球域Bézier曲线的边界

Boundary of ball Bézier curve

针对几何造型和产品测量中的有效误差分析和误差控制,提出了球域Bézier曲线.借助于微分几何中空间曲面族的包络算法和变量替换方法,求得球域Bézier曲线的精确边界表示;进一步利用函数逼近论中Legendre多项式的最佳一致平方逼近方法,把球域Bézier曲线的边界曲面近似地表示为一张Bézier曲面或分片Bézier曲面的组合.利用球面族的隐式方程,得到球域Bézier曲线的边界曲面的隐式方程,进而把边界曲面参数化为显式方程.理论推导和实例运算结果表明,球域Bézier曲线是一种表达方式简洁、存储空间节省、运算速度较快的误差分析和误差控制工具.

Ball Bézier curve was presented for effective error analysls and error control in geometric modeling and product measuring. The envelope algorithm of a family of space surfaces in differential geometry and variable transformation were used to obtain an accurate representation for the boundary of a ball Bézier curve which can be approximately represented as a Bézier surface or a union of Bézier patches by using Legendre polynomial optimal square uniform approximation in function approximation theory. Implicit equations of a family of spheres were used to obtain an implicit equation of boundary surface which was parameterized and transferred to an explicit equation. Theoretical derivation and instance operation show that ball Bézier curve is a kind of effective error analysis and error control tool with brief expression, minimal storing space and fast operating speed.