类,使得算法在工程中的应用与实现能够十分方便而有效.此外,给出了可展曲面的多项式表示,使得算法效率进一步提高.
(6)讨论了广泛应用于工程、能精确表示双曲线、悬链线等超越曲线的H-Bézier曲线的奇异点情况.利用活动控制顶点技术,刻划曲线奇异点与活动控制顶点分布的对应情况,并就判别曲线的类型比较了三次H-Bézier曲线、Bézier曲线、有理Bézier曲线及C-Bézier曲线,并指出了这种曲线刻划方法的重要应用.
Offsets and geodesics are two fundamental curves with great importance in researches of computer aided geometric design(CAGD).The offset approximation in terms of rational representation,error estimation,and low-order approximation, as well as the surface design and modification passing through the given geodesic,has become one of the hotspots of investigation,due to the direct relationship with the function,quality,precision and efficiency of geometric design systems,engineering techniques,and processing industries.However,up to now there are still no satisfying solutions to the issues.Centering on those areas,this thesis carries out an in-depth study,and provides a series of convenient and efficient geometric algorithms.The abundant and innovative results are presented as follows:
(1) Propose a high precision and high-order continuity offset approximation algorithm yielding precise circular offsetting.Based on the thorough analysis of all existing offset approximation methods,we abstract several principles for algorithm evaluation,including geometric interpretation,approximation precision, approximate curve representation and the approximation effect for curves in common use.According to these principles and based on the circle reparametrization theory,we apply a unique parametrization function,and deduce an innovative algorithm for planar offset approximation.The algorithm has the advantages of implementing precise offsetting for circulars without curve identification in advance,generating a much smaller number of curve segmentations and control points in the demand of high precision approximation,and improving the continuity of the resultant curve from G~1 to C~1.So it successfully overcomes different kinds of intrinsic disadvantages of existing methods,which cannot offset circulars precisely,or highly depend on the sampling technique,or cannot achieve global error control.The algorithm has practical significance in saving data storage,raising computing efficiency and improving the whole smoothness, so is especially suitable for geometric design systems.
(2) Carry out a thorough analysis for error estimation of the offset approximation technique based on circle reparametrization.Focusing on the sourse of the approximation error,we discover and explain the duality of the two offset approximation methods based on circle reparametrization and circle approximation respectively.It reveals that the approximation error of the first method origins from the deflection of the approximate offset direction and the normal direction of the base curve,but the distance between the corresponding points of the approximate offset and the base curve is always equal to the given offset radius. In this sense,no precise offset approximation can be achieved until the thorough researches of the geometric meaning and the algebraic form of the deflection angle. We get rid of the error estimation way of measuring the distance between the corresponding points with the same parameter.By skillfully using the geometric information,Hausdorff distance estimation is proposed and successfully computed between the precise offset and the approximate offset.The discussion perfects the theoretical research on the circle reparametrization method, and provides theoretical basis for the comparison between the method and other algorithms.
(3) Develop a new algorithm for low-order offset approximation in terms of fast implementation of curve modeling,and good compatibility of different geometric design systems.By improving the traditional vector Padéapproximation method and combining the curve subdivision and expansion technique at the mid-point,the method yielding arbitrary order rational offset approximation and achieving the given precision is obtained.The method avoids the limits of the sampling technique,which leads to unstable results and cannot achieve global error control.Instead it applies the linear system for the final solution,so the computation and error estimation is convenient and fast in implementation.The method provides a new geometric tool for low-order offset approximation.
(4) Provide a technique for designing and modeling surfaces with the given curve as a geodesic.The previous research only proposed a common principle without the consideration of the surface representation and modification requirements in geometric design systems.Our method solves the problem,developing a new way for design automation in garment manufacture and shoe-making indus- try.Considering the demand of data representation and exchange in computer aided design and computer aided manufacture(CAD/CAM) system,we research the rational surface design,and provide an algorithm for the construction of the cubic rational surface.Variation optimization technique is used,which can be operated directly on the surface in an intuitive way,giving consideration of both surface modeling and smoothing.
(5) Offer a wholly bran-new idea for developable surface design.Based on the view of regarding a surface as a locus of the moving point of the given curve moving along the local frame in the space,we deduce the sufficient and necessary conditions for the surface passing through the given curve to be developable. Inspired by the practice in garment manufacture and shoe-making industry,we give special attention to the situation when the given curve is a geodesic on the surface.According to the types of developable surfaces,the given curve is characterized in order to make the algorithm more convenient and effective in engineering application.Besides,rational representation is provided,which further improves the computation efficiency.
(6) Discuss the singularity distribution of H-Bézier curves,which can precisely represent the widely used curves,hyperbola and catenary.Using the moving control point technique,the correspondence between the distribution of the moving control point and the curve singularities is characterized.Comparison in terms of the type of the discriminant curve is provided between the cubic H-Bézier curve,Bézier curve,rational Bézier curve and C-Bézier curve.We also illustrate the application of this curve characterizing method.
引文
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