有理二次Bèzier曲线的导矢量模长的最优界限

摘要

For the sake of simplification and convenience, the derivative bound estimation problem was usu-ally turned into another estimation problem of parameterl such that tl+¢-R≤PP, wherePi is 1( ) max i i i thei-th control point of a rational Bèzier curveR(t). This paper focuses on the estimation of the derivative bounds of a rational quadratic Bèzier curve, and provides the optimal low bound of the parameterl. Firstly, it divides all of the cases of the three weights ofR(t) into eight cases; secondly, it explicitly expresses the optimal bound ofl in the three weights for each case; finally, it leads to a general conclusion for all of the cases. Numerical examples are also given to illustrate that the bounds of the new method are better than those of prevailing methods.%为了简化与方便估算，有理 Bèzier 曲线 R(t)的导矢量模长估计问题通常转化为t l+¢-R ≤ P P 中常1() max i i i数l的估计问题，其中 Pi为 R(t)对应的第 i 个控制点。针对有理二次 Bèzier 曲线的导矢量模长估计问题，提出参数l的最优下界估算方法。首先将有理二次 Bèzier 曲线的三个权因子的所有情形归结为8种类型；然后分别对每一类情形显式地给出参数l关于三个权因子的表达式，并证明了这是参数l对应的最优下界；最后综合所有的8类情形，给出了相应的结论。通过数值例子，进一步验证了该方法得到结果的最优性。