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 $$\infty$$. The name derives from the Greek λημνισκος (lemniskos, woolen band).

Two forms are common.

Lemniscate of Gerono
This form is named for the French mathematician Camille Christophe Gerono (1799-1891). Its equation in Cartesian coordinates is
 * $$\ x^4 = a^2 (x^2 - y^2)$$.

The figure shows the case a = &radic;2

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 r1 to a focus F1 and a distance r2 to a focus F2, while the product r1&times;r2 is constant. In the figure the foci are on the x-axis at &plusmn;1. The product of the distances is constant and equal to half the distance 2a between the foci squared. For foci on the x-axis at &plusmn;a the equation is,

r_1\,r_2 = a^2 = \left[ (x-a)^2 + y^2\right]^{\frac{1}{2}} \left[(x+a)^2 + y^2\right]^{\frac{1}{2}}. $$ Expanding and simplifying gives

(x^2 + y^2)^2 = 2a^2 (x^2 - y^2).\; $$ The latter equation gives upon substitution of

x=r\cos\theta,\quad y = r\sin\theta $$ the following polar equation

r^4 = 2 a^2 r^2 (\cos^2\theta - \sin^2\theta) \Longrightarrow r^2 = 2 a^2 \cos2\theta. $$ Bernoulli's lemniscate belongs to the more general class of the Cassini ovals.