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.