Bezier curves and C2 interpolation in Riemannian manifolds

In a connected Riemannian manifold, generalised Bezier curves are Coo 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 Bezier curve of arbitrary degree and use the formulae to express the curve's control points in terms of these quantities. These results allow generalised Bezier curves to be pieced together into C-2 splines, and thereby allow C-2 interpolation of a sequence of data points. For the case of uniform splines in symmetric spaces, we show that C-2 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. (c) 2007 Elsevier Inc. All rights reserved.