Lemniscate of Gerono

In algebraic geometry, the lemniscate of Gerono, or lemniscate of Huygens, or figure-eight curve, is a plane algebraic curve of degree four and genus zero and is a lemniscate curve shaped like an formula_1 symbol, or figure eight. It has equation
It was studied by Camille-Christophe Gerono.
Parameterization.
Because the curve is of genus zero, it can be parametrized by rational functions; one means of doing that is
Another representation is
which reveals that this lemniscate is a special case of a Lissajous figure.
Dual curve.
The dual curve (see Plücker formula), pictured below, has therefore a somewhat different character. Its equation is