This paper deals with restricting curve evolution to a finite and not necessarily flat space of curves, obtained as a subspace of the infinite dimensional space of planar curves endowed with the usual but weak parametrization invariant curve L 2-metric.
We first show how to solve differential equations on a finite dimensional Riemannian manifold defined implicitly as a submanifold of a parameterized one, which in turn may be a Riemannian submanifold of an infinite dimensional one, using some optimal control techniques.
We give an elementary example of the technique on a spherical submanifold of a 3-sphere and then a series of examples on a highly non-linear subspace of the space of closed spline curves, where we have restricted mean curvature motion, Geodesic Active contours and compute geodesic between two curves.