View source for Witch of Agnesi

{{short description|Cubic plane curve}}
{{Use dmy dates|date=July 2018}}

[[File:Witch of Agnesi curves.svg|thumb|upright=1.3|right|Selected witch of Agnesi curves (green), and the circles they are constructed from (blue), with radius parameters <math>a=1</math>, <math>a=2</math>, <math>a=4</math>, and <math>a=8</math>.]]

In [[mathematics]], the '''witch of Agnesi''' ({{IPA|it|aɲˈɲeːzi, -eːsi; -ɛːzi}}) is a [[cubic plane curve]] defined from two diametrically opposite points of a circle. 

The curve was studied as early as 1653 by [[Pierre de Fermat]], in 1703 by [[Guido Grandi]], and by [[Isaac Newton]]. It gets its name from Italian mathematician [[Maria Gaetana Agnesi]] who published it in 1748. The Italian name ''{{lang|it|la versiera di Agnesi}}'' is based on [[Latin]] ''{{lang|la|versoria}}'' ([[Sheet (sailing)|sheet]] of sailing ships) and the [[List_of_trigonometric_identities#Half-angle_formulae|sinus versus]]. 
This was read by [[John Colson]] as ''{{lang|it|l’avversiera di Agnesi}}'',  where ''{{lang|it|avversiera}}'' is translated as "woman who is against God"  and interpreted as "witch".<ref>[https://mathworld.wolfram.com/WitchofAgnesi.html Wolfram MathWorld, Witch of Agnesi]</ref><ref>Lynn M. Osen: ''Women in Mathematics.'' MIT Press, Cambridge MA 1975, ISBN 0-262-15014-X, S. 45.</ref><ref>[[Simon Singh]]: ''Fermat’s Enigma. The quest to solve the world’s greatest mathematical problem.'' Walker Books, New York 1997, ISBN 0-471-27047-4, S. 100.</ref><ref>David J. Darling: ''The universal book of mathematics. From Abracadabra to Zeno’s paradoxes.'' Wiley International, Hoboken NJ 2004, ISBN 0-8027-1331-9, S. 8.</ref>

The [[Graph of a function|graph]] of the derivative of the [[arctangent]] function forms an example of the witch of Agnesi. As the [[probability density function]] of the [[Cauchy distribution]], the witch of Agnesi has applications in [[probability theory]]. It also gives rise to [[Runge's phenomenon]] in the [[Linear approximation|approximation of functions]] by [[polynomial]]s, has been used to approximate the energy distribution of [[spectral line]]s, and models the shape of hills.

The witch is [[tangent]] to its defining circle at one of the two defining points, and [[asymptotic]] to the [[Tangent lines to circles|tangent line to the circle]] at the other point. It has a unique [[vertex (curve)|vertex]] (a point of extreme curvature) at the point of tangency with its defining circle, which is also its [[osculating circle]] at that point. It also has two finite [[inflection point]]s and one infinite inflection point. The area between the witch and its asymptotic line is four times the area of the defining circle, and the [[Solid of revolution|volume of revolution]] of the curve around its defining line is twice the volume of the [[torus]] of revolution of its defining circle.

==Construction==
[[File:Witch of Agnesi, construction.svg|thumb|upright=1.3|right|The witch of Agnesi (curve ''MP'') with labeled points]]
[[Image:Agnesi.gif|thumb|right|An animation showing the construction of the witch of Agnesi]]
 
To construct this curve, start with any two points ''O'' and ''M'', and draw a circle with ''OM'' as diameter. For any other point ''A'' on the circle, let ''N'' be the point of intersection of the [[secant line]] ''OA'' and the tangent line at ''M''.
Let ''P'' be the point of intersection of a line perpendicular to ''OM'' through ''A'', and a line parallel to ''OM'' through ''N''. Then ''P'' lies on the witch of Agnesi. The witch consists of all the points ''P'' that can be constructed in this way from the same choice of ''O'' and ''M''.{{r|eagles}} It includes, as a limiting case, the point ''M'' itself.

