Natural parameterizations of closed projective plane curves

TL;DR

This paper introduces a new projectively invariant, balanced parametrization for smooth closed convex projective plane curves, which is globally periodic, unique, and characterizes ellipses via a specific scalar parameter.

Contribution

The authors define a novel balanced parametrization that extends the Forsyth-Laguerre parametrization to closed curves, incorporating a scalar parameter to ensure periodicity and invariance.

Findings

The balanced parametrization is unique up to a shift for non-quadratic curves.

Ellipses are characterized by the scalar parameter  = 1/2.

The scalar parameter  is a global projective invariant.

Abstract

A natural parametrization of smooth projective plane curves which tolerates the presence of sextactic points is the Forsyth-Laguerre parametrization. On a closed projective plane curve, which necessarily contains sextactic points, this parametrization is, however, in general not periodic. We show that by the introduction of an additional scalar parameter $\alpha \leq \frac12$ one can define a projectively invariant $2\pi$-periodic global parametrization on every simple closed convex sufficiently smooth projective plane curve without inflection points. For non-quadratic curves this parametrization, which we call balanced, is unique up to a shift of the parameter. The curve is an ellipse if and only if $\alpha = \frac12$, and the value of $\alpha$ is a global projective invariant of the curve. The parametrization is equivariant with respect to duality.