Hermite Interpolation Using Mobius Transformations of Planar Pythagorean-Hodograph Cubics

Abstract

We present an algorithm for C-1 Hermite interpolation using Mobius 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 Mobius 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 C. Mobius 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.