# Lemniscate

A **lemniscate** is a geometric curve in the form of the digit 8, usually drawn such that the digit is lying on its side, as the infinity symbol . The name derives from the Greek λημνισκος (lemniskos, woolen band).

Two forms are common.

## [edit] Lemniscate of Gerono

This form is named for the French mathematician Camille Christophe Gerono (1799-1891). Its equation in Cartesian coordinates is

- .

The figure shows the case *a* = √2

## [edit] Lemniscate of Bernoulli

This form was discovered by James Bernoulli, who coined the term *Curva Lemniscata*, comparing the curve to a *noeud de ruban* (a ribbon knot) in an article in *Acta Eruditorum* of September 1694 (p. 336). Basically, Bernoulli's lemniscate is the locus of points that have a distance *r*_{1} to a focus *F*_{1} and a distance *r*_{2} to a focus *F*_{2}, while the product *r*_{1}×*r*_{2} is constant. In the figure the foci are on the *x*-axis at ±1. The product of the distances is constant and equal to half the distance 2*a* between the foci squared. For foci on the *x*-axis at ±*a* the equation is,

Expanding and simplifying gives

The latter equation gives upon substitution of

the following polar equation

Bernoulli's lemniscate belongs to the more general class of the Cassini ovals.