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