View source for Lemniscate

{{short description|Figure-eight-shaped curve}}
{{about|figure-eight shaped curves in algebraic geometry|other uses|Lemniscate (disambiguation)}}
[[File:Lemniscate of Bernoulli.svg|thumb|400px|right|The [[lemniscate of Bernoulli]] and its two [[Focus (geometry)|foci]]]]

In [[algebraic geometry]], a '''lemniscate''' ({{IPAc-en|l|ɛ|m|ˈ|n|ɪ|s|k|ɪ|t}} or {{IPAc-en|ˈ|l|ɛ|m|n|ɪ|s|ˌ|k|eɪ|t|,_|-|k|ɪ|t}})<ref>{{cite Dictionary.com|lemniscate}}</ref> is any of several figure-eight or {{math|[[∞]]}}-shaped [[curve]]s.<ref name="lemniscatomy"/><ref name="erickson"/> The word comes from the [[Latin language|Latin]] {{Wikt-lang|la|lēmniscātus}}, meaning "decorated with ribbons",<ref>{{L&S|lemniscatus|ref}}</ref> from the Greek {{Wikt-lang|grc|λημνίσκος}} ({{grc-transl|λημνίσκος}}), meaning "ribbon",<ref name="erickson">{{citation |title=Beautiful Mathematics |series=MAA Spectrum |publisher=[[Mathematical Association of America]] |first=Martin J. |last=Erickson |year=2011 |isbn=9780883855768 |contribution=1.1 Lemniscate |pages=1–3 |url=https://books.google.com/books?id=LgeP62-ZxikC&pg=PA1}}.</ref><ref>{{OEtymD|lemniscus}}</ref><ref>{{L&S|lemniscus|ref}}</ref><ref>{{LSJ|lhmni/skos|λημνίσκος|ref}}.</ref> or which alternatively may refer to the [[wool]] from which the [[ribbons]] were made.<ref name="lemniscatomy"/>

Curves that have been called a lemniscate include three [[quartic plane curve]]s: 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===
[[File:Lemniscate of Booth.png|thumb|[[Lemniscate of Booth]]]]
{{main|Hippopede}}
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 [[Toric section|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 [[Hobble (device)|horse fetter]] (a device for holding two feet of a horse together), or "hippopede" in Greek.<ref>{{LSJ|i(ppope/dh|ἱπποπέδη|shortref}}.</ref> The name "lemniscate of Booth" for this curve dates to its study by the 19th-century mathematician [[James Booth (mathematician)|James Booth]].<ref name="lemniscatomy">{{citation
 | last = Schappacher | first = Norbert
 | contribution = Some milestones of lemniscatomy
 | location = New York
 | mr = 1483331
 | pages = 257–290
 | publisher = Dekker
 | series = Lecture Notes in Pure and Applied Mathematics
 | title = Algebraic Geometry (Ankara, 1995)
 | volume = 193
 | year = 1997}}.</ref>

