混合空间曲线曲面及广义Ball曲线的研究
摘要
本文对CAGD中两类重要的曲线曲面——混合空间曲线曲面和广义Ball曲线进行了深入的研究。其中,混合空间曲线曲面包括基于代数三角多项式的双三次C-Hermite曲面,基于代数双曲多项式的H-Bezier曲线曲面,广义Ball曲线包括Wang-Bezier型广义Ball曲线(WBGB曲线)和IBezier-Said-Wang型广义Ball曲线(BSWGB曲线)。主要研究工作及成果如下：
1、构造了双三次C-Hermite曲面,并且给出了双三次C-Hermite曲面的性质。利用双三次C-Hermite曲面给出了椭球面和圆环面的精确表示,并将其应用于图像的缩放处理。
2、关于H-Bezier曲线曲面,主要做了以下几个方面的工作：
·提出了三次H-Bezier曲线的任意分割算法,即对三次H-Bezier曲线上任意一点p(t*)(0≤t*≤α),求该点把曲线分成的两个子曲线段pi(t)(0≤t≤t*)与pα-i(t)(0≤t≤α-t*)的控制参数和控制顶点;给出了三次H-Bezier曲线与三次Bezier曲线的拼接条件,以及三次H-Bezier曲线在曲面造型中应用的例子。
·运用H-Bezier曲线的升阶公式,结合广义逆矩阵理论给出了H-Bezier曲线一次降多阶的逼近方法；估计了降阶的误差界,并建立了与Bezier曲线降阶的关系。并将该结果推广得到了张量积H-Bezier曲面一次降多阶的算法。实验结果表明,采用该方法可取得较好的逼近效果。
·给出了代数双曲空间的拟Legendre基在反函数逼近和等距曲线逼近上的应用。利用多项式和双曲函数的混合多项式序列来逼近反函数,并通过实例证明给出方法的有效性；对基曲线的法矢曲线进行逼近,构造H-Bezier曲线的等距曲线的最佳逼近,这种方法直接求得逼近曲线的控制顶点,计算简单,截断误差小。
·给出了广义H-Bezier曲面的定义,并研究了它的性质。在此基础上,重点研究了a相等的H-Bezier曲面,给出了曲面拼接和分割的条件,并应用于构造一些特殊的曲面。
3、利用BSWGB曲线的对偶基给出了BSWGB曲线的细分算法。这个方法不同于传统的细分方法,传统的细分方法是把BSWGB基转换成幂基,并利用逆矩阵求解给出的。本文的方法给出了现有的一些广义Ball曲线的细分矩阵的统一表达式,可以很方便的利用此表达式,解决这一类曲线的细分问题。
4、分别应用扰动法和最佳一致逼近法,给出WBGB曲线的降阶算法,并给出了误差估计。实验表明,用最佳一致逼近法效果比扰动法要好,若利用扰动法得到的降阶曲线不能达到预期的误差,则可以先利用细分算法对曲线做细分,再逐段用扰动法降阶。WBGB曲线的降阶算法的给出,丰富了广义Ball曲线曲面的理论体系。
This dissertation summaries our researches on the two important kinds of curves and surfaces---curves and surfaces in the hybrid polynomial space and generalized Ball curves, which include the bi-cubic C-Hermite surfaces based on algebraic trigonometric polynomial, H-Bezier curves and surfaces based on algebraic hyperbolic polynomial, Wang-Bezier type generalized Ball curves (WBGB curves for short) and Bezier-Said-Wang type generalized Ball curves (BSWGB curves for short), respectively. The main results in this dissertation are outlined as follows:
1、 Based on the analysis of the properties of C-Hermite polynomials, Bi-cubic C-Hermite surface has been constructed in this dissertation. Furthermore, the applications of bi-cubic C-Hermite surface in geometric modeling and image interpolation are also given, and experimental results demonstrate that the presented method has good effects.
2、The studies on the H-Bezier curves and surfaces are summaried as follows:· A subdivision algorithm of cubic H-Bezier curves is put forward, which serves to determine the control parameters and control points of the two subcurves pt*(t)(0≤t≤t*) and pα-t*(t)(0≤t≤α-t*) subdivided by any point p(t*)(0≤t*≤a) of cubic H-Bezier curves. The connection conditions of cubic H-Bezier curves and cubic Bezier curves are worked out and the applications of cubic H-Bezier curves in the surface modeling are given. The obtained results, which are simple and intuitionistic, can effectively improve the shape representation and control of cubic H-Bezier curves.
