Lemniscate of Gerono

The 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 ${\displaystyle \infty }$ symbol, or figure eight. It has equation

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

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

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

Another representation is

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

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

Dual curve

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

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

Dual to the lemniscate of Gerono

References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. p. 124. ISBN 0-486-60288-5.

External links

• O'Connor, John J.; Robertson, Edmund F., "Figure Eight Curve", MacTutor History of Mathematics Archive, University of St Andrews