Lemniscate

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Revision as of 13:27, 25 December 2009 by imported>Paul Wormer (→‎Lemniscate of Bernoulli)
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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 . 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

.

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,

Expanding and simplifying gives

The latter equation gives upon substitution of

the following polar equation