曲线曲面的两类几何逼近与两类代数表示
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摘要
曲线曲面的逼近和表示是计算机辅助几何设计的两大基本理论问题.其中,曲线曲面的降阶逼近与导矢逼近、圆锥曲线的有理表示与球域曲面的边界表示由于直接关系到几何设计系统的功能、质量、精度及效率而成为当前的研究热点之一,然而它们迄今未在理论上有所突破.面对这种挑战,作者以应用数学为工具、以现代工业为背景开展深入研究,从根本上攻克了上述难题,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.在曲面降阶逼近方面:发现了三角Jacobi基是统一地实现三角曲面显式、最佳、约束降多阶的一个锐利工具,并成功地把其应用到算法设计.借助于三角Bernstein基与三角Jocobi基的转换关系,将三角Jacobi基的正交代数性质引入到几何逼近之中,自然地诱导出三角Bézier曲面带角点约束和无角点约束的一次性降多阶的简单直观算法,使之具有以往各类曲面降多阶方法所不能同时拥有的四个特点——误差预测、显式表达、机时最少、精度最佳,即:第一,降阶前可迅速判断是否存在满足给定误差的降多阶曲面从而避免了无效降阶;第二,全部降多阶运算可被归结为对曲面的控制顶点按词典顺序排序所写成的列向量执行一个简单的矩阵乘法;第三,此矩阵无需临时计算而是从数据库中直接调用;第四,这张降多阶曲面在L_2范数意义下达到了最佳逼近效果.特别,对于带角点约束的曲面降阶,此算法可保持降阶曲面的边界曲线在角点处达到高阶连续;并且可以利用Foley-Opitz平均方案使降阶曲面片达到全局C~1的连续阶,与曲面细分技术结合应用,更能够适合计算机辅助几何设计(CAGD)系统的造型要求.
2.在曲线降阶逼近方面:发明了广义逆与分块矩阵相结合的代数方法以及正交基运算与二次规划相结合的优化方法,实现了参数曲线或圆域曲线在高精度与高效率下的带端点约束降多阶.对于Said-Bézier型广义Ball曲线(简称SBGB曲线),推导出其升阶矩阵公式,并根据SBGB基的分段表达式,给出了该曲线端点处的各阶导矢公式及相应矩阵表示;在此基础上,应用广义逆矩阵与矩阵分块原理,得到了SBGB曲线在保端点任意阶连续性的条件下一次性降多阶的显式算法.对于圆域Bézier曲线,利用Jacobi多项式的正交性,给出在L_2范数下原圆域Bézier曲线的中心曲线的一次性最佳降多阶逼近,作为降阶圆域Bézier曲线的中心曲线;然后,利用Bernstein基与Legendre基的转换公式以及Legendre基的正交性,把降阶圆域Bézier曲线最佳逼近半径的算法,转化为带约束条件的一个二次规划问题的求解.以上两种方法都具有操作简单、精度高、速度快的特点.
3.在三角曲面导矢逼近方面:发现了升阶公式与差分算子是三角参数曲面导矢逼近的两个犀利武器,并成功地进行了演绎推理.利用一系列恒等式变换及优化的缩写符号,结合缜密的不等式技巧,推导出有理三角Bézier曲面一、二阶偏导矢界的一种精密估计,并证明了新的导矢界在精确性与有效性上优于现有的导矢界,进一步提升且强化了几何设计系统的功能.
4.在圆锥曲线的有理表示方面:创造了按照可降阶与可不适当参数化这两种代数分类条件去研究有理四次Bézier圆锥曲线几何特征的新思想与新方法.将有理四次Bézier圆锥曲线归结为两种特殊类型,即可降阶的以及可不适当参数化的.在此基础上.基于对线性凸组合的代数量及三角形面积的几何量的严密分析,得到了圆锥曲线有理四次Bézier表示的充要条件,使之可被分解成关于Bézier点和权因子这样两部分.利用此条件给出了两种新算法,其一为判断一条有理四次Bézier曲线是否为圆锥曲线,属于何种类型;其二为对于一条已知的圆锥曲线,给出其有理四次Bézier形式下的控制顶点位置和权因子值.这些结果不但丰富了几何计算的学科理论,而且扩充了几何造型与几何设计系统的有效应用范围.在这一研究的基础上,借助低次Bernstein基与同次Said-Ball基或DP-NTP基之间的转化关系,又分别推导出有理低次Said-Ball圆锥曲线和有理低次DP-NTP圆锥曲线表示的充要条件,并给出了相应的曲线造型新算法.
