Lemniscate of Gerono

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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

x⁴ − x² + y² = 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

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 (see Plücker formula), pictured below, has therefore a somewhat different character. Its equation is

(x² − y²)³ + 8y⁴ + 20x²y² − x⁴ − 16y² = 0.