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
![](http://upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Lemniscate-of-Gerono.svg/220px-Lemniscate-of-Gerono.svg.png)
It was studied by Camille-Christophe Gerono.
Parameterization
editBecause the curve is of genus zero, it can be parametrized by rational functions; one means of doing that is
Another representation is
which reveals that this lemniscate is a special case of a Lissajous figure.
Dual curve
editThe dual curve (see Plücker formula), pictured below, has therefore a somewhat different character. Its equation is
References
edit- J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. p. 124. ISBN 0-486-60288-5.
External links
editWikimedia Commons has media related to Lemniscate of Gerono.