规范化三次埃尔米特(Hermite)曲线插值

作者:追风剑情 发布于:2024-12-31 10:58 分类:Algorithms

三次埃尔米特(Hermite)曲线插值

为了使P(t)的定义区间 $t_0 \le t \le t_1$ 变为区间 $0 \le u \le 1$,可以做如下变换 $$ u=\frac{t-t_0}{t_1-t_0} \tag{4-25} $$ 从式(4-25)中解出 $t=t_0+(t_1-t_0)u$,代入式(4-23)中各式,得 \begin{equation} \left\{ \begin{aligned} g_{00}(t)&=g_{00}[t_0+(t_1-t_0)u]=(1+2u)(u-1)^2=2u^3-3u^2+1=q_{00}(u) \\ g_{01}(t)&=g_{01}[t_0+(t_1-t_0)u]=(3-2u)u^2=-2u^3+3u^2=q_{01}(u) \\ h_{00}(t)&=h_{00}[t_0+(t_1-t_0)u]=u(1-u)^2=(u^3-2u^2+u)(t_1-t_0)=q_{10}(u)(t_1-t_0) \\ h_{01}(t)&=h_{01}[t_0+(t_1-t_0)u]=(-1+u)u^2=(u^3-u^2)(t_1-t_0)=q_{11}(u)(t_1-t_0) \end{aligned} \right. \tag{4-26} \end{equation}

将式(4-26)的结果代入式(4-22),使所求的三次多项式成为 $$ \begin{flalign} \widetilde{f}(u)&=\widetilde{P}(u)=P[t_0+(t_1-t_0)u] \\ &=f(t_0)q_{00}(u) + f(t_1)q_{01}(u) + f^{\prime}(t_0)q_{10}(u)(t_1-t_0) + f^{\prime}(t_1)q_{11}(u)(t_1-t_0) \tag{4-27} &\\ \end{flalign} $$ 式中,混合函数 $q_{00}(t)、q_{01}(t)、q_{10}(t)、q_{11}(t)$ 如下,如图 4-5 所示。 $$ \begin{equation} \left\{ \begin{aligned} q_{00}(u)&=2u^3-3u^2+1 \\ q_{01}(u)&=-2u^3 + 3u^2 \\ q_{10}(u)&=u^3 - 2u^2+u \\ q_{11}(u)&=u^3-u^2 \end{aligned} \right. \tag{4-28} \end{equation} $$

插图

根据式(4-27)计算可知 $$ \begin{flalign} \widetilde{f}(0)=f(t_0)&,\widetilde{f}(1)=f(t_1) \\ \widetilde{f}^{\prime}(0)=f^{\prime}(t_0)(t_1-t_0)&,\widetilde{f}^{\prime}(1)=f^{\prime}(t_1)(t_1-t_0) \tag{4-29} \end{flalign} $$ 可以得到 $$ \begin{flalign} \widetilde{P}(u)&=\widetilde{f}(0)q_{00}(u) + \widetilde{f}(1)q_{01}(u) + \widetilde{f}^{\prime}(0)q_{10}(u) + \widetilde{f}^{\prime}(1)q_{11}(u) \\ &= \begin{pmatrix} q_{00}(u) & q_{01}(u) & q_{10}(u) & q_{11}(u) \end{pmatrix} \begin{pmatrix} \widetilde{f}(0) \\ \widetilde{f}(1) \\ \widetilde{f}^{\prime}(0) \\ \widetilde{f}^{\prime}(1) \end{pmatrix} \\ &= \begin{pmatrix} 2u^3-3u^2+1 & -2u^3+3u^2 & u^3-2u^2+u & u^3-u^2 \end{pmatrix} \begin{pmatrix} \widetilde{f}(0) \\ \widetilde{f}(1) \\ \widetilde{f}^{\prime}(0) \\ \widetilde{f}^{\prime}(1) \end{pmatrix} \\ &= \begin{pmatrix} u^3 & u^2 & u & 1 \end{pmatrix} \begin{pmatrix} 2 & -2 & 1 & 1 \\ -3 & 3 & -2 & -1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} \widetilde{f}(0) \\ \widetilde{f}(1) \\ \widetilde{f}^{\prime}(0) \\ \widetilde{f}^{\prime}(1) \end{pmatrix} \tag{4-30} \end{flalign} $$

标签: Algorithms

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