The lemniscate may be defined as an [[algebraic curve]], the zero set of the [[quartic polynomial]] <math>(x^2 + y^2)^2 - cx^2 - dy^2</math> 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===
[[File:Lemniskate bernoulli2.svg|thumb|Lemniscate of Bernoulli]]
{{main|Lemniscate of Bernoulli}}
In 1680, [[Giovanni Domenico Cassini|Cassini]] studied a family of curves, now called the [[Cassini oval]], defined as follows: the [[locus (mathematics)|locus]] of all points, the product of whose distances from two fixed points, the curves' [[focus (geometry)|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 Bernoulli]] studied the lemniscate case of the Cassini oval, now known as the [[lemniscate of Bernoulli]] (shown above), in connection with a problem of "[[Isochrone curve|isochrones]]" that had been posed earlier by [[Leibniz]]. Like the hippopede, it is an algebraic curve, the zero set of the polynomial <math>(x^2 + y^2)^2 - a^2 (x^2 - y^2)</math>. Bernoulli's brother [[Jacob Bernoulli]] also studied the same curve in the same year, and gave it its name, the lemniscate.<ref name="bos">{{citation
 | last = Bos | first = H. J. M.
 | contribution = The lemniscate of Bernoulli
 | location = Dordrecht
 | mr = 774250
 | pages = 3–14
 | publisher = Reidel
 | series = Boston Stud. Philos. Sci., XV
 | title = For Dirk Struik
 | url = https://books.google.com/books?id=OvK9orJNezwC&pg=PA3
 | year = 1974| isbn = 9789027703934
 }}.</ref> 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.<ref>{{citation
 | last1 = Langer | first1 = Joel C.
 | last2 = Singer | first2 = David A.
 | doi = 10.1007/s00032-010-0124-5
 | issue = 2
 | journal = Milan Journal of Mathematics
 | mr = 2781856
 | pages = 643–682
 | title = Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem
 | volume = 78
 | year = 2010| s2cid = 1448521
 }}.</ref> It is a special case of the hippopede (lemniscate of Booth), with <math>d=-c</math>, 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.<ref name="lemniscatomy"/>  The [[lemniscatic elliptic function]]s are analogues of trigonometric functions for the lemniscate of Bernoulli, and the [[Lemniscate constant|lemniscate constants]] arise in evaluating the [[arc length]] of this lemniscate.

===Lemniscate of Gerono===
{{main|Lemniscate of Gerono}}
[[File:Lemniscate-of-Gerono2.svg|thumb|280px|Lemniscate of Gerono: solution set of {{math|''x''<sup>4</sup> − ''x''<sup>2</sup> + ''y''<sup>2</sup> {{=}} 0}}<ref>{{Cite web|url=http://www.mathematische-basteleien.de/acht.htm|title=Acht-Kurve|last=Köller|first=Jürgen|website=www.mathematische-basteleien.de|access-date=2017-11-26}}</ref>]]
Another lemniscate, the [[lemniscate of Gerono]] or lemniscate of Huygens, is the zero set of the quartic polynomial <math>y^2-x^2(a^2-x^2)</math>.<ref>{{citation|title=An elementary treatise on cubic and quartic curves|first=Alfred Barnard|last=Basset|publisher=Deighton, Bell|year=1901|pages=171–172|url=https://books.google.com/books?id=T40LAAAAYAAJ&pg=PA171|contribution=The Lemniscate of Gerono}}.</ref><ref>{{citation|title=Newton's Principia for the common reader|first=S|last=Chandrasekhar|publisher=Oxford University Press|year=2003|isbn=9780198526759|page=133|url=https://books.google.com/books?id=qomP58txKQwC&pg=PA133}}.</ref> [[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.<ref>{{citation|first1=Luisa Rossi|last1=Costa|first2=Elena|last2=Marchetti|contribution=Mathematical and Historical Investigation on Domes and Vaults|pages=73–80|title=Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004|year=2005|location=Mammendorf|publisher=Pro Literatur|editor1-last=Weber|editor1-first=Ralf|editor2-last=Amann|editor2-first=Matthias Albrecht}}.</ref>

===Others===

Other figure-eight shaped algebraic curves include
* The [[Devil's curve]], a curve defined by the quartic equation <math>y^2 (y^2 - a^2) = x^2 (x^2 - b^2)</math> in which one connected component has a figure-eight shape,<ref>{{citation|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes|first=David|last=Darling|publisher=John Wiley & Sons|year=2004|isbn=9780471667001|contribution=devil's curve|pages=91–92|url=https://books.google.com/books?id=HrOxRdtYYaMC&pg=PA91}}.</ref>
* [[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 <math>(x^2+y^2)(x^2+y^2-d^2)^2+4a^2y^2(x^2+y^2-b^2)=0</math> 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 as [[generalized conic]]s
* [[Lorenz attractor]], a three-dimensional dynamic system exhibiting a lemniscate shape
* [[Polynomial lemniscate]], a level set of the absolute value of a complex polynomial

==References==
{{Reflist|30em}}

==External links==
{{commons category|Lemniscate}}
* {{springer|title=Lemniscates|id=p/l058130}}

[[Category:Mathematical terminology]]
[[Category:Plane curves]]