5.在球域曲面的边界表示方面:创造了微分几何的包络原理与Legendre代数式的正交原理综合运用的新的分析方法.借助经典微分几何中双参数曲面族的包络原理,运用球面参数坐标和Cramer法则,首先给出了球域Bézier曲面边界的精确的显式表达式.再利用Legendre多项式的正交性,得到其精确边界用多项式形式表示的最佳平方逼近.进一步利用Legendre基与Bernstein基的转换公式,将这种曲面的近似边界用CAGD系统中最常用的Bézier形式表示,因而更适合应用到外形设计系统中.
Approximation and representation of curves and surfaces are the two fundamental theoretical questions in computer aided geometric design(CAGD).Therein approximation of degree reduction and derivatives for curves and surfaves,rational representation of conics and boundary representation of ball surface have become one of the hotspots of investigation, since they directly relate to the function,quality,precision,and efficiency of geometric design systems.However,up to now there is still no theoretical breakthrough in these areas.Facing these challenges,we take applied mathematics as a tool and rely on modern industry as the background to do an in-depth study.Then we overcome these key technological problems and provide a series of convenient and efficient geometric algorithms.These abundant and innovative results are represented as follows:
1.In respect of degree-reduced approximation of surfaces:we discover a sharp tool, triangular Jacobi basis,to uniformly achieve explicit,optimal,and constrained multi-degree reduction of triangular surfaces,and then apply it into algorithm design.The algebraic property of triangular orthonormal Jacobi polynomials is introduced into geometric approximation.And based on the transformation formulae between bivariate Bernstein basis and bivariate Jacobi basis,a simple and intuitive algorithm is deduced for multi-degree reduction of triangular Bézier surfaces with or without corner constraints. The algorithm has following four advantages which known methods for multi-degree reduction of surfaces cannot own at the same time.That is,ability of error forecast, explicit expression,less time consumption,and best precision.It means,firstly,we can judge beforehand that whether there exists a multi-degree reduced surface within a prescribed tolerance in order to avoid useless operations;Secondly,all the operations of multi-degree reduction are just to multiply the column vector generated by arranging the series of control points of the surface in iexicographic order by a simple matrix;Thirdly, this matrix can be computed at one time and then stored in database before processing degree reduction to be summon for using at any time;Fourthly,the multi-degree reduced surface achieves an optimal approximation in the norm L_2.Specially,for the case of corner constraints,this algorithm can keep the boundary curves of degree-reduced surface high order continuity at corners.Also applying Foley-Opitz averaging scheme,the piecewise degree-reduced patches possess global C~1 continuity.Combined with surface subdivision technique,it will suit better for modeling demands in CAGD.
2.In respect of degree-reduced approximation of curves:we invent an algebraic method combining generalized inverse matrix with partitioned matrix and an optimized method combining orthonormal basis operations with constrained quadratic programming.Then we achieve constrained multi-degree reduction of parametric curves or disk curves with high precision and high efficiency.As for the Said-Bézier generalized Ball(SBGB) curve, we deduce a matrix for its degree elevation,and present some formulae for its derived vectors at two endpoints and the corresponding matrix representations according to the piecewise expressions of SBGB basis.Based on these matrices,we apply the principles of both generalized inverse matrix and partitioned matrix to deduce an explicit algorithm for multi-degree reduction of SBGB curves with endpoints constraints of arbitrary order continuity.As for the disk Bézier curve,applying the orthogonality of Jacobi polynomials, we find out an optimal multi-degree reduced polynomial approximation of the center curve of the original disk Bézier curve in the norm L_2,and regard it as the center curve of the degree-reduced disk Bézier curve;Next,applying the conversion formulae between Bernstein basis and Legendre basis as well as the orthogonality of Legendre basis,we transform the task,optimally multi-degree reduced approximating the error radius curve of the disk Bézier curve,into solving a constraned quadratic programming(QP) problem. Both above-mentioned techniques have entirely three fine characters,easy processing, high precision and high speed.
