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

TL;DR

This paper explores the relationship between centro-affine invariants of convex plane curves and the canonical Lorentz metric on the space of centered ellipses, providing a Lorentzian geometric perspective on affine curve invariants.

Contribution

It introduces a novel approach linking centro-affine invariants to the Lorentz structure on the space of ellipses, extending conformal invariants concepts to affine geometry.

Findings

Centro-affine curvature and arc length are described via null curves of osculating ellipses.

Establishes a Lorentzian geometric framework for centro-affine invariants.

Analogy with conformal invariants in the sphere is developed.

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.