A Note on Modified Hermite Interpolation

R. Kozera, M. Wilkołazka

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2 Citations (Scopus)

Abstract

We discuss the problem of fitting a smooth regular curve γ: [0 , T] → En based on reduced dataQm={qi}i=0m in arbitrary Euclidean space En. The respective interpolation knots T={ti}i=0m satisfying qi= γ(ti) are assumed to be unknown. In our setting the substitutes Tλ={t^i}i=0m of T are selected according to the so-called exponential parameterization governed by Qm and λ∈ [0 , 1]. A modified Hermite interpolant γ^ H introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used here to fit (T^ λ, Qm). The case of λ= 1 (i.e. for cumulative chords) for general class of admissible samplings yields a sharp quartic convergence order in estimating γ∈ C4 by γ^ H [see Kozera (Stud Inf 25(4B–61):1–140, 2004) and Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004)]. As recently shown in Kozera and Wilkołazka (Math Comput Sci, 2018. https://doi.org/10.1007/s11786-018-0362-4) the remaining λ∈ [0 , 1) render a linear convergence order in γ^ H≈ γ for any Qm sampled more-or-less uniformly. The related analysis relies on comparing the difference γ- γ^ H∘ ϕH in which ϕH forms a special mapping between [0, T] and [0 , T^] with T^ = t^ m. In this paper: (a) several sufficient conditions enforcing ϕH to yield a genuine reparameterization are first formulated and then analytically and symbolically simplified. The latter covers also the asymptotic case expressed in a simple form. Ultimately, the reformulated conditions can be algebraically verified and/or geometrically visualized, (b) additionally in Sect. 3, the sharpness of the asymptotics of γ- γ^ H∘ ϕH [from Kozera and Wilkołazka (Math Comput Sci, 2018. https://doi.org/10.1007/s11786-018-0362-4)] is proved upon applying symbolic and analytic calculations in Mathematica.

Original languageEnglish
JournalMathematics in Computer Science
DOIs
Publication statusE-pub ahead of print - 10 Dec 2019

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