Lemniscates

A lemniscate is a plane curve with a figure-eight or infinity-symbol shape. The term comes from the Latin lemniscatus meaning “decorated with ribbons,” and it is used for several curves that resemble an eight. The most famous example is the lemniscate of Bernoulli.

Lemniscate of Bernoulli: It can be defined as the locus of points for which the product of the distances to two fixed points (the foci) is constant. With foci at (±a, 0), its Cartesian equation is (x^2 + y^2)^2 = a^2 (x^2 − y^2). In polar coordinates, relative to the origin, it is r^2 = a^2 cos 2θ. The curve consists of two symmetric lobes that meet at the origin, and it is a special case of a Cassini oval.

Area and symmetry: The two lobes are symmetric about both axes. The area of each lobe is a^2/2, so the total area is a^2. The Bernoulli lemniscate is a fourth-degree algebraic curve, illustrating a standard example of a curve with a self-intersection.

Other lemniscates: The term also covers other figure-eight curves, such as the lemniscate of Gerono and various other quartic curves described by different parametric or implicit equations. In mathematics, the word is used to denote curves with a similar eight-shaped form, encountered in geometry, algebraic geometry, and complex analysis as convenient examples and objects of study.