Witch of Agnesi

The Witch of Agnesi with parameters a = 1, a = 2, a = 4, and a = 8

In mathematics, the witch of Agnesi (Italian pronunciation: [a.ˈɲe.zi]) is a cubic plane curve defined from two diametrically opposite points of a circle.It gets its name from Italian mathematician Maria Gaetana Agnesi, and from a mistranslation of an Italian word for a sailing sheet. Before Agnesi, the same curve was studied by Fermat, Grandi, and Newton.

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,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

The witch of Agnesi (curve MP) with labeled points
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.[1] It includes, as a limiting case, the point M itself.

Other Languages
Afrikaans: Heks van Agnesi
български: Версиера
bosanski: Versiera
español: Curva de Agnesi
italiano: Versiera
norsk nynorsk: Agnesis heks
Piemontèis: Masca d'Agnesi
polski: Lok Agnesi
português: Curva de Agnesi
slovenščina: Agnesin koder
svenska: Agnesis häxa
українська: Локон Аньєзі
中文: 箕舌线