Lemniscate of Bernoulli
The curve has a shape similar to the numeral 8 and to the ∞ symbol.
Its name is from lemniscatus, which is Latin for "decorated with hanging ribbons".
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).
[1] The equations can be stated in terms of the focal distance c or the half-width a of a lemniscate.
Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his Disquisitiones Arithmeticae).
The period lattices are of a very special form, being proportional to the Gaussian integers.
[3] The following theorem about angles occurring in the lemniscate is due to German mathematician Gerhard Christoph Hermann Vechtmann, who described it 1843 in his dissertation on lemniscates.
[4] Dynamics on this curve and its more generalized versions are studied in quasi-one-dimensional models.
