螺线和正交多项式在CAGD中的应用
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摘要
本文主题为计算机辅助几何设计中几类曲线的表示、拼接与逼近,特别对螺线和正交多项式在计算机辅助几何设计中的应用进行了深入的研究,主要获得了以下一些创新成果.
(1)基于道路设计的工程需要,构造了曲率单调且保号的起点曲率为零的平面三次C-Bezier螺线.利用这条螺线,详细推导了在道路设计等工业应用中,直线和圆弧之间的过渡曲线算法.如同工程中已使用回旋曲线来过渡一样,直线和圆弧之间用一条螺线过渡,圆弧与圆弧之间用一对C型或S型螺线过渡,两条直线之间用一对螺线过渡,圆包含圆弧时用一条螺线过渡.在前4种情况中均给出了螺线的具体表达式,第5种情况下不一定有解.由于直线、圆弧也能用C-Bezier曲线精确表示,可以在C-Bezier模式下统一处理整条道路设计问题,避免了以往采用Fresnel积分所表示的回旋曲线,不适合于在计算机辅助设计系统使用的情况.
(2)构造了带有可调参数的单段C-Bezier曲线光滑地拼接前后两条圆弧的算法.在S型过渡的场合,给出了一条带参数的曲率单调的C-Bezier曲线,讨论了这条曲线能拼接两条圆弧的半径范围,其结果优于使用Bezier螺线的老方法;在C型过渡的场合,给出了一条带参数的内部仅含一个曲率极值点的C-Bezier曲线.所有过渡曲线的具体算法均有详细介绍.用本方法设计的道路线型,与老方法相比具有以下三个优越性：模式统一；可用参数调节形状；拼接曲线仅有一段,且算法归结为求解低次方程的正根,故计算简单,容易实现.
(3)为了适合当前计算机辅助设计系统中的曲线形式和工业设计中的美学需要,提出了对数螺线段的两种逼近方法.第一种利用s-Power级数,首先给出了s-Power系数的计算公式,提出了对数螺线段的快速多项式逼近算法,同时也给出了对数螺线的等距曲线的具体表达式及其s-Power逼近算法.第二种首先推导出两端点C-Bezier形式的G2 Hermite插值公式,提出了对数螺线的C-Bezie形式的G2 Hermite插值逼近算法.
(4)为了在计算机辅助几何设计中,有效地求解曲线或曲面的最小平方逼近问题,给出了具有边界约束特征的加权正交基与Bernstein基之间的转换矩阵,并且利用该矩阵,给出了两个具体应用：(ⅰ)得到了在Jacobi加权L2范数下基于正交基的Bezier曲线约束最佳降多阶逼近算法,给出了具体的端点约束最佳降多阶矩阵,且给出了该降阶逼近的可预报的误差公式.同时提出了在L2,L1,L∞范数下适合于最佳降阶逼近的相应Jacobi基的权函数的选取方案.(ⅱ)求解多项式反函数是CAGD中的一个基本问题.利用约束Jacobi基作为有效工具,提出带端点Ck约束的反函数逼近算法.该算法稳定、简易,克服了以往计算反函数的系数时每次逼近系数需全部重新计算的缺陷,同时给出了该算法在PH曲线准弧长参数化中的应用.
(5)为了在计算机辅助几何设计中,有效的求解三角域上Bezier曲面的最小平方逼近问题,给出了三角域上双变量Jacobi和Bernstein基的相互转换矩阵.首先利用Bernstein基构造了三角域上的Jacobi多项式,然后利用单变量Jacobi和Bernstein基的转换关系,给出了三角域上双变量Bernstein的Jacobi基的相互转换矩阵,并且利用该矩阵,得到了在加权L2范数下基于正交基的Bezier曲面最佳降多阶逼近算法,给出了具体的最佳降多阶矩阵,且给出了该降阶逼近的可预报的误差公式.
In this paper, the theme is everal kind of curve's expressions, splices and approximations, in particular, we have made an in-depth research on spiral and orthogonal polynomials in the CAGD and have obtained new creative production on the following five aspects:
(1) Based on the requirement of highway design, a planar cubic C-Bezier spiral with monotone curvature of constant sign and a starting point of zero curvature is constructed, then an algorithm of the transition curve between straight lines and/or circular arcs is derived in detail which can be applied in highway design. As was done with clothoids in engineering, a single spiral is used for straight line to circular arc, two spirals suiting C-shaped or S-shaped transition are used for circular arc to circular arc, two spirals are used for straight line to straight line, and a single spiral is used for circular to circular arc when the latter is contained within the former. The concrete expressions for the first four cases are given; in the fifth case the solution cannot always be obtained. Because straight line segments and circular arcs can be represented precisely by C-Bezier curves, the issues such as highway design can be handled in the system by C-Bezier model, avoiding the difficult situation for computer-aided design system to use the clothoids defined in terms of the Fresnel integral.
