- (x2 + y2)2 = a2(x2 − y2)
Graphing this equation produces a curve similar to . The curve has become a symbol of infinity and is widely used in math. The symbol itself is sometimes referred to as the lemniscate. Its Unicode representation is ∞ (
The 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 lemniscate, by contrast, is the locus of points for which the product of these distances is constant. Bernoulli called it the lemniscus, which is Latin for 'pendant ribbon'.
A lemniscate may also be described by the polar equation
- r2 = a2cos2φ
or the bipolar equation
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 Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his Disquisitiones Arithmeticae).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 the square root of minus one is called the lemniscatic case in some sources.