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An approach to data parametrization in parametric cubic spline interpolation problems. (English) Zbl 0558.41006

From the author’s summary: ”A new approach to the problem of parametrizing data in parametric cubic spline interpolation problems is discussed. Parametrizations \(0=t_ 0<t_ 1<...<t_ N=1\) of K- dimensional data \(\{Z_ i\}^ N_{i=0}\), \(Z_ i=(z^ 1_ i,...,z^ k_ i)\) are chosen by minimizing \(\sum^{k}_{\ell =1}(1/\alpha \ell^ 2)\int^{1}_{0}(d^ 2\theta^{\ell}/dt^ 2)^ 2dt\), where \(\theta^{\ell}(t)\) is the natural cubic spline with broken points \(t_ 0\), \(t_ 1,...,t_ N\) satisfying \(\theta^{\ell}(t_ i)=z_ i^{\ell}\), \(i=0,...,N\) and \(\alpha_{\ell}\), \(\ell =1,...,K\), are positive numbers. This approach yields parametrizations which, by complementing the well-known smoothest interpolation property of natural cubic splines, leads to smoother component functions. The improvements are, in part, evidenced by reduced position overshoots and lower second derivatives. A closed form solution of the problem is derived for one- dimensional data. In higher dimensions the gradient projection method is used to obtain approximate numerical solutions. Geometric curve fitting problems and an example involving the design of a trajectory for a robot manipulator are used to illustrate the method.”
Reviewer: R.B.Saxena

MSC:

41A05 Interpolation in approximation theory
41A15 Spline approximation
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