(2) A single C-Bezier curve with a shape parameter for G2 joining two circular arcs is constructed. It is shown that a S-shaped transition curve which is able to better manage the broader scope about two circle radii than the Bezier curves has no curvature extrema, and a C-shaped transition curve has a single curvature extremum. Regarding the two kinds of curves, specific algorithms is presented in detail, strict mathematical proof was given, and the effectiveness of the method is showed by examples. This method has the following three advantages:the pattern is unified; the parameters able to adjust the shape of the transition curves are available; the transition curve is only a single segment, and the algorithm can be formulated as a low order equation to be solved for its positive root. Therefore it is simple and easy to implement.
(3) To fit the form of curve in the current Computer Aided Design system and aesthetic needs in industrial design, two approximation algorithms for logarithmic spiral segments is proposed. In the first method, the calculation formula for s-Power series is firstly derived and a fast polynomial approximation algorithm is presented, and then the calculation formula of the offset curves of the logarithmic spiral and corresponding approximation algorithm by s-Power series is presented. In the second method, The G2 Hermite interpolation formula of the two end points by the C-Bezier form is firstly derived, and then the G2 Hermite interpolation approximation algorithm by the C-Bezier form is presented. The results of example show that the algorithms are correct and effective, suitable for the use of the CAD system.
(4)To solve least squares approximation problem effectively in CAGD, the transformation matrices between the weighted orthogonal basis which possesses boundary constraints characteristic and Bernstein basis are derived. Then using the matrices, two specific applications are given.(i) the optimal algorithm based on Jacobi weighted L2 norm for constrained multi-degree reducing Bezier curve, including its matrix representation, is presented. Also the degree reduction error that can be forecasted is given. the Jacobi weighted function adapting to optimal degree reduction is selected with respect to L2, L1, L∞norm, respectively. (ⅱ) To solve the inverse function of polynomial is a basic problem in CAGD, an algorithm about approximating the inverse function with Ck constrains by using the constrained Jacobi basis is proposed. The approximation method is easy and steady. Moreover, the defect that the corresponding coefficients must be recalculated when approximating every inverse function one by one is overcame. As an application, how to generate quasi arc-length parameterization of PH curves is shown.
(5) For solve least squares approximation problem simply and effectively on triangular domains in CAGD, The matrices of transformation of the bivariate Bernstein basis form into the Jacobi basis of the same degree and vice versa are derived. A method for constructing bivariate Jacobi-weighted orthogonal polynomials in the Bernstein form on triangular domains was formulated firstly. And then, by using connection coefficients between the univariate Bernstein and Jacobi basis, the transformation matrices between bivariate Jacobi and Bernstein basis were presented. Finally, Then using the matrices, an explicit form of the multi-degree reduction matrix for Bezierr surface on triangular domains with respect to Jacobi weighted L2 norm was presented, and the error of the degree reduction was given.
(1)基于道路设计的工程需要,构造了曲率单调且保号的起点曲率为零的平面三次C-Bezier螺线.利用这条螺线,详细推导了在道路设计等工业应用中,直线和圆弧之间的过渡曲线算法.如同工程中已使用回旋曲线来过渡一样,直线和圆弧之间用一条螺线过渡,圆弧与圆弧之间用一对C型或S型螺线过渡,两条直线之间用一对螺线过渡,圆包含圆弧时用一条螺线过渡.在前4种情况中均给出了螺线的具体表达式,第5种情况下不一定有解.由于直线、圆弧也能用C-Bezier曲线精确表示,可以在C-Bezier模式下统一处理整条道路设计问题,避免了以往采用Fresnel积分所表示的回旋曲线,不适合于在计算机辅助设计系统使用的情况.
(2)构造了带有可调参数的单段C-Bezier曲线光滑地拼接前后两条圆弧的算法.在S型过渡的场合,给出了一条带参数的曲率单调的C-Bezier曲线,讨论了这条曲线能拼接两条圆弧的半径范围,其结果优于使用Bezier螺线的老方法;在C型过渡的场合,给出了一条带参数的内部仅含一个曲率极值点的C-Bezier曲线.所有过渡曲线的具体算法均有详细介绍.用本方法设计的道路线型,与老方法相比具有以下三个优越性：模式统一；可用参数调节形状；拼接曲线仅有一段,且算法归结为求解低次方程的正根,故计算简单,容易实现.
(3)为了适合当前计算机辅助设计系统中的曲线形式和工业设计中的美学需要,提出了对数螺线段的两种逼近方法.第一种利用s-Power级数,首先给出了s-Power系数的计算公式,提出了对数螺线段的快速多项式逼近算法,同时也给出了对数螺线的等距曲线的具体表达式及其s-Power逼近算法.第二种首先推导出两端点C-Bezier形式的G2 Hermite插值公式,提出了对数螺线的C-Bezie形式的G2 Hermite插值逼近算法.
