The cubic de Casteljau construction and Riemannian cubics

The classical de Casteljau construction of cubic polynomials in Euclidean space Em has been generalised to Riemannian settings since the 1980's. The generalisations replace line segments by geodesic arcs, yielding elegant methods for Hermite interpolation in Riemannian manifolds M, such as spheres and rotation groups. Unlike the classical algorithm however, it is not so easy to analyse the resulting curves. Even when M is the unit m-sphere Sm in Em+1, the mathematical properties of generalised cubic de Casteljau curves are not well understood.