| In Euclidean space, a cubic spline provides the "lowest energy" curve passing through a given set of points. However, if you want an even lower energy curve, and are willing to settle for a curve which just comes close to the given points (instead of passing through them), the answer is again a cubic spline, but a different one. If one specifies points on a Riemannian manifold, and once again looks for an optimal curve on the manifold that comes close to those points while keeping its energy as low as possible, this curve still exists, but finding it presents a challenge, since it is no longer "cubic". |