  We study the relation between the centro-affine geometry of star-shaped
planar curves and the projective geometry of parametrized maps into $\RP^1$. We
show that projectivization induces a map between differential invariants and a
bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian
evolution equation for closed star-shaped planar curves, discovered by Pinkall,
has the Schwarzian KdV equation as its projectivization. (For both flows, the
curvature evolves by the KdV equation.) Using algebro-geometric methods and the
relation of group-based moving frames to AKNS-type representations, we
construct examples of closed solutions of Pinkall's flow associated with
periodic finite-gap KdV potentials.
