On parametric smoothness of generalised B-spline curves

In a Riemannian manifold, generalised B-spline curves are piecewise C-infinity curves defined by a generalisation of the classical Cox-de Boor algorithm, in which line segments are replaced by minimal geodesics. Their applications include rigid body motion planning and computer graphics. We prove that, like classical B-spline curves, they are C-1 at knots of multiplicity at most m-1, where m is the degree. We then compute the difference between their left and right (covariant) accelerations at knots of multiplicity at most m-2. Unlike classical B-spline curves, generalised B-spline curves are not in general C-2 at such knots (c) 2006 Elsevier B.V. All rights reserved.