Lemniscate of Gerono

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

  • J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. p. 124. ISBN 0-486-60288-5.

External links

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