Lemniscate

A lemniscate is a plane curve shaped like a figure eight. The name most often denotes the Bernoulli lemniscate, defined by a quartic equation and related to an ellipse.

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Bernoulli lemniscate: A figure-eight curve defined by the equation (x² + y²)² = a²(x² − y²). This is the specific curve most commonly meant by lemniscate.
Cassini oval: A quartic plane curve defined by a constant product of distances from two fixed points. Cassini's work on these ovals preceded Jakob Bernoulli's study of the figure-eight case.
Plane curve: A one-dimensional geometric set lying in a plane, often described by equations or parametrizations. A lemniscate is a particular plane curve with two loops.
Arc length: The distance measured along a curve between two points. Computing the lemniscate's arc length leads to a distinctive elliptic integral.
Polar coordinate system: A coordinate system that locates points by distance and angle from an origin. In polar coordinates, the Bernoulli lemniscate has the compact equation r² = a² cos 2θ.
Jakob Bernoulli: A Swiss mathematician whose work helped establish the early calculus and the study of curves. He named and analyzed the lemniscate in a 1694 publication.
Quartic plane curve: A plane algebraic curve defined by a polynomial equation of degree four. The Bernoulli lemniscate is a quartic curve.
Fagnano's problem: A classical problem asking for the shortest inscribed polygon in an acute triangle. It illustrates the wider geometric tradition of optimizing lengths along curves.
Cartesian coordinate system: A coordinate system that locates points using perpendicular axes and signed coordinates. Its quartic Cartesian equation makes the curve's symmetry and two loops explicit.
Acta Eruditorum: A Latin scholarly journal published in Leipzig from 1682 to 1782. Jakob Bernoulli introduced the lemniscate in this journal.