Said-Bézier型广义Ball曲线显式降多阶
Explicit Multi-Degree Reduction of Said-Bézier Generalized Ball Curves
修订日期：2009-07-07
DOI：

 作者 单位 刘 刚 浙江大学 计算机图像图形研究所,浙江 杭州 310027 王国瑾 浙江大学 计算机图像图形研究所,浙江 杭州 310027 浙江大学 CAD&CG国家重点实验室,浙江 杭州 310027

给出了计算Said-Bézier型广义Ball曲线(SBGB曲线)在L2范数下保持端点约束的一种最佳降多阶算法.基于SBGB基函数、幂基函数和Jacobi基函数之间的相互转换关系,得到了SBGB基函数和Jacobi基函数之间的显式转换矩阵;进一步利用Jacobi基的正交性和上述转换矩阵的逆矩阵,导出了SBGB曲线在L2范数下的显式约束降多阶算法.此算法蕴含了Said-Ball曲线、Bézier曲线以及位置介于这两类曲线之间的一大类参数曲线的相应降多阶算法.证明了这是一种可以预报最佳误差且满足端点高阶约束的一次性降多阶算法.最后用数值实例说明了算法的正确性和优越性.

This paper presents an optimal algorithm to compute multi-degree reduction of Said-Bézier generalized Ball curves (SBGB) with endpoints constraints in the L2-norm. Based on the relations between Said-Bézier basis, Power basis and Jacobi basis, this paper deduces the explicit transformation matrix from SBGB basis to Jacobi basis and in reverse order. Then based on the inverse matrix of the above matrix and the orthogonality of Jacobi basis, an explicit constrained algorithm for multi-degree reduction of SBGB curves in the L2-norm is put forward. This algorithm can be used in not only Said-Ball curve and Bézier curve but also the large class curves located between the two curves. This paper proves that the algorithm has some superiorities, including approximating optimal error of the degree reduction estimated beforehand, high order interpolation in the endpoints and multi-degree reduction in one time. Numerical examples demonstrate its validity and superiorities.
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