Polar set (potential theory)

The lemniscate of Gerono

In algebraic geometry, the lemniscate of Gerono, or lemnicate 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 $\infty$ 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

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

Another representation is

$x=\cos \varphi ,\ y=\sin \varphi \,\cos \varphi =\sin(2\varphi )/2$

which reveals that this lemniscate is a special case of a lissajous figure.

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

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

Dual to the lemniscate of Gerono

References

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