Asymptotics for Length and Trajectory from Cumulative Chord Piecewise-Quartics

Article type: Research Article

Authors: Kozera, Ryszard

Affiliations: School of Computer Science and Software Engineering, The University of Western Australia, 35 Stirling Highway, Crawley W.A. 6009, Perth, Australia

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 Qm={γ(t0),γ(t1),...,γ(tm)}, with tabular points ti'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.

Keywords: interpolation, unparameterized data, cumulative chord piecewise-quartics, curve and length estimation, orders of approximation

Journal: Fundamenta Informaticae, vol. 61, no. 3-4, pp. 267-283, 2004