Lemniscate of Bernoulli

In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points "F"1 and "F"2, known as foci, at distance 2"c" from each other as the locus of points "P" so that "PF"1·"PF"2 = "c"2. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from , 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). 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 parallelogram.
Equations.
The equations can be stated in terms of the focal distance "c" or the half-width "a" of a lemniscate.
These parameters are related as formula_1
Arc length and elliptic functions.
The determination of the arc length of arcs of the lemniscate leads to elliptic integrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his "Disquisitiones Arithmeticae").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. For this reason the case of elliptic functions with complex multiplication by is called the "lemniscatic case" in some sources.
Using the elliptic integral
the formula of the arc length formula_10 can be given as
Angles.
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. 
Applications.
Dynamics on this curve and its more generalized versions are studied in quasi-one-dimensional models.