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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 560246, 15 pages
http://dx.doi.org/10.1155/2012/560246
Research Article

Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics

Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 17 January 2012; Accepted 11 February 2012

Academic Editor: Saminathan Ponnusamy

Copyright © 2012 Sunhong Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present an algorithm for 𝐶 1 Hermite interpolation using Möbius transformations of planar polynomial Pythagoreanhodograph (PH) cubics. In general, with PH cubics, we cannot solve 𝐶 1 Hermite interpolation problems, since their lack of parameters makes the problems overdetermined. In this paper, we show that, for each Möbius transformation, we can introduce an extra parameter determined by the transformation, with which we can reduce them to the problems determining PH cubics in the complex plane . Möbius transformations preserve the PH property of PH curves and are biholomorphic. Thus the interpolants obtained by this algorithm are also PH and preserve the topology of PH cubics. We present a condition to be met by a Hermite dataset, in order for the corresponding interpolant to be simple or to be a loop. We demonstrate the improved stability of these new interpolants compared with PH quintics.