  We consider smooth plane curves which are convex with respect to the origin.
We describe centro-affine invariants (that is, GL_+(2,R)-invariants), such as
centro-affine curvature and arc length, in terms of the canonical Lorentz
structure on the three dimensional space of all the ellipses centered at zero,
by means of null curves of osculating ellipses. This is the centro-affine
analogue of the approach to conformal invariants of curves in the sphere
introduced by Langevin and O'Hara, using the canonical pseudo Riemannian metric
on the space of circles.

Differential Geometry