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| In [[mathematics]], the '''witch of Agnesi''' ({{IPA-it|a.ˈɲe.zi}}), sometimes called the '''witch of [[Maria Gaetana Agnesi|Maria Agnesi]]''' is the curve defined as follows.
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| [[Image:WitchOfAgnesi03a.png|thumb|300px|right|The Witch of Agnesi with labeled points]]
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| Starting with a fixed circle, a point O on the circle is chosen. For any other point A on the circle, the secant line OA is drawn. The point M is diametrically opposite to O. The line OA intersects the tangent of M at the point N. The line parallel to OM through N, and the line perpendicular to OM through A intersect at P. As the point A is varied, the path of P is the witch.
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| The curve is [[asymptote|asymptotic]] to the line tangent to the fixed circle through the point O.
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| ==Equations==
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| [[Image:Agnesi.gif|thumb|300px|right|An animation showing the construction of the Witch of Agnesi]]
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| Suppose the point O is the origin, and that M is on the positive ''y''-axis.
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| Suppose the radius of the circle is ''a''.
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| Then the curve has Cartesian equation
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| : <math>\!y = \frac{8a^3}{x^2+4a^2}.</math>
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| Note that if ''a=1/2'', then this equation becomes rather simple:
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| : <math>\!y = \frac{1}{x^2+1}.</math>
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| Parametrically, if <math>\theta\,</math> is the angle between OM and OA, measured clockwise, then
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| the curve is defined by the equations
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| : <math>\!x = 2a \tan \theta,\ y = 2a \cos ^2 \theta.\,</math>
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| Another parameterization, with <math>\theta\,</math> being the angle between OA and the ''x''-axis, increasing anti-clockwise is
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| : <math>\!x = 2a \cot \theta,\ y=2a\sin ^2 \theta.\,</math>
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| ==Properties==
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| [[Image:WitchOfAgnesi04.png|thumb|350px|right|The Witch of Agnesi with parameters ''a''=1, ''a''=2, ''a''=4, and ''a''=8]]
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| *The area between the Witch and its asymptote is four times the area of the fixed circle (i.e., <math>4 \pi a^2</math>).
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| *The [[centroid]] of this region is located at <math>\!\left(0,a/2\right)</math>.
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| This [[centroid]] of the generating circle (diameter = 2a) is located at <math>\!\left(0,a\right)</math>.
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| *The [[volume of revolution]] of the Witch, about its asymptote, is <math>4\pi^2 a^3</math>.
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| ==History==
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| The curve was studied by [[Pierre de Fermat]] in 1630. In 1703, [[Guido Grandi]] gave a construction for the curve. In 1718 Grandi suggested the name '[[wikt:versoria|versoria]]' for the curve, the Latin term for [[Sheet (sailing)|sheet]], the rope which turns (adjusts the [[wikt:trim|trim]] of) the sail, and used the Italian word for it, 'versiera', a hint to [[sinus versus]] that appeared in his construction.<ref name=truesdell/>
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| In 1748, [[Maria Gaetana Agnesi|Maria Agnesi]] published her famous summation treatise ''Instituzioni analitiche ad uso della gioventù italiana'', in which the curve was named according to Grandi, 'versiera'.<ref name=truesdell>C. Truesdell, "Correction and Additions for 'Maria Gaetana Agnesi'", ''Archive for History of Exact Science'' 43 (1991), 385-386. {{doi|10.1007/BF00374764}}
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| *Per Grandi: "...nata da' seni versi, che da me suole chiamarsi la ''Versiera'' in latino pero ''Versoria''..."</ref> Coincidentally, the contemporary Italian word 'Aversiera'/'Versiera', derived from Latin 'Adversarius', a nickname for [[Devil]], "Adversary of God", was synonymous with "witch".<ref>[[Pietro Fanfani]], ''Vocabolario dell' uso toscano'', [http://books.google.com/books?id=SOczmy2F2y0C&pg=PA334&lpg=PA334&dq=versicra&source=bl&ots=uR83eWWp30&sig=efg5mhXOHtdB4XNw0FnSgOAf6zM&hl=en&ei=7zWPTvHTLsLpsQLbqsidAQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CCQQ6AEwATgK#v=onepage&q=versicra&f=false p. 334]</ref> Probably for this reason Cambridge professor [[John Colson]] mistranslated the name of the curve thus. Different modern works about Agnesi and about the curve suggest slightly different guesses how exactly this mistranslation happened.<ref>''Women in Mathematics'' By Lynn M. Osen (1975) p. 45</ref><ref>"[[Fermat's Enigma]]" by [[Simon Singh]] p. 100</ref><ref>''The universal book of mathematics: from Abracadabra to Zeno's paradoxes'' By David J. Darling (2004) p. 8</ref> [[Dirk Jan Struik|Struik]] mentions that:
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| {{quote|The word [''versiera''] is derived from Latin ''vertere'', to turn, but is also an abbreviation of Italian ''avversiera'', female devil. Some wit in England once translated it 'witch', and the silly pun is still lovingly preserved in most of our textbooks in English language. ... The curve had already appeared in the writings of [[Pierre de Fermat|Fermat]] (''Oeuvres'', I, 279-280; III, 233-234) and of others; the name ''versiera'' is from Guido Grandi (''Quadratura circuli et hyperbolae'', Pisa, 1703). The curve is type 63 in [[Newton's]] classification. ... The first to use the term 'witch' in this sense may have been B. Williamson, ''Integral calculus'', 7 (1875), 173;<ref>"173 Find the area between the witch of Agnesi <math>x y^2 = 4 a^2 (2 a -x)</math> and its asymptote." (''Oxford English Dictionary'')</ref> see ''Oxford English Dictionary''.}}