==Equations==
Suppose that point ''O'' is at the [[Origin (mathematics)|origin]] and point ''M'' lies on the positive <math>y</math>-axis, and that the circle with diameter ''OM'' has {{nowrap|radius <math>a</math>.}}
Then the witch constructed from ''O'' {{nowrap|and ''M''}} has the Cartesian equation{{r|lawrence|yates}}
<math display="block">y = \frac{8a^3}{x^2+4a^2} =\frac{(2a)^3}{(x)^2+(2a)^2}.</math>
This equation can be simplified, by choosing {{nowrap|<math>a=\tfrac12</math>,}} to the form
<math display=block>y = \frac{1}{x^2+1}.</math>
or equivalently, by [[clearing denominators]], as the cubic [[algebraic equation]]
<math display=block>(x^2+1)y=1.</math>
In its simplified form, this curve is the [[Graph of a function|graph]] of the [[derivative]] of the [[arctangent]] function.{{r|calc}}

<math display=block>\frac{dy}{dx}=-\frac{16a^3x}{(x^2+4a^2)^2} </math>

Area below curve <math display=block>= 4a^2(arctan(\frac{c}{2a})-arctan(\frac{b}{2a})) </math>
for: <math display=block>b <= x <= c </math>

The witch of Agnesi can also be described by [[parametric equation]]s whose parameter {{mvar|θ}} is the angle between ''OM'' and ''OA'', measured clockwise:{{r|lawrence|yates}}
<math display=block>\begin{align}
x &= 2a \tan \theta, \\
y &= 2a \cos ^2 \theta.
\end{align}</math>

==Properties==
The main properties of this curve can be derived from [[integral calculus]].
The area between the witch and its asymptotic line is four times the area of the fixed circle, {{nowrap|<math>4\pi a^2</math>.{{r|lawrence|yates|larsen}}}}
The [[volume of revolution]] of the witch of Agnesi about its asymptote {{nowrap|is <math>4\pi^2a^3</math>.{{r|lawrence}}}} This is two times the volume of the [[torus]] formed by revolving the defining circle of the witch around the same line.{{r|larsen}}

The curve has a unique [[Vertex (curve)|vertex]] at the point of tangency with its defining circle. That is, this point is the only point where the [[curvature]] reaches a local minimum or local maximum.{{r|egdc}} The defining circle of the witch is also its [[osculating circle]] at the vertex,{{r|haftendorn}} the unique circle that "kisses" the curve at that point by sharing the same orientation and curvature.{{r|kiss}} Because this is an osculating circle at the vertex of the curve, it has [[Contact (mathematics)|third-order contact]] with the curve.{{r|omnibus}}

The curve has two [[inflection point]]s, at the points
<math display=block>\left( \pm\frac{2a}{\sqrt{3}}, \frac{3a}{2}\right)</math>
corresponding to the {{nowrap|angles <math>\theta=\pm\pi/6</math>.{{r|lawrence|yates}}}} When considered as a curve in the [[projective plane]] there is also a third infinite inflection point, at the point where the [[line at infinity]] is crossed by the asymptotic line.{{r|arnold}} It has an isolated [[Singularity (mathematics)|singular point]], at the point where the line at infinity is crossed by its axis of symmetry.{{r|mathcurve}} As the graph of a [[rational function]] <math>1/(x^2+1)</math>, it is a rational [[parametric curve]], but not one that can be given a polynomial parameterization.{{r|fomshu}}

The largest area of a [[rectangle]] that can be inscribed between the witch and its asymptote {{nowrap|is <math>4a^2</math>,}} for a rectangle whose height is the radius of the defining circle and whose width is twice the diameter of the {{nowrap|circle.{{r|larsen}}}}

==History==
===Early studies===
[[File:Witch of Agnesi (Agnesi, 1748).jpg|thumb|Agnesi's 1748 illustration of the curve and its construction{{r|agnesi}}]]
The curve was studied by [[Pierre de Fermat]] in his 1659 treatise on [[Quadrature (geometry)|quadrature]]. In it, Fermat computes the area under the curve and (without details) claims that the same method extends as well to the [[cissoid of Diocles]]. Fermat writes that the curve was suggested to him "''ab erudito geometra''" [by a learned geometer].{{r|fermat}} {{harvtxt|Paradís|Pla|Viader|2008}} speculate that the geometer who suggested this curve to Fermat might have been [[Antoine de Laloubère]].{{r|fmq}}

