External versus Internal Parameterizations for Lengths of Curves with Nonuniform Samplings

Abstract

This paper studies differences in estimating length (and also trajectory) of an unknown parametric curve from an ordered collection of data points qi = GAMA (ti), with either the ti's known or unknown. For the ti's uniform (known or unknown) piecewise Lagrange interpolation provides efficient length estimates, but in other cases it may fail. In this paper, we apply this classical algorithm when the ti's are sampled according to first ALPHA-order and then when sampling is EPSILON-uniform. The latter was introduced in [20] for the case where ti's are unknown. In the present paper we establish new results for the case when the ti's are known for both types of samplings. For curves sampled EPSILON-uniformly comparison is also made between the cases, where the tabular parameters ti's are known and unknown. Numerical experiments are carried out to investigate sharpness of our theoretical results. The work may be of interest in computer vision and graphics, approximation and complexity theory, digital and computational geometry, and digital image analysis.