In mathematics, a lemniscate is a type of curve described by a Cartesian equation of the form:

(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 (an ellipse is the locus of points which are equidistant from two given points, i.e. the sum of the two distances is constant for all points on an ellipse, but in the case of the lemniscate, the product of these distances is constant). He called it the lemniscus, which is Latin for 'pendant ribbon'.

The lemniscate 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).

Other equations

A lemniscate may also be described by the polar equation

r² = a²cos2φ

or the bipolar equation