· This dissertation gives an approximation method of multidegree reduction of H-Bezier curves by making use of the elevation property of H-Bezier curves, and the theory of generalized inverse matrix. The degree reduction error bound is estimated, the relationship between the reduction of H-Bezier curves and that of Bezier curves is established, and the method of multidegree reduction of tensor H-Bezier surfaces is also got. Numerical examples are given to show that the presented method has good approximation effects, and hence the theory of H-Bezier curves is enriched.
· The applications of the quasi-Legendre basis defined in the algebraic hyperbolic space in the inversion approximation and offsetting are given in this dissertation. Inversion approximation is constructed by using the blending of polynomial and hyperbolic functions, and the experimental results show that the approximation method is effective; We present an approach to approximate the offset curves of the H-Bezier curve based on the ideal of the approximation for the normal curve. The algebraic approximation algorithms which can obtain the control points of the approximation curves directly are simple, intuitive and of high precision.
· Generalized H-Bezier surfaces are defined. On the basis of the properties of generalized H-Bezier surfaces, the study on the α-equaled H-Bezier surfaces is focused. The conditions for geometric continuity and the algorithm for subdivision are given, and some specific surfaces are constructed.
3、A subdivision algorithm of Bezier-Said-Wang type generalized Ball curves is presented. This method gives an explicit subdivision matrix by using the dual bases of BSWGB bases, which is different from the traditional method where an algorithm is needed to convert BSWGB bases into power bases and solve the inverse matrix. This algorithm gives the unifying representation of subdivision matrices of the existing generalized Ball curves, which can be used in solving the subdivision of this kind of curves. Numerical examples are also given to show the effectiveness of our methods.
4、The degree reduction of generalized Ball curves of Wang-Bezier type is discussed by perturbation and the best uniform approximation respectively. The approximation error is given. The experimental results demonstrate that the effect of the best uniform approximation is better than that of perturbation. If we cannot get the error in advance, we can subdivide the original curve firstly, and then reduce the degree piecewisely. The degree reduction method of WBGB curves enriches the theory system of generalizen Ball curves effectively.
1、构造了双三次C-Hermite曲面,并且给出了双三次C-Hermite曲面的性质。利用双三次C-Hermite曲面给出了椭球面和圆环面的精确表示,并将其应用于图像的缩放处理。
2、关于H-Bezier曲线曲面,主要做了以下几个方面的工作：
·提出了三次H-Bezier曲线的任意分割算法,即对三次H-Bezier曲线上任意一点p(t*)(0≤t*≤α),求该点把曲线分成的两个子曲线段pi(t)(0≤t≤t*)与pα-i(t)(0≤t≤α-t*)的控制参数和控制顶点;给出了三次H-Bezier曲线与三次Bezier曲线的拼接条件,以及三次H-Bezier曲线在曲面造型中应用的例子。
·运用H-Bezier曲线的升阶公式,结合广义逆矩阵理论给出了H-Bezier曲线一次降多阶的逼近方法；估计了降阶的误差界,并建立了与Bezier曲线降阶的关系。并将该结果推广得到了张量积H-Bezier曲面一次降多阶的算法。实验结果表明,采用该方法可取得较好的逼近效果。
·给出了代数双曲空间的拟Legendre基在反函数逼近和等距曲线逼近上的应用。利用多项式和双曲函数的混合多项式序列来逼近反函数,并通过实例证明给出方法的有效性；对基曲线的法矢曲线进行逼近,构造H-Bezier曲线的等距曲线的最佳逼近,这种方法直接求得逼近曲线的控制顶点,计算简单,截断误差小。
·给出了广义H-Bezier曲面的定义,并研究了它的性质。在此基础上,重点研究了a相等的H-Bezier曲面,给出了曲面拼接和分割的条件,并应用于构造一些特殊的曲面。
3、利用BSWGB曲线的对偶基给出了BSWGB曲线的细分算法。这个方法不同于传统的细分方法,传统的细分方法是把BSWGB基转换成幂基,并利用逆矩阵求解给出的。本文的方法给出了现有的一些广义Ball曲线的细分矩阵的统一表达式,可以很方便的利用此表达式,解决这一类曲线的细分问题。
4、分别应用扰动法和最佳一致逼近法,给出WBGB曲线的降阶算法,并给出了误差估计。实验表明,用最佳一致逼近法效果比扰动法要好,若利用扰动法得到的降阶曲线不能达到预期的误差,则可以先利用细分算法对曲线做细分,再逐段用扰动法降阶。WBGB曲线的降阶算法的给出,丰富了广义Ball曲线曲面的理论体系。
This dissertation summaries our researches on the two important kinds of curves and surfaces---curves and surfaces in the hybrid polynomial space and generalized Ball curves, which include the bi-cubic C-Hermite surfaces based on algebraic trigonometric polynomial, H-Bezier curves and surfaces based on algebraic hyperbolic polynomial, Wang-Bezier type generalized Ball curves (WBGB curves for short) and Bezier-Said-Wang type generalized Ball curves (BSWGB curves for short), respectively. The main results in this dissertation are outlined as follows:
