page_title: Witch of Agnesi
section_title: Construction
hierarchical_section_title: Witch of Agnesi / Construction
caption_attribution_description: Curva de Agnesi Français&#160;: Construction de la courbe d'Agnesi.
context_page_description: In mathematics, the witch of Agnesi 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 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 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.
context_section_description: 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. It includes, as a limiting case, the point M itself.
caption: An animation showing the construction of the witch of Agnesi