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Blue: lemniscate of Bernoulli (a=1); red: lemniscate of Gerono (a2=2)

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.

[edit] 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 = √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 r1 to a focus F1 and a distance r2 to a focus F2, while the product r1×r2 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 2a between the foci squared. For foci on the x-axis at ±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.

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