Rahul Sharma (Editor)

Lemniscate of Gerono

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit
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 ∞ symbol, or figure eight. It has equation

x 4 − x 2 + y 2 = 0.

It was studied by Camille-Christophe Gerono.

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 ⁡ φ ,   y = sin ⁡ φ cos ⁡ φ = sin ⁡ ( 2 φ ) / 2

which reveals that this lemniscate is a special case of a Lissajous figure.

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.

References

Lemniscate of Gerono Wikipedia
(Text) CC BY-SA