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| [[Image:Lemniscate of Bernoulli.svg|thumb|400px|right|A lemniscate of Bernoulli and its two foci]]
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| [[Image:Lemniscate of Bernoulli.gif|thumb|300px|right|Lemniscate of Bernoulli is the [[pedal curve]] of rectangular [[hyperbola]]]]
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| {{Sinusoidal_spirals.svg}}
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| In [[geometry]], the '''lemniscate of Bernoulli''' is a plane curve defined from two given points ''F''<sub>1</sub> and ''F''<sub>2</sub>, known as '''foci''', at distance 2''a'' from each other as the locus of points ''P'' so that ''PF''<sub>1</sub>·''PF''<sub>2</sub> = ''a''<sup>2</sup>. The curve has a shape similar to the numeral 8 and to the [[Infinity|∞]] symbol. Its name is from ''lemniscus'', which is [[Latin]] for "pendant ribbon". It is a special case of the [[Cassini oval]] and is a rational [[algebraic curve]] of degree 4.
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| The 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.
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| 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 square]].<ref>{{citation|title=How round is your circle? Where Engineering and Mathematics Meet|first1=John|last1=Bryant|first2=Christopher J.|last2=Sangwin|publisher=Princeton University Press|year=2008|isbn=978-0-691-13118-4|pages=58–59|url=http://books.google.com/books?id=iIN_2WjBH1cC&pg=PA58}}.</ref>
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| == Equations ==
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| * Its [[Cartesian coordinate system|Cartesian]] [[equation]] is (up to translation and rotation):
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| :<math>(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)\,</math>
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| * In [[polar coordinates]]:
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| :<math>r^2 = 2a^2 \cos 2\theta\,</math>
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| * As [[parametric equation]]:
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| :<math>x = \frac{a\sqrt{2}\cos(t)}{\sin(t)^2 + 1}; \qquad y = \frac{a\sqrt{2}\cos(t)\sin(t)}{\sin(t)^2 + 1} </math>
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| In [[two-center bipolar coordinates]]:
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| :<math>rr' = a^2\,</math>
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| In [[rational trigonometry|rational polar coordinates]]:
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| :<math>Q = 2s-1\,</math>
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| == Derivatives ==
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| Each first derivative below was calculated using [[Implicit function#Implicit_differentiation| implicit differentiation]].
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| === With ''y'' as a function of ''x'' ===
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| :<math>\frac{dy}{dx} = \begin{cases}
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| \mbox{unbounded} & \mbox{if } y = 0 \mbox{ and } x \ne 0 \\
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| \pm1 & \mbox{if } y = 0 \mbox{ and } x = 0 \\
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| \frac{x(a^2 - x^2 - y^2)}{y(a^2 + x^2 + y^2)} & \mbox{if } y \ne 0
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| \end{cases}</math>
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| :<math>\frac{d^2y}{dx^2} = \begin{cases}
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| \mbox{unbounded} & \mbox{if } y = 0 \mbox{ and } x \ne 0 \\
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| 0 & \mbox{if } y = 0 \mbox{ and } x = 0 \\
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| \frac{3a^6(y^2 - x^2)}{y^3(a^2 + 2x^2 + 2y^2)^3} & \mbox{if } y \ne 0
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| \end{cases}</math>
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| === With ''x'' as a function of ''y'' ===
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| :<math>\frac{dx}{dy} = \begin{cases}
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| \mbox{unbounded} & \mbox{if } 2x^2 + 2y^2 = a^2 \\
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| \pm1 & \mbox{if } x = 0 \mbox{ and } y = 0 \\
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| \frac{y(a^2 + 2x^2 + 2y^2)}{x(a^2 - 2x^2 - 2y^2)} & \mbox{else }
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| \end{cases}</math>
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| :<math>\frac{d^2x}{dy^2} = \begin{cases}
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| \mbox{unbounded} & \mbox{if } 2x^2 + 2y^2 = a^2 \\
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| 0 & \mbox{if } x = 0 \mbox{ and } y = 0 \\
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| \frac{3a^6(x^2 - y^2)}{x^3(a^2 - 2x^2 - 2y^2)^3} & \mbox{else }
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| \end{cases}</math>
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| == Curvature ==
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| Once the first two derivatives are known, [[Curvature#Local_expressions| curvature]] is easily calculated:
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| :<math>\kappa = \pm3(x^2 + y^2)^{1/2}a^{-2} \,</math>
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| the sign being chosen according to the direction of motion along the curve. The lemniscate has the property that the magnitude of the curvature at any point is proportional to that point's distance from the origin.
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| ==Arc length and elliptic functions==
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| 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 the [[square root of minus one]] is called the ''[[lemniscatic case]]'' in some sources.
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| == See also ==
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| *[[Lemniscate of Booth]]
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| *[[Lemniscate of Gerono]]
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| *[[Gauss's constant]]
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| *[[Lemniscatic elliptic function]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=4–5,121–123,145,151,184 }}
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| == External links ==
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| {{commonscat|Lemniscate of Bernoulli}}
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| * {{MathWorld|title=Lemniscate|urlname=Lemniscate}}
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| * [http://www-history.mcs.st-andrews.ac.uk/history/Curves/Lemniscate.html "Lemniscate of Bernoulli" at The MacTutor History of Mathematics archive]
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| * [http://mathcurve.com/courbes2d/lemniscate/lemniscate.shtml "Lemniscate de Bernoulli" at Encyclopédie des Formes Mathématiques Remarquables] (in French)
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| [[Category:Curves]]
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| [[Category:Algebraic curves]]
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| [[Category:Spiric sections]]
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My hobby is mainly Vintage Books.
I try to learn Swedish in my free time.
Here is my web blog - Hostgator Coupon Code (https://mylescollingsla.wordpress.com)