# Length Estimation for Exponential Parameterization and ε-Uniform Samplings

• Ryszard Kozera
• Lyle Noakes
• Piotr Szmielew
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8334)

## 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].

## Keywords

Length estimation interpolation numerical analysis computer graphics and vision

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## Authors and Affiliations

• Ryszard Kozera
• 1
• Lyle Noakes
• 2
• Piotr Szmielew
• 1
1. 1.Faculty of Applied Informatics and MathematicsWarsaw University of Life Sciences - SGGWWarsawPoland
2. 2.Department of Mathematics and StatisticsThe University of Western AustraliaPerthAustralia