Lemniscate

In algebraic geometry, a lemniscate ( or ) is any of several figure-eight or ∞-shaped curves. The word comes from the Latin lēmniscātus, meaning "decorated with ribbons", from the Greek λημνίσκος (lēmnískos), meaning "ribbon", or which alternatively may refer to the wool from which the ribbons were made.

Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli, and the lemniscate of Gerono. The hippopede was studied by Proclus (5th century), but the term "lemniscate" was not used until the work of Jacob Bernoulli in the late 17th century.

History and examples

Lemniscate of Booth

The consideration of curves with a figure-eight shape can be traced back to Proclus, a Greek Neoplatonist philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a torus by a plane parallel to the axis of the torus. As he observed, for most such sections the cross section consists of either one or two ovals; however, when the plane is tangent to the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called a horse fetter (a device for holding two feet of a horse together), or "hippopede" in Greek. The name "lemniscate of Booth" for this curve dates to its study by the 19th-century mathematician James Booth.

The lemniscate may be defined as an algebraic curve, the zero set of the quartic polynomial

(

x

2

+

y

2

)

2

−

c

x

2

−

d

y

2

{\displaystyle (x^{2}+y^{2})^{2}-cx^{2}-dy^{2}}

when the parameter d is negative (or zero for the special case where the lemniscate becomes a pair of externally tangent circles). For positive values of d one instead obtains the oval of Booth.

Lemniscate of Bernoulli

In 1680, Cassini studied a family of curves, now called the Cassini oval, defined as follows: the locus of all points, the product of whose distances from two fixed points, the curves' foci, is a constant.