Lemniscate of Gerono explained

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

inf ty 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 = (t ^{2-1})/(t ^{2+1}) , y = (2 t(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} + 8 y ^{4+20x} ^{2y} ^{2-x} ^{4-16y} ^{2=0.}

References

• Book: J. Dennis Lawrence . A catalog of special plane curves . . 1972 . 0-486-60288-5 . 124 . registration .