Length Estimation for Exponential Parameterization and ε-Uniform Samplings

Abstract

This paper discusses the problem of estimating the length of the unknown curve γ in Euclidean space, from ε-uniformly (for ε ≥ 0) sampled reduced data \(Q_m=\{q_i\}_{i=0}^m\), where γ(t _i ) = q _i . The interpolation knots \(\{t_i\}_{i=0}^m\) are assumed here to be unknown (yielding the so-called non-parametric interpolation). We fit Q _m with the piecewise-quadratic interpolant \(\hat \gamma_2\) combined with the so-called exponential parameterization (characterized by the parameter λ ∈ [0,1]). Such parameterization (applied e.g. in computer graphics for curve modeling [1], [2]) uses estimates of the missing knots \(\{t_i\}_{i=0}^m\approx\{\hat t_i\}_{i=0}^m\). The asymptotic orders β _ε (λ) for length estimation \(d(\gamma)\approx d(\hat \gamma_2)\) in case of λ = 0 (uniformly guessed knots) read as β _ε (0) =  min {4,4ε} (for ε > 0) - see [3]. On the other hand λ = 1 (cumulative chords) renders β _ε (1) =  min {4,3 + ε} (see [4]). A recent result [5] proves that for all λ ∈ [0,1) and ε-uniform samplings, the respective orders amount to β _ε (λ) =  min {4,4ε}. As such β _ε (λ) are independent of λ ∈ [0,1). In addition, the latter renders a discontinuity in asymptotic orders β _ε (λ) at λ = 1. In this paper we verify experimentally the above mentioned theoretical results established in [5].