Misplaced Pages

Lemniscate of Gerono

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
(Redirected from Huygens lemniscate) Plane algebraic curve
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

x 4 x 2 + y 2 = 0. {\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

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

Another representation is

x = cos φ ,   y = sin φ cos φ = sin ( 2 φ ) / 2 {\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

( x 2 y 2 ) 3 + 8 y 4 + 20 x 2 y 2 x 4 16 y 2 = 0. {\displaystyle (x^{2}-y^{2})^{3}+8y^{4}+20x^{2}y^{2}-x^{4}-16y^{2}=0.}
Dual to the lemniscate of Gerono

References

External links

Category: