What is "lemniscate of bernoulli"

n. (context mathematics English) Any closed curve described by a Cartesian equation of the form $(x^2\; +\; y^2)^2\; =\; 2a^2\; (x^2\; -\; y^2)$.

thumb|upright=1.5|right|A lemniscate of Bernoulli and its two foci F and F In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F and F, known as foci, at distance 2a from each other as the locus of points P so that PF·PF = a. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscus, which is Latin for "pendant ribbon". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4.

This lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed focal points is a constant. A Cassini oval, by contrast, is the locus of points for which the product of these distances is constant. In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli.

This curve can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola (bisector of its two foci). It may also be drawn by a mechanical linkage in the form of Watt's linkage, with the lengths of the three bars of the linkage and the distance between its endpoints chosen to form a crossed square.