View source for Lemniscate of Bernoulli

{{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]]]]
{{Sinusoidal_spirals.svg}}
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>

== Equations ==
The equations can be stated in terms of the focal distance {{mvar|c}} or the half-width {{mvar|a}} of a lemniscate. These parameters are related as {{math|''a'' {{=}} ''c''{{sqrt|2}}}}.
* Its [[Cartesian coordinate system|Cartesian]] [[equation]] is (up to translation and rotation):
*:<math>\begin{align} \left(x^2 + y^2\right)^2 &= a^2 \left(x^2 - y^2\right) \\ &= 2 c^2 \left(x^2 - y^2\right) \end{align}</math>
* As a square function of x:
*:<math>y^2 = \left(\sqrt{8 x^2 + a^2} - a\right) \frac{a}{2} - x^2</math>
* As a [[parametric equation]]:
*:<math>x = \frac{a\cos t}{1 + \sin^2 t}; \qquad y = \frac{a\sin t \cos t}{1 + \sin^2 t} </math>
* A rational parametrization:<ref>{{cite arXiv |last=Lemmermeyer |first=Franz |title=Parametrizing Algebraic Curves |eprint=1108.6219 |year=2011 |class=math.NT }}</ref>
*:<math>x = a \frac{t+t^3}{1+t^4}; \qquad y = a\frac{t-t^3}{1 + t^4} </math>
* In [[polar coordinates]]:
*:<math>r^2 = a^2 \cos{2\theta}</math>
* In the [[complex plane]]:
*:<math>|z-c||z+c|=c^2</math>
* In [[two-center bipolar coordinates]]:
*:<math>rr' = c^2</math>

==Arc length and elliptic functions==
{{main|Lemniscate elliptic functions}}
[[File:The lemniscate sine and cosine related to the arclength of the lemniscate of Bernoulli.png|thumb|upright=1.8|The [[lemniscate elliptic functions|lemniscate sine and cosine]] relate the arc length of an arc of the lemniscate to the distance of one endpoint from the origin.]]
The determination of the [[arc length]] of arcs of the lemniscate leads to [[elliptic integral]]s, as was discovered in the eighteenth century. Around 1800, the [[elliptic function]]s inverting those integrals were studied by [[C. F. Gauss]] (largely unpublished at the time, but allusions in the notes to his ''[[Disquisitiones Arithmeticae]]''). The [[period lattice]]s are of a very special form, being proportional to the [[Gaussian integer]]s. For this reason the case of elliptic functions with [[complex multiplication]] by [[square root of minus one|{{sqrt|−1}}]] is called the ''[[lemniscatic case]]'' in some sources.

Using the elliptic integral
:<math>\operatorname{arcsl}x \stackrel{\text{def}}{{}={}} \int_0^x\frac{dt}{\sqrt{1-t^4}}</math>
the formula of the arc length {{mvar|L}} can be given as
:<math>\begin{align} L &= 4a \int_{0}^1\frac{dt}{\sqrt{1-t^4}} = 4a\,\operatorname{arcsl}1 = 2\varpi a \\[6pt] &= \frac{\Gamma (1/4)^2}{\sqrt\pi}\,c =\frac{2\pi}{\operatorname{M}(1,1/\sqrt{2})}c\approx 7{.}416 \cdot c \end{align}</math>
where <math>c</math> and <math>a = \sqrt{2}c</math> are defined as above, <math>\varpi = 2 \operatorname{arcsl}{1}</math> is the [[lemniscate constant]], <math>\Gamma</math> is the [[gamma function]] and <math>\operatorname{M}</math> is the [[arithmetic–geometric mean]].

==Angles==
Given two distinct points <math>\rm A</math> and <math>\rm B</math>, let <math>\rm M</math> be the midpoint of <math>\rm AB</math>. Then the lemniscate of [[Diameter of a set|diameter]] <math>\rm AB</math> can also be defined as the set of points <math>\rm A</math>, <math>\rm B</math>, <math>\rm M</math>, together with the locus of the points <math>\rm P</math> such that <math>|\widehat{\rm APM}-\widehat{\rm BPM}|</math> is a right angle (cf. [[Thales' theorem]] and its converse).<ref>{{Cite book |last1=Eymard |first1=Pierre |last2=Lafon| first2=Jean-Pierre |title=The Number Pi |publisher=American Mathematical Society |year=2004 |isbn=0-8218-3246-8}} p. 200</ref>

