Length Estimation for Exponential Parameterization and ε-Uniform Samplings Ryszard Kozera 1 , Lyle Noakes 2 , and Piotr Szmielew 1 1 Warsaw University of Life Sciences - SGGW Faculty of Applied Informatics and Mathematics Nowoursynowska str. 159, 02-776 Warsaw, Poland 2 Department of Mathematics and Statistics The University of Western Australia 35 Stirling Highway, Crawley W.A. 6009, Perth, Australia ryszard kozera@sggw.pl, lyle.noakes@maths.uwa.edu.au, piotr szmielew@sggw.pl 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 Qm = {qi } m i=0 , where γ(ti )= qi . The interpola- tion knots {ti } m i=0 are assumed here to be unknown (yielding the so-called non-parametric interpolation). We fit Qm with the piecewise-quadratic interpolant ˆ γ2 combined with the so-called exponential parameterization (characterized by the parameter λ ∈ [0, 1]). Such parameterization (ap- plied e.g. in computer graphics for curve modeling [1], [2]) uses estimates of the missing knots {ti } m i=0 ≈{ ˆ ti } m i=0 . The asymptotic orders βε(λ) for length estimation d(γ) ≈ d(ˆ γ2) in case of λ = 0 (uniformly guessed knots) read as βε(0) = min{4, 4ε} (for ε> 0) - see [3]. On the other hand λ = 1 (cumulative chords) renders βε(1) = min{4, 3+ ε} (see [4]). A re- cent result [5] proves that for all λ ∈ [0, 1) and ε-uniform samplings, the respective orders amount to βε(λ) = min{4, 4ε}. As such βε(λ) are in- dependent of λ ∈ [0, 1). In addition, the latter renders a discontinuity in asymptotic orders βε(λ) at λ = 1. In this paper we verify experimentally the above mentioned theoretical results established in [5]. Keywords: Length estimation, interpolation, numerical analysis, com- puter graphics and vision. 1 Introduction In classical non-parametric interpolation (see e.g. [6]) the sampled data points Q m = {q i } m i=0 satisfying γ (t i )= q i ∈ IR n yield the following pair ({t i } m i=0 ,Q m ) commonly known as non-reduced data. For the need of this paper, we also stip- ulate that t i <t i+1 , q i = q i+1 and that γ : [0,T ] → IR n (with 0 <T< ∞) is a sufficiently smooth (specified later) regular curve ˙ γ (t) = 0. Recall that the length of the curve γ is defined as: d(γ )=  T 0 ‖ ˙ γ (t)‖dt. (1) F. Huang and A. Sugimoto (Eds.): PSIVT 2013 Workshops, LNCS 8334, pp. 33–46, 2014. c  Springer-Verlag Berlin Heidelberg 2014