Lemniscate of Bernoulli

(mathematics) The locus of points (x,y) in the plane satisfying the equation (x² + y²)² = a² (x² - y²) or, in polar coordinates (r,θ), the equation r² = a² cos 2θ.

A curve shaped like the figure eight (see illustration), referred to by Jacques Bernoulli in 1694. Let F₁, F₂ be points of a plane π with F₁F₂ = 2a, a > 0. The locus of a point P of π which moves so that PF₁ · PF₂ = b₂, where b is a positive constant, is called an oval of Cassini. The lemniscate is obtained when b = a. Its equation in rectangular coordinates is (x² + y²)²= a²(x² − y²) and in polar coordinates ρ² = a² cos 2θ. See also Analytic geometry.

Curve known as lemniscate.