Choosing nodes in parametric curve interpolation
Computer-Aided Design
The NURBS book
Length Estimation for Curves with Different Samplings
Digital and Image Geometry, Advanced Lectures [based on a winter school held at Dagstuhl Castle, Germany in December 2000]
C 1 Interpolation with Cumulative Chord Cubics
Fundamenta Informaticae
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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(ti)=qi. 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.