Lemniscate

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

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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 = √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×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

$r^{4}=2a^{2}r^{2}(\cos ^{2}\theta -\sin ^{2}\theta )\Longrightarrow r^{2}=2a^{2}\cos 2\theta .$