Lemniscate of Gerono

The Lemniscate of Gerono

In algebraic geometry, the Lemniscate of Gerono, or figure-eight curve, is a plane algebraic curve of degree four and genus zero shaped like an ∞ symbol, or figure eight. It has equation

$x^{4}-x^{2}+y^{2}=0.$

It was studied by Camille-Christophe Gerono.

Because the curve is of genus zero, it can be parametrized by rational functions; one means of doing that is

$x={\frac {t^{2}-1}{t^{2}+1}},\ y={\frac {2t(t^{2}-1)}{(t^{2}+1)^{2}}}.$

Unlike the Lemniscate of Bernoulli or the Lemniscate of Booth, the double point at the origin of the Lemniscate of Gerono is not an ordinary double point, having a delta invariant of two. The dual curve, pictured below, has therefore a somewhat different character. Its equation is

$(x^{2}-y^{2})^{3}+8y^{4}+20x^{2}y^{2}-x^{4}-16y^{2}=0.$

Dual to the Lemniscate of Gerono