Lemniscate of Bernoulli

In geometry, the lemniscate of Bernoulli is a plane curve whose shape resembles the numeral 8 or the ∞ symbol. It can be defined from two given points ⁠${\displaystyle F_{1}}$⁠ and ⁠${\displaystyle F_{2}}$⁠, called the foci, as the locus of points ⁠${\displaystyle P}$⁠ satisfying the relation ${\displaystyle |PF_{1}|\cdot |PF_{2}|=c^{2},}$ where the notation ${\displaystyle |AB|}$ means the distance between two points ⁠${\displaystyle A}$⁠ and ⁠${\displaystyle B}$⁠, and ${\displaystyle c={\tfrac {1}{2}}|F_{1}F_{2}|}$ is half the distance between foci. The name lemniscate derives from the Latin word lemniscatus, meaning "decorated with hanging ribbons". The lemniscate of Bernoulli is a special case of the Cassini oval and is a rational algebraic curve of degree four.

The curve was first studied in 1694 by Jakob Bernoulli, who introduced it as a modification of an ellipse. An ellipse is defined as the locus of points for which the sum of the distances to two fixed focal points is constant, whereas a Cassini oval is defined as the locus of points for which the product of these distances is constant. The lemniscate of Bernoulli is the special case of a Cassini oval which passes through the midpoint between its foci.

The lemniscate of Bernoulli results from applying a circle inversion transformation to a hyperbola, where the center of inversion is the midpoint of the hyperbola's foci. It can also be drawn mechanically using a mechanical linkage known as Watt's linkage, provided that the lengths of the three bars and the distance between the fixed endpoints are chosen to form an crossed parallelogram.