algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a lemniscate ( or ) is any of several figure-eight or -shaped

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

s. The word comes from the

Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...

, meaning "decorated with ribbons", from the Greek (), meaning "ribbon",. or which alternatively may refer to the

wool Wool is the textile fiber obtained from sheep and other mammals, especially goats, rabbits, and camelids. The term may also refer to inorganic materials, such as mineral wool and glass wool, that have some properties similar to animal w ...

from which the

ribbons A ribbon or riband is a thin band of material, typically cloth but also plastic or sometimes metal, used primarily as decorative binding and tying. Cloth ribbons are made of natural materials such as silk, cotton, and jute and of synthetic mate ...

were made. Curves that have been called a lemniscate include three

quartic plane curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one of ...

s: the

hippopede In geometry, a hippopede () is a plane curve determined by an equation of the form :(x^2+y^2)^2=cx^2+dy^2, where it is assumed that and since the remaining cases either reduce to a single point or can be put into the given form with a rotation. ...

or lemniscate of Booth, the

lemniscate of Bernoulli In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. I ...

, and the

lemniscate of Gerono In algebraic geometry, the lemniscate of Gerono, or lemniscate of Huygens, or figure-eight curve, is a plane algebraic curve of degree four and genus zero and is a lemniscate curve shaped like an \infty symbol, or figure eight. It has equation ...

. The hippopede was studied by

Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...

(5th century), but the term "lemniscate" was not used until the work of

Jacob Bernoulli Jacob Bernoulli (also known as James in English or Jacques in French; – 16 August 1705) was a Swiss mathematician. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibniz ...

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

, a Greek

Neoplatonist Neoplatonism is a version of Platonic philosophy that emerged in the 3rd century AD against the background of Hellenistic philosophy and religion. The term does not encapsulate a set of ideas as much as a series of thinkers. Among the common id ...

philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a

torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

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 James Booth (born David Noel Geeves; 19 December 1927 – 11 August 2005) was an English film, stage and television actor and screenwriter. He is best known for his role as Private Henry Hook in '' Zulu.'' ''Variety'' called him "a punchy b ...

.. The lemniscate may be defined as an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

, the zero set of the

quartic polynomial In algebra, a quartic function is a function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A ''quartic equation'', or equation of the fourth de ...

(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 In geometry, a hippopede () is a plane curve determined by an equation of the form :(x^2+y^2)^2=cx^2+dy^2, where it is assumed that and since the remaining cases either reduce to a single point or can be put into the given form with a rotation. ...

Lemniscate of Bernoulli

In 1680, Cassini studied a family of curves, now called the

Cassini oval In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points (foci) is constant. This may be contrasted with an ellipse, for which the ''sum'' of th ...

, defined as follows: the locus of all points, the product of whose distances from two fixed points, the curves'

foci Focus (: foci or focuses) may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in East Australia Film * ''Focus'' (2001 film), a 2001 film based on the Arthur Miller novel * ''Focus'' (2015 film), a 201 ...

, is a constant. Under very particular circumstances (when the half-distance between the points is equal to the square root of the constant) this gives rise to a lemniscate. In 1694,

Johann Bernoulli Johann Bernoulli (also known as Jean in French or John in English; – 1 January 1748) was a Swiss people, Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infin ...

studied the lemniscate case of the Cassini oval, now known as the

(shown above), in connection with a problem of " isochrones" that had been posed earlier by

Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many ...

. Like the hippopede, it is an algebraic curve, the zero set of the polynomial

(x^2 + y^2)^2 - a^2 (x^2 - y^2)

. Bernoulli's brother

also studied the same curve in the same year, and gave it its name, the lemniscate.. It may also be defined geometrically as the locus of points whose product of distances from two foci equals the square of half the interfocal distance. It is a special case of the hippopede (lemniscate of Booth), with

d=-c

, and may be formed as a cross-section of a torus whose inner hole and circular cross-sections have the same diameter as each other. The

lemniscatic elliptic function In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among oth ...

s are analogues of trigonometric functions for the lemniscate of Bernoulli, and the lemniscate constants arise in evaluating the

arc length Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...

of this lemniscate.

Lemniscate of Gerono

Another lemniscate, the

or lemniscate of Huygens, is the zero set of the quartic polynomial

y^2-x^2(a^2-x^2)

Viviani's curve In mathematics, Viviani's curve, also known as Viviani's window, is a Lemniscate, figure-eight-shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder (geometry), cylinder that ...

, a three-dimensional curve formed by intersecting a sphere with a cylinder, also has a figure eight shape, and has the lemniscate of Gerono as its planar projection.

Others

Other figure-eight shaped algebraic curves include * The

Devil's curve In geometry, a Devil's curve, also known as the Devil on Two Sticks, is a curve defined in the Cartesian plane by an equation of the form y^2(y^2 - b^2) = x^2(x^2 - a^2) The polar equation of this curve is of the form r = \sqrt = \sqrt Devil ...

, a curve defined by the quartic equation

y^2 (y^2 - a^2) = x^2 (x^2 - b^2)

in which one connected component has a figure-eight shape,. *

Watt's curve In mathematics, Watt's curve is a circular algebraic curve, tricircular algebraic curve, plane algebraic curve of sextic, degree six. It is generated by two circles of radius ''b'' with centers distance 2''a'' apart (taken to be at (±''a'', 0)) ...

, a figure-eight shaped curve formed by a mechanical linkage. Watt's curve is the zero set of the degree-six polynomial equation

(x^2+y^2)(x^2+y^2-d^2)^2+4a^2y^2(x^2+y^2-b^2)=0

and has the lemniscate of Bernoulli as a special case.

References

External links

* {{springer, title=Lemniscates, id=p/l058130 Mathematical terminology Plane curves