Lemniscate of Gerono

File:Lemniscate-of-Gerono.svg

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

$x^4-x^2+y^2=0.$

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

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

Another representation is

$$\begin{gathered} x=\cos \phi , \\ y=\sin \phi \cos \phi =\sin (2\phi )/2 \end{gathered}$$

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

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

File:Dualger.png

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