Convergence orders in length estimation for exponential parameterization and ε-uniform samplings

Abstract

We discuss the problem of approximating the length of a curve γ in arbitrary euclidean space En, from ε-uniformly sampled reduced data Qm={qi}i=0m, where γ(ti) = qi. The interpolation knots {ti}i=0m are assumed in this paper to be unknown (the so-called non-parametric interpolation). We fit Qm with the piecewise-quadratic interpolant γ^2 based on the so-called exponential parameterization (depending on parameter λ ∈ [0, 1]) which estimates the missing knots {ti}i=0m≈{t^i}i=0m.The asymptotic orders βε (λ) for length estimation d{γ}≈d{γ^2} in case of λ = 0 (uniformly guessed knots) read as βε (0) = min{4,4ε} (for ε>0) - see [1]. On the other hand λ = 1 (cumulative chords) yields βε (1) = min{4,3+ε} (see [2]). This paper establishes a more general result holding for all λ ∈ [0,1] and ε-uniform samplings - see Th. 5. More specifically the respective convergence orders amount to βε (λ) = min{4,4ε} for λ ∈ [0,1). Consequently βε (λ) are independent of λ ∈ [0,1) and the discontinuity in asymptotic orders βε (λ) at λ = 1 occurs, for all ε ∈ (0,1). The full proof of Th. 5 is presented in ICNAAM’14 post-conference journal publication together with more exhaustive relevant experimentation.