Length Estimation for the Adjusted Exponential Parameterization

Abstract

In this paper we discuss the problem of interpolating the so-called reduced data $Q_m=\{q_i\}_{i=0}^m$ to estimate the length d(γ) of the unknown curve γ sampled in accordance with γ(t _i ) = q _i . The main issue for such non-parametric data fitting (given a fixed interpolation scheme) is to complement the unknown knots $\{t_i\}_{i=0}^m$ with $\{\hat t_i\}_{i=0}^m$, so that the respective convergence prevails and yields possibly fast orders. We invoke here the so-called exponential parameterizations (including centripetal) combined with piecewise-quadratics (and -cubics). Such family of guessed knots $\{\hat{t}_i^{\lambda}\}_{i=0}^m$ (with 0 ≤ λ ≤ 1) comprises well-known cases. Indeed, for λ = 0 a blind uniform guess is selected. When λ = 1/2 the so-called centripetal parameterization is invoked. On the other hand, if λ = 1 cumulative chords are applied. The first case yields a bad length estimation (with possible divergence). In opposite, cumulative chords match the convergence orders established for the non-reduced data i.e. for $(\{t_i\}_{i=0}^m, Q_m)$. In this paper we show that, for exponential parameterization, while λ ranges from one to zero, diminishing convergence rates in length approximation occur. In addition, we discuss and verify a method of possible improvement for such decreased rates based on iterative knot adjustment.