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

:$(x^2-y^2)^3\; +\; 8y^4+20x^2y^2-x^4-16y^2=0.$

References

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