The construction given above for this curve was found by {{harvtxt|Grandi|1718}}; the same construction was also found earlier by [[Isaac Newton]], but only published posthumously later, in 1779.{{r|stigler}}
{{harvtxt|Grandi|1718}} also suggested the name ''versiera'' (in Italian) or ''versoria'' (in Latin) for the curve.<ref>In his notes to Galileo's "Trattato del moto naturalmente accelerato," Grandi had referred to "quella curva che io descrivo nel mio libro delle quadrature [1703], alla prop. IV, nata da' seni versi, che da me suole chiamarsi ''Versiera'', in latino però ''Versoria''." See Galilei, ''Opere'', 3: 393. One finds the new term in Lorenzo Lorenzini, ''Exercitatio geometrica'', xxxi: "sit pro exemplo curva illa, quam Doctissimus magnusque geometra Guido Grandus versoria nominat."</ref> The Latin term is also used for a [[Sheet (sailing)|sheet]], the rope which turns the sail, but Grandi may have instead intended merely to refer to the [[versine]] function that appeared in his construction.{{r|larsen|stigler|truesdell|grandi}}

In 1748, [[Maria Gaetana Agnesi]] published ''Instituzioni analitiche ad uso della gioventù italiana'', an early textbook on [[calculus]].{{r|agnesi}} In it, after first considering two other curves, she includes a study of this curve. She defines the curve geometrically as the locus of points satisfying a certain proportion, determines its algebraic equation, and finds its vertex, asymptotic line, and inflection points.{{r|struik}}

===Etymology===
[[Maria Gaetana Agnesi]] named the curve according to Grandi, ''versiera''.{{r|truesdell|struik}} Coincidentally, at that time in Italy it was common to speak of the [[Devil]] through other words like ''aversiero'' or ''versiero'', derived from Latin ''adversarius'', the "adversary" of God. ''Versiera'', in particular, was used to indicate the wife of the devil, or "witch".{{r|fanfani}} Because of this, Cambridge professor [[John Colson]] mistranslated the name of the curve as "witch".{{r|mulcrone}} Different modern works about Agnesi and about the curve suggest slightly different guesses how exactly this mistranslation happened.{{r|singh|darling}} [[Dirk Jan Struik|Struik]] mentions that:{{r|struik}}
{{blockquote|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 [[Isaac Newton|Newton]]'s classification. ... The first to use the term 'witch' in this sense may have been B. Williamson, ''Integral calculus'', 7 (1875), 173;{{r|oed}} see ''Oxford English Dictionary''.}}
On the other hand, [[Stephen Stigler]] suggests that  Grandi himself "may have been indulging in a play on words", a double pun connecting the devil to the versine and the sine function to the shape of the female breast (both of which can be written as "seno" in Italian).{{r|stigler}}

==Applications==
A scaled version of the curve is the [[probability density function]] of the [[Cauchy distribution]]. This is the probability distribution on the [[random variable]] <math>x</math> determined by the following [[Experiment (probability theory)|random experiment]]: for a fixed point <math>p</math> above the {{nowrap|<math>x</math>-axis,}} choose uniformly at random a line {{nowrap|through <math>p</math>,}} and let <math>x</math> be the coordinate of the point where this random line crosses the axis. The Cauchy distribution has a peaked distribution visually resembling the [[normal distribution]], but its [[Heavy-tailed distribution|heavy tails]] prevent it from having an [[expected value]] by the usual definitions, despite its symmetry. In terms of the witch itself, this means that the {{nowrap|<math>x</math>-coordinate}} of the [[centroid]] of the region between the curve and its asymptotic line is not well-defined, despite this region's symmetry and finite area.{{r|stigler|alexander}}

