Lemniscate

This article is about figure-eight shaped curves in algebraic geometry. For other uses, seeLemniscate (disambiguation).

The lemniscate of Bernoulli and its two foci

Inalgebraic geometry,alemniscate(/lɛmˈnɪskɪt/or/ˈlɛmnɪsˌkeɪt,-kɪt/)is any of several figure-eight or∞-shapedcurves.The word comes from theLatinlēmniscātus,meaning "decorated with ribbons",from the Greekλημνίσκος(lēmnískos), meaning "ribbon",or which alternatively may refer to thewoolfrom which theribbonswere made.

Curves that have been called a lemniscate include threequartic plane curves:thehippopedeorlemniscate of Booth,thelemniscate of Bernoulli,and thelemniscate of Gerono.The hippopede was studied byProclus(5th century), but the term "lemniscate" was not used until the work ofJacob Bernoulliin the late 17th century.

History and examples

Lemniscate of Booth

Lemniscate of Booth

Main article:Hippopede

The consideration of curves with a figure-eight shape can be traced back toProclus,a GreekNeoplatonistphilosopher and mathematician who lived in the 5th century AD. Proclus considered thecross-sectionsof atorusby 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 istangentto the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called ahorse 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 mathematicianJames Booth.

The lemniscate may be defined as analgebraic curve,the zero set of thequartic polynomial$(x^{2}+y^{2})^{2}-cx^{2}-dy^{2}$when the parameterdis negative (or zero for the special case where the lemniscate becomes a pair of externally tangent circles). For positive values ofdone instead obtains theoval of Booth.

Lemniscate of Bernoulli

Lemniscate of Bernoulli

Main article:Lemniscate of Bernoulli

In 1680,Cassinistudied a family of curves, now called theCassini oval,defined as follows: thelocusof all points, the product of whose distances from two fixed points, the curves'foci,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 Bernoullistudied the lemniscate case of the Cassini oval, now known as thelemniscate of Bernoulli(shown above), in connection with a problem of "isochrones"that had been posed earlier byLeibniz.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 brotherJacob Bernoullialso 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.Thelemniscatic elliptic functionsare analogues of trigonometric functions for the lemniscate of Bernoulli, and thelemniscate constantsarise in evaluating thearc lengthof this lemniscate.

Lemniscate of Gerono

Main article:Lemniscate of Gerono

Lemniscate of Gerono: solution set ofx⁴−x²+y²= 0

Another lemniscate, thelemniscate of Geronoor lemniscate of Huygens, is the zero set of the quartic polynomial$y^{2}-x^{2}(a^{2}-x^{2})$.Viviani's curve,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

• TheDevil's curve,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,a figure-eight shaped curve formed by a mechanical linkage. Watt's curve is the zero set of the degree-six polynomial equation$(x^{2}+y^{2})(x^{2}+y^{2}-d^{2})^{2}+4a^{2}y^{2}(x^{2}+y^{2}-b^{2})=0$and has the lemniscate of Bernoulli as a special case.

See also

• Analemma,the figure-eight shaped curve traced by the noontime positions of the sun in the sky over the course of a year
• Infinity symbol
• Lemniscates asgeneralized conics
• Lorenz attractor,a three-dimensional dynamic system exhibiting a lemniscate shape
• Polynomial lemniscate,a level set of the absolute value of a complex polynomial

External links

• "Lemniscates",Encyclopedia of Mathematics,EMS Press,2001 [1994]