Lagrange Piecewise-Quadratic Interpolation Based on Planar Unordered Reduced Data

Abstract

This paper discusses the problem of fitting non-parametric unordered reduced data (i.e. a collection of interpolation points) with piecewise-quadratic interpolation to estimate an unknown curve $\gamma$ in Euclidean space $E^2$. The term reduced data stands for the situation in which the corresponding interpolation knots are unavailable. The construction of ordering algorithm based on e-graph of points (i.e. a complete weighted graph using euclidean distances between points as respective weights) is introduced and tested here. The unordered set of input points is transformed into an ordered one upon using a minimal spanning tree (applicable for open curves). Once the order on points is imposed a piecewise-quadratic interpolation $\hat{\gamma }_2$ combined with the so-called cumulative chords is used to fit unordered reduced data. The entire scheme is tested initially on sparse data. The experiments carried out for dense set of interpolation points and designed to test the asymptotics in $\gamma$ approximation by $\hat{\gamma }_2$ result in numerically computed cubic convergence order. The latter coincides with already established asymptotics derived for $\gamma$ estimation via piecewise-quadratic interpolation based on ordered reduced data and cumulative chords.

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