3.In respect of derivative approximation of triangular surfaces:we find two acuminous implements,degree elevation and difference operators,to successfully deduce the results for derivative approximation of triangular parameter surfaces.Applying a series of identical formulae and abbreviated notations also combined with refined inequality skills, we derive sharp bound estimates on first and second partial derivatives of rational triangular Bézier surfaces.We also prove that the new bounds are superior to the known ones in terms of precision and validity.The results have practical value for promoting and reinforcing functions in geometric design systems.
4.In respect of rational representations of conics:we present a new idea to study geometric characters of rational quartic Bézier conics according to two kinds of algebraic class conditions,i.e.,degree-reducible and improperly parameterized.We first regard rational quartic Bézier conics as two special kinds,i.e.,degree-reducible and improperly parameterized;and then on the basis of strictly analyzing the algebraic quantity of linear convex combinations and geometric quantity of triangle areas,we derive the necessary and sufficient conditions for rational quartic Bézier representation of conics.They can be divided into two parts:Bézier control points and weights.By using these conditions,two algorithms are provided to construct and design rational conics.One is to judge whether a rational quartic Bézier curve is a conic section,and which type it is.Another is to present positions of the control points and values of the weights of a given conic section in form of a rational quartic Bézier curve.These results not only enrich the theory of CAGD,but also enlarge applied range of geometric modeling and geometric design systems.Based on these results,we apply the transformation relations between lower degree Bernstein basis and Said-Ball basis or DP-NTP basis to deduce the necessary and sufficient conditions for rational lower degree Said-Ball or DP-NTP representations of conics respectively.Also the corresponding algorithms for curve modeling are given.
5.In respect of boundary representation of ball surfaces:a new analytical method is created which is combining the envelop theory in differential geometry and the orthogonality of Legendre algebraic expression.Recurring to the envelope theory of the family of bivariate parametric surfaces,spherical coordinates and Cramer's rule,we give an explicit representation of the exact boundary of a ball Bézier surface.Then according to the orthogonality of Legendre polynomials,the exact boundary of the ball Bézier surface is optimal squarely approximated in polynomial form.Furthermore,by the conversion formulae between Legendre basis and Bernstein basis,the approximate boundary of the ball Bézier surface is expressed in Bernstein form.It is more suited for use in shape design systems.
1.在曲面降阶逼近方面:发现了三角Jacobi基是统一地实现三角曲面显式、最佳、约束降多阶的一个锐利工具,并成功地把其应用到算法设计.借助于三角Bernstein基与三角Jocobi基的转换关系,将三角Jacobi基的正交代数性质引入到几何逼近之中,自然地诱导出三角Bézier曲面带角点约束和无角点约束的一次性降多阶的简单直观算法,使之具有以往各类曲面降多阶方法所不能同时拥有的四个特点——误差预测、显式表达、机时最少、精度最佳,即:第一,降阶前可迅速判断是否存在满足给定误差的降多阶曲面从而避免了无效降阶;第二,全部降多阶运算可被归结为对曲面的控制顶点按词典顺序排序所写成的列向量执行一个简单的矩阵乘法;第三,此矩阵无需临时计算而是从数据库中直接调用;第四,这张降多阶曲面在L_2范数意义下达到了最佳逼近效果.特别,对于带角点约束的曲面降阶,此算法可保持降阶曲面的边界曲线在角点处达到高阶连续;并且可以利用Foley-Opitz平均方案使降阶曲面片达到全局C~1的连续阶,与曲面细分技术结合应用,更能够适合计算机辅助几何设计(CAGD)系统的造型要求.
2.在曲线降阶逼近方面:发明了广义逆与分块矩阵相结合的代数方法以及正交基运算与二次规划相结合的优化方法,实现了参数曲线或圆域曲线在高精度与高效率下的带端点约束降多阶.对于Said-Bézier型广义Ball曲线(简称SBGB曲线),推导出其升阶矩阵公式,并根据SBGB基的分段表达式,给出了该曲线端点处的各阶导矢公式及相应矩阵表示;在此基础上,应用广义逆矩阵与矩阵分块原理,得到了SBGB曲线在保端点任意阶连续性的条件下一次性降多阶的显式算法.对于圆域Bézier曲线,利用Jacobi多项式的正交性,给出在L_2范数下原圆域Bézier曲线的中心曲线的一次性最佳降多阶逼近,作为降阶圆域Bézier曲线的中心曲线;然后,利用Bernstein基与Legendre基的转换公式以及Legendre基的正交性,把降阶圆域Bézier曲线最佳逼近半径的算法,转化为带约束条件的一个二次规划问题的求解.以上两种方法都具有操作简单、精度高、速度快的特点.