1、 Based on the analysis of the properties of C-Hermite polynomials, Bi-cubic C-Hermite surface has been constructed in this dissertation. Furthermore, the applications of bi-cubic C-Hermite surface in geometric modeling and image interpolation are also given, and experimental results demonstrate that the presented method has good effects.
2、The studies on the H-Bezier curves and surfaces are summaried as follows:· A subdivision algorithm of cubic H-Bezier curves is put forward, which serves to determine the control parameters and control points of the two subcurves pt*(t)(0≤t≤t*) and pα-t*(t)(0≤t≤α-t*) subdivided by any point p(t*)(0≤t*≤a) of cubic H-Bezier curves. The connection conditions of cubic H-Bezier curves and cubic Bezier curves are worked out and the applications of cubic H-Bezier curves in the surface modeling are given. The obtained results, which are simple and intuitionistic, can effectively improve the shape representation and control of cubic H-Bezier curves.
· This dissertation gives an approximation method of multidegree reduction of H-Bezier curves by making use of the elevation property of H-Bezier curves, and the theory of generalized inverse matrix. The degree reduction error bound is estimated, the relationship between the reduction of H-Bezier curves and that of Bezier curves is established, and the method of multidegree reduction of tensor H-Bezier surfaces is also got. Numerical examples are given to show that the presented method has good approximation effects, and hence the theory of H-Bezier curves is enriched.
· The applications of the quasi-Legendre basis defined in the algebraic hyperbolic space in the inversion approximation and offsetting are given in this dissertation. Inversion approximation is constructed by using the blending of polynomial and hyperbolic functions, and the experimental results show that the approximation method is effective; We present an approach to approximate the offset curves of the H-Bezier curve based on the ideal of the approximation for the normal curve. The algebraic approximation algorithms which can obtain the control points of the approximation curves directly are simple, intuitive and of high precision.
· Generalized H-Bezier surfaces are defined. On the basis of the properties of generalized H-Bezier surfaces, the study on the α-equaled H-Bezier surfaces is focused. The conditions for geometric continuity and the algorithm for subdivision are given, and some specific surfaces are constructed.
3、A subdivision algorithm of Bezier-Said-Wang type generalized Ball curves is presented. This method gives an explicit subdivision matrix by using the dual bases of BSWGB bases, which is different from the traditional method where an algorithm is needed to convert BSWGB bases into power bases and solve the inverse matrix. This algorithm gives the unifying representation of subdivision matrices of the existing generalized Ball curves, which can be used in solving the subdivision of this kind of curves. Numerical examples are also given to show the effectiveness of our methods.
4、The degree reduction of generalized Ball curves of Wang-Bezier type is discussed by perturbation and the best uniform approximation respectively. The approximation error is given. The experimental results demonstrate that the effect of the best uniform approximation is better than that of perturbation. If we cannot get the error in advance, we can subdivide the original curve firstly, and then reduce the degree piecewisely. The degree reduction method of WBGB curves enriches the theory system of generalizen Ball curves effectively.
引文
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