lemniscate of Gerono

Image:Lemniscate-of-Gerono.svg

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 $\infty$ 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

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

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.

Dual curve

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

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

Image:Dualger.png

References

{{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | page=[https://archive.org/details/catalogofspecial00lawr/page/124 124] | url-access=registration | url=https://archive.org/details/catalogofspecial00lawr/page/124 }}

External links

{{MacTutor|class=Curves|id=Eight|title=Figure Eight Curve}}

Category:Quartic curves