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| On the other hand, [[Stephen Stigler]] suggests that Grandi himself "may have been indulging in a play on words".<ref>S.M.Stigler, "Cauchy and the witch of Agnesi: An historical note on the Cauchy distribution", ''[[Biometrika]]'', 1974, vol. 61, no.2 p. 375-380</ref>
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| ''The Witch of Agnesi'' is also a fiction novel by Robert Spiller, in which a teacher gives a version of the history of the term.<ref>{{cite book
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| |title= The Witch of Agnesi
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| |last= Spiller
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| |first= Robert
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| |year= 2006
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| |publisher= Medallion Press
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| |location= Palm Beach, FL
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| |isbn= 978-1-932815-72-6
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| |oclc= 71259167
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| |accessdate=}}</ref>
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| == Application ==
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| [[File:Shallow water wave.png|thumb|The cross-section of a single water wave has a shape similar to the Witch of Agnesi.]]<!-- or animation Shallow water wave.gif-->
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| In addition to its theoretical properties, the curve has applications to real-life phenomena, applications which have only been discovered fairly recently during the late 20th and early 21st centuries. The Cartesian equation (above) has appeared in the modeling of some [[Physics|physical phenomena]] [[mathematical model]]s:<ref>http://www.mathsci.appstate.edu/~sjg/wmm/final/agnesifinal/applications.pdf</ref>
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| the equation approximates the spectral line distribution of optical lines and x-rays, as well as the amount of power dissipated in resonant circuits.
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| Formally, the curve is equivalent to the [[probability density function]] of the [[Cauchy distribution]].
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| The cross-section of a [[hill|smooth hill]] also has a similar shape. It has been used as the generic topographic obstacle in a flow in mathematical modeling.<ref>{{cite journal
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| | last =Snyder | first = William H. | coauthors = et al
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| | title = The structure of strongly stratified flow over hills: dividing-streamline concept
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| | journal = J. Fluid Mech. | volume = 152 | pages = 249–288
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| | url = http://www.cpom.org/people/jcrh/jfm-152.pdf
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| | accessdate = 12-Jan-2014 }}</ref><ref>{{cite journal
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| | last =Lamb | first = Kevin G.
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| | title = Numerical simulations of stratified inviscid flowover a smooth obstacle
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| | journal = J. Fluid Mech. | volume = 260 | pages = 1–22
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| | url = http://mseas.mit.edu/download/evheubel/LambJFM1994.pdf
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| | accessdate = 12-Jan-2014 }}</ref>
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| ==See also==
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| * [[Cauchy distribution]]
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| == Notes ==
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| {{Reflist}}
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| ==References==
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| {{Wikisource1911Enc|Witch of Agnesi}}
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| *{{MathWorld | urlname=WitchofAgnesi | title=Witch of Agnesi}}
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| *[http://www-groups.dcs.st-and.ac.uk/~history/Curves/Witch.html "Witch of Agnesi" at MacTutor's Famous Curves Index]
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| *[http://www.mathcurve.com/courbes2d/agnesi/agnesi.shtml "Cubique d'Agnesi" at Encyclopédie des Formes Mathématiques Remarquables] (in French)
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| *{{cite web |url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Agnesi.html | title=MacTutor biography of Agnesi}}
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| *{{cite episode |transcripturl=http://www.uh.edu/engines/epi1741.htm |series=The Engines of Our Ingenuity |serieslink=The Engines of Our Ingenuity |number=1741 |title=The Witch of Agnesi |credits=John H. Lienhard |network=NPR |station=KUHF-FM Houston |airdate=2002}}
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| ==External links==
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| {{commons|Witch of Agnesi}}
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| *[http://demonstrations.wolfram.com/WitchOfAgnesi/ Witch of Agnesi] by Chris Boucher based on work by [[Eric W. Weisstein]], [[The Wolfram Demonstrations Project]].
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| *[http://mathforum.org/dynamic/java_gsp/witch.html The Witch of Agnesi] - Mathforum.org Java applet
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| [[Category:Curves]]
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