approximation and complexity theory, digital and computational geometry, and ...... Again, Taylor's Theorem applied to each factor of the integrand of (25) com-.
Computer Science Department of The University of Auckland CITR at Tamaki Campus (http://www.citr.auckland.ac.nz/)
CITR-TR-119
Nov 2002
External versus Internal Parameterizations for Lengths of Curves with Nonuniform Samplings Ryszard Kozera 1, Lyle Noakes 2 and Reinhard Klette 3
Abstract This paper studies differences in estimating length (and also trajectory) of an unknown parametric curve γ : [0, 1] → IRn from an ordered collection of data points qi = γ (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 α-order and then when sampling is ε-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 ε-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. 1
The University of Western Australia, School of Computer Science and Software Engineering, 35 Stirling Highway, Crawley WA 6009, Australia.
2
The University of Western Australia, School of Mathematics and Statistics, 35 Stirling Highway, Crawley WA 6009, Australia.
3
The University of Auckland, Centre for Image Technology and Robotics, Tamaki Campus, Building 731, Auckland, New Zealand
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