[[File:Lemniskate vechtmann.svg|thumb|upright=1.75|relation between angles at Bernoulli's lemniscate]]
The following theorem about angles occurring in the lemniscate is due to German mathematician [[Gerhard Christoph Hermann Vechtmann]], who described it 1843 in his dissertation on lemniscates.<ref>Alexander Ostermann, Gerhard Wanner: ''Geometry by Its History.'' Springer, 2012, pp. [https://books.google.com/books?id=eOSqPHwWJX8C&pg=PA207 207-208]</ref> 

:{{math|''F''<sub>1</sub>}} and {{math|''F''<sub>2</sub>}} are the foci of the lemniscate, {{math|''O''}} is the midpoint of the line segment {{math|''F''<sub>1</sub>''F''<sub>2</sub>}} and {{math|''P''}} is any point on the lemniscate outside the line connecting {{math|''F''<sub>1</sub>}} and {{math|''F''<sub>2</sub>}}. The normal {{math|''n''}} of the lemniscate in {{math|''P''}} intersects the line connecting {{math|''F''<sub>1</sub>}} and {{math|''F''<sub>2</sub>}} in {{math|''R''}}. Now the interior angle of the triangle {{math|''OPR''}} at {{math|''O''}} is one third of the triangle's exterior angle at {{math|''R''}} (see also [[angle trisection]]). In addition the interior angle at {{math|''P''}} is twice the interior angle at {{math|''O''}}.

==Further properties==
[[File:Lemniskate hyperbel.svg|thumb|upright=1.25|The inversion of hyperbola yields a lemniscate]]
*The lemniscate is symmetric to the line connecting its foci {{math|''F''<sub>1</sub>}} and {{math|''F''<sub>2</sub>}} and as well to the perpendicular bisector of the line segment {{math|''F''<sub>1</sub>''F''<sub>2</sub>}}.
*The lemniscate is symmetric to the midpoint of the line segment {{math|''F''<sub>1</sub>''F''<sub>2</sub>}}.
*The area enclosed by the lemniscate is {{math|''a''<sup>2</sup> {{=}} 2''c''<sup>2</sup>}}.
*The lemniscate is the [[Inversive_geometry#Circle_inversion|circle inversion]] of a [[hyperbola]] and vice versa.
*The two tangents at the midpoint {{math|''O''}} are perpendicular, and each of them forms an angle of {{math|{{sfrac|{{pi}}|4}}}} with the line connecting {{math|''F''<sub>1</sub>}} and {{math|''F''<sub>2</sub>}}.
*The planar cross-section of a standard [[torus]] tangent to its inner equator is a lemniscate.
*The [[curvature]] at <math>(x,y)</math> is <math>{3\over a^2}\sqrt{x^2+y^2}</math>. The maximum curvature, which occurs at <math>(\pm a,0)</math>, is therefore <math>3/a</math>.

==Applications==
Dynamics on this curve and its more generalized versions are studied in quasi-one-dimensional models.

== See also ==
*[[Lemniscate of Booth]]
*[[Lemniscate of Gerono]]
*[[Lemniscate constant]]
*[[Lemniscatic elliptic function]]
*[[Cassini oval]]

==Notes==
{{reflist}}

==References==
* {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=[https://archive.org/details/catalogofspecial00lawr/page/4 4–5,121–123,145,151,184] | url-access=registration | url=https://archive.org/details/catalogofspecial00lawr/page/4 }}

== External links ==
{{commons category|Lemniscate of Bernoulli}}
* {{MathWorld|title=Lemniscate|urlname=Lemniscate}}
* [https://mathshistory.st-andrews.ac.uk/Curves/Lemniscate/ "Lemniscate of Bernoulli" at The MacTutor History of Mathematics archive]
* [http://mathcurve.com/courbes2d.gb/lemniscate/lemniscate.shtml "Lemniscate of Bernoulli"] at MathCurve.
* [http://images.math.cnrs.fr/Coup-d-oeil-sur-la-lemniscate-de.html Coup d'œil sur la lemniscate de Bernoulli] (in French)

[[Category:Plane curves]]
[[Category:Quartic curves]]
[[Category:Spiric sections]]