(4)为了在计算机辅助几何设计中,有效地求解曲线或曲面的最小平方逼近问题,给出了具有边界约束特征的加权正交基与Bernstein基之间的转换矩阵,并且利用该矩阵,给出了两个具体应用：(ⅰ)得到了在Jacobi加权L2范数下基于正交基的Bezier曲线约束最佳降多阶逼近算法,给出了具体的端点约束最佳降多阶矩阵,且给出了该降阶逼近的可预报的误差公式.同时提出了在L2,L1,L∞范数下适合于最佳降阶逼近的相应Jacobi基的权函数的选取方案.(ⅱ)求解多项式反函数是CAGD中的一个基本问题.利用约束Jacobi基作为有效工具,提出带端点Ck约束的反函数逼近算法.该算法稳定、简易,克服了以往计算反函数的系数时每次逼近系数需全部重新计算的缺陷,同时给出了该算法在PH曲线准弧长参数化中的应用.
(5)为了在计算机辅助几何设计中,有效的求解三角域上Bezier曲面的最小平方逼近问题,给出了三角域上双变量Jacobi和Bernstein基的相互转换矩阵.首先利用Bernstein基构造了三角域上的Jacobi多项式,然后利用单变量Jacobi和Bernstein基的转换关系,给出了三角域上双变量Bernstein的Jacobi基的相互转换矩阵,并且利用该矩阵,得到了在加权L2范数下基于正交基的Bezier曲面最佳降多阶逼近算法,给出了具体的最佳降多阶矩阵,且给出了该降阶逼近的可预报的误差公式.
In this paper, the theme is everal kind of curve's expressions, splices and approximations, in particular, we have made an in-depth research on spiral and orthogonal polynomials in the CAGD and have obtained new creative production on the following five aspects:
(1) Based on the requirement of highway design, a planar cubic C-Bezier spiral with monotone curvature of constant sign and a starting point of zero curvature is constructed, then an algorithm of the transition curve between straight lines and/or circular arcs is derived in detail which can be applied in highway design. As was done with clothoids in engineering, a single spiral is used for straight line to circular arc, two spirals suiting C-shaped or S-shaped transition are used for circular arc to circular arc, two spirals are used for straight line to straight line, and a single spiral is used for circular to circular arc when the latter is contained within the former. The concrete expressions for the first four cases are given; in the fifth case the solution cannot always be obtained. Because straight line segments and circular arcs can be represented precisely by C-Bezier curves, the issues such as highway design can be handled in the system by C-Bezier model, avoiding the difficult situation for computer-aided design system to use the clothoids defined in terms of the Fresnel integral.
(2) A single C-Bezier curve with a shape parameter for G2 joining two circular arcs is constructed. It is shown that a S-shaped transition curve which is able to better manage the broader scope about two circle radii than the Bezier curves has no curvature extrema, and a C-shaped transition curve has a single curvature extremum. Regarding the two kinds of curves, specific algorithms is presented in detail, strict mathematical proof was given, and the effectiveness of the method is showed by examples. This method has the following three advantages:the pattern is unified; the parameters able to adjust the shape of the transition curves are available; the transition curve is only a single segment, and the algorithm can be formulated as a low order equation to be solved for its positive root. Therefore it is simple and easy to implement.
(3) To fit the form of curve in the current Computer Aided Design system and aesthetic needs in industrial design, two approximation algorithms for logarithmic spiral segments is proposed. In the first method, the calculation formula for s-Power series is firstly derived and a fast polynomial approximation algorithm is presented, and then the calculation formula of the offset curves of the logarithmic spiral and corresponding approximation algorithm by s-Power series is presented. In the second method, The G2 Hermite interpolation formula of the two end points by the C-Bezier form is firstly derived, and then the G2 Hermite interpolation approximation algorithm by the C-Bezier form is presented. The results of example show that the algorithms are correct and effective, suitable for the use of the CAD system.
(4)To solve least squares approximation problem effectively in CAGD, the transformation matrices between the weighted orthogonal basis which possesses boundary constraints characteristic and Bernstein basis are derived. Then using the matrices, two specific applications are given.(i) the optimal algorithm based on Jacobi weighted L2 norm for constrained multi-degree reducing Bezier curve, including its matrix representation, is presented. Also the degree reduction error that can be forecasted is given. the Jacobi weighted function adapting to optimal degree reduction is selected with respect to L2, L1, L∞norm, respectively. (ⅱ) To solve the inverse function of polynomial is a basic problem in CAGD, an algorithm about approximating the inverse function with Ck constrains by using the constrained Jacobi basis is proposed. The approximation method is easy and steady. Moreover, the defect that the corresponding coefficients must be recalculated when approximating every inverse function one by one is overcame. As an application, how to generate quasi arc-length parameterization of PH curves is shown.
(5) For solve least squares approximation problem simply and effectively on triangular domains in CAGD, The matrices of transformation of the bivariate Bernstein basis form into the Jacobi basis of the same degree and vice versa are derived. A method for constructing bivariate Jacobi-weighted orthogonal polynomials in the Bernstein form on triangular domains was formulated firstly. And then, by using connection coefficients between the univariate Bernstein and Jacobi basis, the transformation matrices between bivariate Jacobi and Bernstein basis were presented. Finally, Then using the matrices, an explicit form of the multi-degree reduction matrix for Bezierr surface on triangular domains with respect to Jacobi weighted L2 norm was presented, and the error of the degree reduction was given.
引文
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