【作者】 江平；

【导师】 檀结庆；

【作者基本信息】 合肥工业大学 ， 计算机应用技术， 2006， 博士

【摘要】 在计算机辅助几何设计中,定义在千变万化的拓扑结构上的自由曲线曲面,存在着千变万化的形式,而广义Ball曲线曲面则是其中一种在曲线求值及升降阶的计算速度方面明显优于Bézier曲线的曲线曲面。本文主要是基于不同形式曲线曲面之间的转换,并结合区间(圆域)算法、曲线曲面的降阶等问题,对广义Ball曲线曲面几何造型的相关问题进行了较深入的研究。研究成果主要体现在以下几个方面: 1.在WSGB基函数的对偶基的基础上,得到了WSGB曲线与Bézier曲线之间的互换关系式,同时也就得到了Bézier曲线与Said-Ball曲线、Wang-Ball曲线之间的互换。另外,还给出了一种WSGB曲线的显式细分算法,从而避免了转换成幂基及求逆的过程。还给出了几个相关的组合恒等式以及幂函数在WSGB基下的Marsden恒等式。同时,由WSGB基与Bernstein基之间的转换公式,还给出了WSGB曲线的包络算法(几何生成算法)。 2.刘松涛和刘根洪([刘96])、邬弘毅([邬98])曾分别利用菱形算法与直接展开法给出了三角域上Said-Ball曲面与Bézier曲面之间的转换公式。而本文通过引入一族三角域上带位置参数H的广义Ball基和广义Ball曲面,利用相邻两曲面的基函数之间的关系,给出三角域上Said-Ball曲面与Bézier曲面之间互相转换的递归算法。该算法计算量小,编程简单,更有助于广义Ball曲面的推广应用。最后还在计算复杂性方面与[刘96]的菱形算法与[邬98]的直接展开法这两种不同的算法进行了比较。 3.目前,Bézier曲线曲面降多阶方法中多采用求逆矩阵的方法得到逼近曲线的控制点表达式,这无疑会导致计算的复杂性。Tchebyshev多项式的最小零偏差性质在研究曲线曲面降阶时起到了非常重要的作用,有鉴于此,本文给出了Tchebyshev多项式与Bernstein基函数之间的转换递推算法,将其应用于Bézier曲线曲面的降阶处理,避免了求近似最佳一致逼近曲线时需要求逆矩阵的麻烦,且该算法稳定、计算量小。 4.给出了区间Said-Ball曲线的边界表示,并分别用线性规划法及最佳一致逼近法讨论了区间Ball曲线的降阶算法。实验结果表明,用最佳一致逼近法效果显然比线性规划法好。若利用线性规划法得到的区间曲线不能达到预期的误差,则可以先对曲线在t=1/2处做细分,再逐段用线性规划法降阶,而且用线性规划法对n(n≥3)次区间Ball曲线降阶时,降阶后的曲线必定插值端点,而利用最佳一致逼近法则不一定,若要实现插值端点,则必须增加约束条件。 5.讨论了圆域Said-Ball曲线的降阶问题。首先给出圆域Said-Ball曲线的定义,讨论了圆域更多还原

【Abstract】 Free curves/ surfaces defined in different topological structure have different expressions in CAGD, and generalized Ball curves are more efficient than Bezier curves in calculation, the degree elevation and reduction. Based on the transformation between the different curves/ surfaces, and the interval (disc) arithmetical and the degree reduction of the curves and surfaces, this dissertation focuses on the research about the geometrical modeling of the generalized Ball curves/ surfaces. The achieved main results are as follows:1. By means of the dual functionals of the generalized Ball basis of Wang-Said type, we investigated the basis transformation formula between Bernstein basis and WSGB basis, and at the same time, worked out the basis transformation formulae between Bernstein basis and Wang-Ball and Said-Ball basis. We discussed the explicit subdivision algorithms for WSGB curves of degree n = 2m + 1, including the subdivision algorithm for Wang-Ball curves with L = m-1 and the subdivision algorithm for Said-Ball curves with L = 0. In the last part, we also constructed the envelope algorithm of the WSGB curves. As a result, the curves of higher degrees can be generated from the curves of lower degrees with envelope technique.2. By introducing a family of generalized Ball bases and generalized Ball surfaces with position parameter H and using the relation between the bases of adjacent surfaces, this dissertation presents a new algorithm for conversion of Said-Ball surfaces and Bezier surfaces on triangular domain, which is easy for programming.3. Considering the fact that orthogonal Tchebyshev polynomials, which produce the least zero-deviation approximation, play important roles in the degree reduction of curves and surfaces, we worked out a recursive conversion algorithm between the Tchebyshev polynomials and Bernstein polynomials instead of computing the inverse matrices in the process of degree reduction of Bezier curves and surfaces as required in many current existing methods, which proves more stable and more economical than other methods.4. The boundary of interval Ball curves of degree n is discussed and two algorithms for the degree reduction of interval Ball curves are given by means of the linear programming method and best approximation method respectively. If the interval curves obtained by linear programming method fail to be controlled within the expected error, then the subdivision techniques can be used.5. The basic concepts of disk Said-Ball curves are given in a form suitable for the study. The control points of disk Said-Ball curves are disks, rather than rectangles as in the interval Said-Ball curves. To solve the degree reduction problem of disk Said-Ball curves, we design an algorithm involving both cases: non-interpolating endpoints and interpolating endpoints of disk Said-Ball curves. Some examples are provided to demonstrate the efficiency of the algorithm.6. The degree reduction of generalized Ball curves of Wang-Said type by perturbation and the best uniform approximation are respectively discussed, the approximation error and relative approximation error are analyzed, and the comparison between these two methods is made by some examples.更多还原