In [[numerical analysis]], when approximating functions using [[polynomial interpolation]] with equally spaced interpolation points, it may be the case for some functions that using more points creates worse approximations, so that the interpolation diverges from the function it is trying to approximate rather than converging to it. This paradoxical behavior is called [[Runge's phenomenon]]. It was first discovered by [[Carl David Tolmé Runge]] for Runge's function {{nowrap|<math>y=1/(1+25x^2)</math>,}} another scaled version of the witch of Agnesi, when interpolating this function over the {{nowrap|interval <math>[-1,1]</math>.}} The same phenomenon occurs for the witch <math>y=1/(1+x^2)</math> itself over the wider {{nowrap|interval <math>[-5,5]</math>.{{r|runge}}}}

The witch of Agnesi approximates the [[spectral energy distribution]] of [[spectral lines]], particularly [[X-ray]] lines.{{r|spencer}}

The cross-section of a smooth [[hill]] has a similar shape to the witch.{{r|coppin}} Curves with this shape have been used as the generic topographic obstacle in a flow in mathematical modeling.{{r|snyder|lamb}}
[[Soliton|Solitary wave]]s in deep water can also take this shape.{{r|benjamin|noonan}}

A version of this curve was used by [[Gottfried Wilhelm Leibniz]] to derive the [[Leibniz formula for π|Leibniz formula for {{pi}}]]. This formula, the [[infinite series]]
<math display=block>\frac{\pi}{4} = 1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots,</math>
can be derived by equating the area under the curve with the integral of the {{nowrap|function <math>1/(1+x^2)</math>,}}
using the [[Taylor series]] expansion of this function as the infinite [[geometric series]] {{nowrap|<math>1-x^2+x^4-x^6+\cdots</math>,}} and integrating term-by-term.{{r|yates}}

==In popular culture== 
''The Witch of Agnesi'' is the title of a novel by Robert Spiller. It includes a scene in which a teacher gives a version of the history of the term.{{r|gazette}}
 	
''Witch of Agnesi'' is also the title of a music album by jazz quartet Radius. The cover of the album features an image of the construction of the witch.{{r|discogs}}

==References==
<references>

<ref name=agnesi>{{citation
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 | url = https://archive.org/details/A298183
 | year = 1748}} See in particular Problem 3, [https://archive.org/details/A298183/page/n403 pp. 380–382], and [https://archive.org/details/A298183/page/n507 Fig. 135].</ref>

<ref name=alexander>{{citation
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<ref name=arnold>{{citation
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 | series = London Mathematical Society Lecture Note Series
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<ref name=benjamin>{{citation
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<ref name=darling>{{citation
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<ref name=discogs>{{citation|url=https://www.discogs.com/Radius-Witch-Of-Agnesi/release/1335001|publisher=Discogs|title=Radius – Witch Of Agnesi (Plutonium Records, 2002)|access-date=28 May 2018}}</ref>

<ref name=eagles>{{citation|title=Constructive Geometry of Plane Curves: With Numerous Examples|first=Thomas Henry|last=Eagles|publisher=Macmillan and Company|year=1885|contribution=The Witch of Agnesi|pages=313–314|contribution-url=https://archive.org/stream/constructivegeom00eagluoft#page/312/mode/2up}}</ref>

<ref name=egdc>{{citation
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<ref name=fanfani>[[Pietro Fanfani]], ''Vocabolario dell'uso toscano'', [https://books.google.com/books?id=SOczmy2F2y0C&pg=PA334 p. 334]</ref>

<ref name=fermat>{{citation|first=Pierre|last=de Fermat|author-link=Pierre de Fermat|title=Oevres|volume=1|publisher=Gauthier-Villars et fils|year=1891|url=https://archive.org/stream/uvresdefermat00natigoog#page/n330/mode/2up|language=la|pages=280–285}}</ref>

<ref name=fmq>{{citation
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<ref name=fomshu>{{citation
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<ref name=gazette>{{citation|newspaper=[[The Gazette (Colorado Springs)|The Gazette]]|title=Local teacher, author figures math into books|first=Dave|last=Phillips|date=12 September 2006|url=http://gazette.com/local-teacher-author-figures-math-into-books/article/6514}}</ref>

<ref name=grandi>{{citation|first=G.|last=Grandi|author-link=Luigi Guido Grandi|year=1718|contribution=Note al trattato del Galileo del moto naturale accellerato|title=Opera Di Galileo Galilei|volume=III|page=393|location=Florence|language=it}}. As cited by {{harvtxt|Stigler|1974}}.</ref>

