In algebraic geometry, the lemniscate of Gerono, or lemniscate of Huygens, or figure-eight curve, is a

plane algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...

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 Camille-Christophe Gerono (1799 in Paris, France – 1891 in Paris) was a French mathematician. He concerned himself above all with geometry. The Lemniscate of Gerono or ''figure-eight curve'' was named after him. With Olry Terquem, he was foundin ...

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 A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations : x=A\sin(at+\delta),\quad y=B\sin(bt), which describe the superposition of two perpendicular oscillations in x and y dir ...

Dual curve

The

dual curve In projective geometry, a dual curve of a given plane curve is a curve in the dual projective plane consisting of the set of lines tangent to . There is a map from a curve to its dual, sending each point to the point dual to its tangent line. I ...

(see

Plücker formula In mathematics, a Plücker formula, named after Julius Plücker, is one of a family of formulae, of a type first developed by Plücker in the 1830s, that relate certain numeric invariants of algebraic curves to corresponding invariants of their du ...

), 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