On parametric smoothness of generalised B-spline curves

In a Riemannian manifold, generalised B-spline curves are piecewise C^∞ 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 C¹ 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 C² at such knots.