Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics

Abstract

We present an algorithm for $C^1$ Hermite interpolationusing Möbius transformations of planar polynomial Pythagoreanhodograph(PH) cubics. In general, with PH cubics, we cannotsolve $C^1$ Hermite interpolation problems, since their lack of parametersmakes the problems overdetermined. In this paper, weshow that, for each Möbius transformation, we can introduce anextra parameter determined by the transformation, with which wecan reduce them to the problems determining PH cubics in thecomplex plane $ℂ$. Möbius transformations preserve the PH propertyof PH curves and are biholomorphic. Thus the interpolantsobtained by this algorithm are also PH and preserve the topologyof PH cubics. We present a condition to be met by a Hermitedataset, in order for the corresponding interpolant to be simple orto be a loop. We demonstrate the improved stability of these newinterpolants compared with PH quintics.