Natural parameterizations of closed projective plane curves

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 $\alpha \le \frac{1}{2}$ 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 $\alpha = \frac{1}{2}$, and the value of $\alpha$ is a global projective invariant of the curve. The parametrization is equivariant with respect to duality.