Lemniscate of Bernoulli


In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F1 and F2, known as foci, at distance 2a from each other as the locus of points P so that PF1·PF2 = a2. 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. 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 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. 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 square root of minus one| is called the lemniscatic case in some sources.
Using the elliptic integral
the formula of the arc length 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.

Further properties

Dynamics on this curve and its more generalized versions are studied in quasi-one-dimensional models.