Interpolation on the hypersphere with Thiele type rational interpolants

Abstract

In order to interpolate 2n + 1 points on the unit hypersphere $\mathcal{S}^{d-1}$ with a vector-valued rational function, we use the Generalised Inverse Rational Interpolants (GIRI) of Graves–Morris. The construction process of these Thiele type rational interpolants is based on the Samelson’s inverse for vectors. We show that in general any GIRI of 2n + 1 points of $\mathcal{S}^{d-1}$ lies on $\mathcal{S}^{d-1}$. We also show that the stereographic projection induces a one-to-one correspondence between the set of vector-valued rational functions lying on $\mathcal{S}^{d-1}$ and the set of Generalised Inverse Rational Fractions in the equator plane.