Asymptotics for Length and Trajectory from Cumulative Chord Piecewise-Quartics

Abstract

We discuss the problem of estimating the trajectory of a regular curve γ:[0,T]→Rⁿ and its length d(γ) from an ordered sample of interpolation points Q_m={γ(t₀),γ(t₁),...,γ(t_m)}, with tabular points t_i's unknown, coined as interpolation of unparameterized data. The respective convergence orders for estimating γ and d(γ) with cumulative chord piecewise-quartics are established for different types of unparameterized data including egr-uniform and more-or-less uniform samplings. The latter extends previous results on cumulative chord piecewise-quadratics and piecewise-cubics. As shown herein, further acceleration on convergence orders with cumulative chord piecewise-quartics is achievable only for special samplings (e.g. for ε-uniform samplings). On the other hand, convergence rates for more-or-less uniform samplings coincide with those already established for cumulative chord piecewise-cubics. The results are experimentally confirmed to be sharp for m large and n=2,3. A good performance of cumulative chord piecewise-quartics extends also to sporadic data (m small) for which our asymptotical analysis does not apply directly.