The cubic de Casteljau construction and Riemannian cubics

The classical de Casteljau construction of cubic polynomials in Euclidean space E^m 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 S^m in E^m+1, the mathematical properties of generalised cubic de Casteljau curves are not well understood. There is another class of curves called Riemannian cubics which can also be used for Hermite interpolation in Riemannian manifolds. Unlike generalised cubic de Casteljau curves, Riemannian cubics are defined as critical curves for a variational problem, and their mathematical properties are much better understood. Riemannian cubics are also more difficult to construct, whereas generalised cubic de Casteljau curves have a simple geometrical construction. It is well-known that, when M is curved, generalised cubic de Casteljau curves and Riemannian cubics are different. On the other hand their general appearance is somewhat similar. In the classical situation where M=E^m, both cubic de Casteljau curves and Riemannian cubics reduce to curves that are cubic polynomial in each coordinate. The present paper analyses the differences between generalised cubic de Casteljau curves and Riemannian cubics. We also modify the generalised cubic de Casteljau algorithm to yield curves that are much closer to Riemannian cubics. In this way the elegant geometry of the de Casteljau algorithm is adjusted to better approximate a class of curves that are better understood from a mathematical point of view. Examples are given of the modified cubic de Casteljau algorithm for curves in the 2-dimensional unit sphere S²⊂E³, and in the special orthogonal group SO(3) with bi-invariant Riemannian metric.