几何连续的多项式插值逼近与Hermite插值的比较(英文)

Polynomial Interpolant Approximation With Geometric Continuity is Much Better Than Hermite Interpolation

一般而言 ,几何连续的Bzier插值曲线比Hermite插值曲线具有更多的自由度 ,因此 ,若插值多项式的阶数相同 ,前者的逼近误差通常比后者更小 .但有时 ,这种差异会非常大 .本文以四分之一圆的逼近为例 ,发现了三次Hermite插值曲线的误差ε 3 (t)是基于几何连续的三次Bzier插值曲线误差ε3 (t)的 4 0 0多倍 ,即 ε 3 (t) ∞≥ 4 0 0 · ε3 (t) ∞ .即使考察具有一个自由参数c的 4阶Hermite插值曲线 ,其误差函数ε 4(t,c)仍然满足minc∈R maxt∈ [0 ,1]ε 4(t,c) ≥ 6 .3· maxt∈ [0 ,1]ε3 (t) .

Generaly speaking, Bzier curve interpolation with geometric continuity has more free parameter than corresponding Hermite interpolation, therefore with the same interpolation polynomial degree, the approximation error of the former is smaller. But sometimes there is a surprising big difference between them. In this paper, as approximating a quarter unit circle, we show that the error ε *_3(t) of cubic Hermite interpolation is of 400 times the error ε_3(t) for cubic Bzier curve interpolation with geometric continuity, i.e., ε *_3(t) _∞ ≥400 ·ε_3(t)_∞. Furthermore, for the error ε *_4(t,c) of Hermite interpolation of degree 4 with a free parameter c, we still have \%min\% c∈R \%max\% t∈ ε *_4(t,c) ≥6.3 · \%max\% t∈ ε_3(t) .