Bézier curves and C2 interpolation in Riemannian manifolds

In a connected Riemannian manifold, generalised Bézier curves are C∞ curves defined by a generalisation, in which line segments are replaced by minimal geodesics, of the classical de Casteljau algorithm. As in Euclidean space, these curves join their first and last control points. We compute the endpoint velocities and (covariant) accelerations of a generalised Bézier curve of arbitrary degree and use the formulae to express the curve's control points in terms of these quantities. These results allow generalised Bézier curves to be pieced together into C2 splines, and thereby allow C2 interpolation of a sequence of data points. For the case of uniform splines in symmetric spaces, we show that C2 continuity is equivalent to a simple relationship, involving the global symmetries at knot points, between the control points of neighbouring curve segments. We also present some examples in hyperbolic 2-space.