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Witch of Agnesi
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From Wikipedia, the free encyclopedia
Cubic plane curve
Selected witch of Agnesi curves (green), and the circles they are constructed from (blue), with radius parameters
{\displaystyle a=1}
{\displaystyle a=2}
{\displaystyle a=4}
, and
{\displaystyle a=8}
In mathematics, the witch of Agnesi (Italian pronunciation: [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 la versiera di Agnesi is based on Latin versoria (sheet of sailing ships) and the sinus versus.<br>This was read by John Colson as l'avversiera di Agnesi, where avversiera is translated as "woman who is against God" and interpreted as "witch".[1][2][3][4]
The 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 approximation of functions by polynomials, has been used to approximate the energy distribution of spectral lines, 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 line to the circle at the other point. It has a unique 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 points 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 volume of revolution of the curve around its defining line is twice the volume of the torus of revolution of its defining circle.
Construction<br>[edit]
The witch of Agnesi (curve MP) with labeled points<br>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.<br>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.[5] It includes, as a limiting case, the point M itself.
Equations<br>[edit]
Suppose that point O is at the origin and point M lies on the positive
{\displaystyle y}
-axis, and that the circle with diameter OM has a."}},"i":0}}]}'>radius
{\displaystyle a}
Then the witch constructed from O and M has the Cartesian equation[6][7]
{\displaystyle y={\frac {8a^{3}}{x^{2}+4a^{2}}}={\frac {(2a)^{3}}{(x)^{2}+(2a)^{2}}}.}
This equation can be simplified, by choosing a=\\tfrac12,"}},"i":0}}]}'>
{\displaystyle a={\tfrac {1}{2}}}
, to the form
{\displaystyle y={\frac {1}{x^{2}+1}}.}
or equivalently, by clearing denominators, as the cubic algebraic equation
1.
{\displaystyle (x^{2}+1)y=1.}
In its simplified form, this curve is the graph of the derivative of the arctangent function.[8]
The witch of Agnesi can also be described by parametric equations whose parameter θ is the angle between OM and OA, measured clockwise:[6][7]
tan
cos
{\displaystyle {\begin{aligned}x&=2a\tan \theta ,\\y&=2a\cos ^{2}\theta .\end{aligned}}}
Properties<br>[edit]
The main properties of this curve can be derived from integral calculus.<br>The area between the witch and its asymptotic line is four times the area of the fixed circle, 4\\pi a^2.{{r|lawrence|yates|larsen}}"}},"i":0}}]}'>
{\displaystyle 4\pi a^{2}}
.[6][7][9]<br>The volume of revolution of the witch of Agnesi about its asymptote 4\\pi^2a^3.{{r|lawrence}}"}},"i":0}}]}'>is
{\displaystyle 4\pi ^{2}a^{3}}
.[6] This is two times the volume of the torus formed by revolving the defining circle of the witch around the same line.[9]
The curve has a unique 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.[10] The defining circle of the witch is also its osculating circle at the vertex,[11] the unique circle that "kisses" the curve at that point by sharing the same orientation and curvature.[12] Because this is an osculating circle at the vertex of the curve, it has third-order contact with...