<ref name=haftendorn>{{citation
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<ref name=kiss><!-- ridiculous dictionary definition of "kiss" reference requested by Gatoclass for DYK -->{{citation|title=Multivariable Calculus with MATLAB®: With Applications to Geometry and Physics|first1=Ronald L.|last1=Lipsman|first2=Jonathan M.|last2=Rosenberg|publisher=Springer|year=2017|isbn=9783319650708|page=42|url=https://books.google.com/books?id=PABCDwAAQBAJ&pg=PA42|quote=The circle "kisses" the curve accurately to second order, thus is given the name osculating circle (from the Latin word for "kissing").}}</ref>

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<ref name=oed>{{citation|title=Oxford English Dictionary|url=http://www.oed.com/view/Entry/229575|publisher=Oxford University Press|year=2018|at=witch, ''n''.2, 4(e)|access-date=3 July 2018|quote=1875 B. Williamson ''Elem. Treat. Integral Calculus'' vii. 173   Find the area between the witch of Agnesi <math>xy^2 = 4a^2(2a-x)</math> and its asymptote.}}</ref>

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 | page = 415
 | title = Properties of the Witch of Agnesi—Application to Fitting the Shapes of Spectral Lines
 | volume = 30| bibcode = 1940JOSA...30..415S
 }}</ref>

<ref name=stigler>{{citation
 | last = Stigler | first = Stephen M. | author-link = Stephen Stigler
 | date = August 1974
 | doi = 10.1093/biomet/61.2.375
 | issue = 2
 | journal = [[Biometrika]]
 | jstor = 2334368
 | mr = 0370838
 | pages = 375–380
 | title = Studies in the History of Probability and Statistics. XXXIII. Cauchy and the Witch of Agnesi: An Historical Note on the Cauchy Distribution
 | volume = 61}}</ref>

<ref name=struik>A translation of Agnesi's work on this curve can be found in: {{citation|title=A Source Book in Mathematics, 1200–1800|last=Struik|first=Dirk J.|author-link=Dirk Jan Struik|year=1969|publisher=Harvard University Press|location=Cambridge, Massachusetts|pages=178–180|url=https://archive.org/stream/B-001-001-112#page/n199/mode/2up}}</ref>

<ref name=truesdell>{{citation |first=C. |last=Truesdell |author-link=Clifford Truesdell|title=Correction and Additions for "Maria Gaetana Agnesi" |journal=Archive for History of Exact Sciences |volume=43 |issue=4 |year=1991 |pages=385–386 |doi=10.1007/BF00374764 |quote=[…] nata da' seni versi, che da me suole chiamarsi la ''Versiera'' in latino però ''Versoria'' […]|doi-access=free }}</ref>

<ref name=yates>{{citation|first=Robert C.|last=Yates|title=Curves and their Properties|publisher=National Council of Teachers of Mathematics|series=Classics in Mathematics Education|volume=4|year=1954|url=https://files.eric.ed.gov/fulltext/ED100648.pdf|contribution=Witch of Agnesi|pages=237–238}}</ref>

</references>

==External links==
{{EB1911 poster|Witch of Agnesi}}
{{commons|Witch of Agnesi}}
*[http://www-groups.dcs.st-and.ac.uk/~history/Curves/Witch.html "Witch of Agnesi" at MacTutor's Famous Curves Index]
*{{MathWorld | urlname=WitchofAgnesi | title=Witch of Agnesi|mode=cs2}}
*[http://demonstrations.wolfram.com/WitchOfAgnesi/ Witch of Agnesi] by Chris Boucher based on work by [[Eric W. Weisstein]], [[The Wolfram Demonstrations Project]].
*{{citation|first=Evelyn|last=Lamb|publisher=[[Scientific American]]|url=https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-witch-of-agnesi/|title=A Few of My Favorite Spaces: The Witch of Agnesi|work=Roots of Unity|date=28 May 2018}}

{{good article}}

[[Category:Cubic curves]]
[[Category:Plane curves]]