Editing Lemniscate of Bernoulli (section)

{{short description|Plane algebraic curve}}
[[Image:Lemniskate bernoulli2.svg|thumb|upright=1.5|right|A lemniscate of Bernoulli and its two foci {{math|''F''<sub>1</sub>}} and {{math|''F''<sub>2</sub>}}]]
[[Image:Lemniscate of Bernoulli.gif|thumb|300px|right|The lemniscate of Bernoulli is the [[pedal curve]] of a rectangular [[hyperbola]]]]
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In [[geometry]], the '''lemniscate of Bernoulli''' is a [[plane curve]] defined from two given points {{math|''F''<sub>1</sub>}} and {{math|''F''<sub>2</sub>}}, known as '''foci''', at distance {{math|2''c''}} from each other as the locus of points {{math|''P''}} so that {{math|''PF''<sub>1</sub>·''PF''<sub>2</sub> {{=}} ''c''<sup>2</sup>}}. The curve has a shape similar to the [[8 (number)|numeral 8]] and to the [[Infinity|∞]] symbol. Its name is from {{wikt-lang|la|lemniscatus}}, which is [[Latin]] for "decorated with hanging ribbons". It is a special case of the [[Cassini oval]] and is a rational [[algebraic curve]] of degree 4.

This [[lemniscate]] was first described in 1694 by [[Jakob Bernoulli]] as a modification of an [[ellipse]], which is the [[Locus (mathematics)|locus]] of points for which the sum of the [[distance]]s to each of two fixed ''focal points'' is a [[mathematical constant|constant]]. A [[Cassini oval]], by contrast, is the locus of points for which the ''product'' of these distances is constant.  In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli.

This curve can be obtained as the [[inversive geometry|inverse transform]] of a [[hyperbola]], with the inversion [[circle]] centered at the center of the hyperbola (bisector of its two foci). It may also be drawn by a [[mechanical linkage]] in the form of [[Watt's linkage]], with the lengths of the three bars of the linkage and the distance between its endpoints chosen to form a [[antiparallelogram|crossed parallelogram]].<ref>{{citation|title=How round is your circle? Where Engineering and Mathematics Meet|title-link=How Round Is Your Circle|first1=John|last1=Bryant|first2=Christopher J.|last2=Sangwin|publisher=Princeton University Press|year=2008|isbn=978-0-691-13118-4|at=[https://books.google.com/books?id=iIN_2WjBH1cC&pg=PA58 pp. 58–59]}}.</ref>