有理二次Bèzier曲线的导矢量模长的最优界限

Optimal Derivative Bounds of Rational Quadratic Bèzier Curves

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||R'(t)||≤λmaxi||P_i-P_(i+1)||中常数l的估计问题,其中P_i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.

For the sake of simplification and convenience, the derivative bound estimation problem was usuallyturned into another estimation problem of parameter ? such that ( ) max i i 1 it ? ? R? ≤ P ? P , where Pi isthe i-th control point of a rational Bèzier curve R(t). This paper focuses on the estimation of the derivativebounds of a rational quadratic Bèzier curve, and provides the optimal low bound of the parameter ?. Firstly,it divides all of the cases of the three weights of R(t) into eight cases; secondly, it explicitly expresses theoptimal bound of ? in the three weights for each case; finally, it leads to a general conclusion for all of thecases. Numerical examples are also given to illustrate that the bounds of the new method are better thanthose of prevailing methods.