algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, the lemniscate of Gerono, or lemniscate of Huygens, or figure-eight curve, is a plane algebraic curve of degree four and

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...

zero and is a

lemniscate In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves. The word comes from the Latin "''lēmniscātus''" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons",. or which alternative ...

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,\ y = \frac.

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.

References

External links

*{{MacTutor, class=Curves, id=Eight, title=Figure Eight Curve Algebraic curves Christiaan Huygens