Witch of Agnesi

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. 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.