3.在三角曲面导矢逼近方面:发现了升阶公式与差分算子是三角参数曲面导矢逼近的两个犀利武器,并成功地进行了演绎推理.利用一系列恒等式变换及优化的缩写符号,结合缜密的不等式技巧,推导出有理三角Bézier曲面一、二阶偏导矢界的一种精密估计,并证明了新的导矢界在精确性与有效性上优于现有的导矢界,进一步提升且强化了几何设计系统的功能.
4.在圆锥曲线的有理表示方面:创造了按照可降阶与可不适当参数化这两种代数分类条件去研究有理四次Bézier圆锥曲线几何特征的新思想与新方法.将有理四次Bézier圆锥曲线归结为两种特殊类型,即可降阶的以及可不适当参数化的.在此基础上.基于对线性凸组合的代数量及三角形面积的几何量的严密分析,得到了圆锥曲线有理四次Bézier表示的充要条件,使之可被分解成关于Bézier点和权因子这样两部分.利用此条件给出了两种新算法,其一为判断一条有理四次Bézier曲线是否为圆锥曲线,属于何种类型;其二为对于一条已知的圆锥曲线,给出其有理四次Bézier形式下的控制顶点位置和权因子值.这些结果不但丰富了几何计算的学科理论,而且扩充了几何造型与几何设计系统的有效应用范围.在这一研究的基础上,借助低次Bernstein基与同次Said-Ball基或DP-NTP基之间的转化关系,又分别推导出有理低次Said-Ball圆锥曲线和有理低次DP-NTP圆锥曲线表示的充要条件,并给出了相应的曲线造型新算法.
5.在球域曲面的边界表示方面:创造了微分几何的包络原理与Legendre代数式的正交原理综合运用的新的分析方法.借助经典微分几何中双参数曲面族的包络原理,运用球面参数坐标和Cramer法则,首先给出了球域Bézier曲面边界的精确的显式表达式.再利用Legendre多项式的正交性,得到其精确边界用多项式形式表示的最佳平方逼近.进一步利用Legendre基与Bernstein基的转换公式,将这种曲面的近似边界用CAGD系统中最常用的Bézier形式表示,因而更适合应用到外形设计系统中.
Approximation and representation of curves and surfaces are the two fundamental theoretical questions in computer aided geometric design(CAGD).Therein approximation of degree reduction and derivatives for curves and surfaves,rational representation of conics and boundary representation of ball surface have become one of the hotspots of investigation, since they directly relate to the function,quality,precision,and efficiency of geometric design systems.However,up to now there is still no theoretical breakthrough in these areas.Facing these challenges,we take applied mathematics as a tool and rely on modern industry as the background to do an in-depth study.Then we overcome these key technological problems and provide a series of convenient and efficient geometric algorithms.These abundant and innovative results are represented as follows:
1.In respect of degree-reduced approximation of surfaces:we discover a sharp tool, triangular Jacobi basis,to uniformly achieve explicit,optimal,and constrained multi-degree reduction of triangular surfaces,and then apply it into algorithm design.The algebraic property of triangular orthonormal Jacobi polynomials is introduced into geometric approximation.And based on the transformation formulae between bivariate Bernstein basis and bivariate Jacobi basis,a simple and intuitive algorithm is deduced for multi-degree reduction of triangular Bézier surfaces with or without corner constraints. The algorithm has following four advantages which known methods for multi-degree reduction of surfaces cannot own at the same time.That is,ability of error forecast, explicit expression,less time consumption,and best precision.It means,firstly,we can judge beforehand that whether there exists a multi-degree reduced surface within a prescribed tolerance in order to avoid useless operations;Secondly,all the operations of multi-degree reduction are just to multiply the column vector generated by arranging the series of control points of the surface in iexicographic order by a simple matrix;Thirdly, this matrix can be computed at one time and then stored in database before processing degree reduction to be summon for using at any time;Fourthly,the multi-degree reduced surface achieves an optimal approximation in the norm L_2.Specially,for the case of corner constraints,this algorithm can keep the boundary curves of degree-reduced surface high order continuity at corners.Also applying Foley-Opitz averaging scheme,the piecewise degree-reduced patches possess global C~1 continuity.Combined with surface subdivision technique,it will suit better for modeling demands in CAGD.
