- Art Gallery -


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 \infty 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

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

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.

The dual curve (see Pl├╝cker formula), pictured below, has therefore a somewhat different character. Its equation is

\( (x^2-y^2)^3 + 8y^4+20x^2y^2-x^4-16y^2=0. \)

Dual to the lemniscate of Gerono


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

External links

O'Connor, John J.; Robertson, Edmund F., "Figure Eight Curve", MacTutor History of Mathematics archive, University of St Andrews.

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License

Home - Hellenica World