Centro-affine invariants and the canonical Lorentz metric on the space of centered ellipses

Abstract

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.