2.In respect of degree-reduced approximation of curves:we invent an algebraic method combining generalized inverse matrix with partitioned matrix and an optimized method combining orthonormal basis operations with constrained quadratic programming.Then we achieve constrained multi-degree reduction of parametric curves or disk curves with high precision and high efficiency.As for the Said-Bézier generalized Ball(SBGB) curve, we deduce a matrix for its degree elevation,and present some formulae for its derived vectors at two endpoints and the corresponding matrix representations according to the piecewise expressions of SBGB basis.Based on these matrices,we apply the principles of both generalized inverse matrix and partitioned matrix to deduce an explicit algorithm for multi-degree reduction of SBGB curves with endpoints constraints of arbitrary order continuity.As for the disk Bézier curve,applying the orthogonality of Jacobi polynomials, we find out an optimal multi-degree reduced polynomial approximation of the center curve of the original disk Bézier curve in the norm L_2,and regard it as the center curve of the degree-reduced disk Bézier curve;Next,applying the conversion formulae between Bernstein basis and Legendre basis as well as the orthogonality of Legendre basis,we transform the task,optimally multi-degree reduced approximating the error radius curve of the disk Bézier curve,into solving a constraned quadratic programming(QP) problem. Both above-mentioned techniques have entirely three fine characters,easy processing, high precision and high speed.
3.In respect of derivative approximation of triangular surfaces:we find two acuminous implements,degree elevation and difference operators,to successfully deduce the results for derivative approximation of triangular parameter surfaces.Applying a series of identical formulae and abbreviated notations also combined with refined inequality skills, we derive sharp bound estimates on first and second partial derivatives of rational triangular Bézier surfaces.We also prove that the new bounds are superior to the known ones in terms of precision and validity.The results have practical value for promoting and reinforcing functions in geometric design systems.
4.In respect of rational representations of conics:we present a new idea to study geometric characters of rational quartic Bézier conics according to two kinds of algebraic class conditions,i.e.,degree-reducible and improperly parameterized.We first regard rational quartic Bézier conics as two special kinds,i.e.,degree-reducible and improperly parameterized;and then on the basis of strictly analyzing the algebraic quantity of linear convex combinations and geometric quantity of triangle areas,we derive the necessary and sufficient conditions for rational quartic Bézier representation of conics.They can be divided into two parts:Bézier control points and weights.By using these conditions,two algorithms are provided to construct and design rational conics.One is to judge whether a rational quartic Bézier curve is a conic section,and which type it is.Another is to present positions of the control points and values of the weights of a given conic section in form of a rational quartic Bézier curve.These results not only enrich the theory of CAGD,but also enlarge applied range of geometric modeling and geometric design systems.Based on these results,we apply the transformation relations between lower degree Bernstein basis and Said-Ball basis or DP-NTP basis to deduce the necessary and sufficient conditions for rational lower degree Said-Ball or DP-NTP representations of conics respectively.Also the corresponding algorithms for curve modeling are given.
5.In respect of boundary representation of ball surfaces:a new analytical method is created which is combining the envelop theory in differential geometry and the orthogonality of Legendre algebraic expression.Recurring to the envelope theory of the family of bivariate parametric surfaces,spherical coordinates and Cramer's rule,we give an explicit representation of the exact boundary of a ball Bézier surface.Then according to the orthogonality of Legendre polynomials,the exact boundary of the ball Bézier surface is optimal squarely approximated in polynomial form.Furthermore,by the conversion formulae between Legendre basis and Bernstein basis,the approximate boundary of the ball Bézier surface is expressed in Bernstein form.It is more suited for use in shape design systems.
引文
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