Witch of Agnesi

Witch of Agnesi

Published by Maria Gaëtana Agnesi (1718-1799) in her best-known work Instituzioni analitiche ad uso della gioventù italiana (translated: Analytical Institutions for the Use of Italian Youth, 1748) the Witch of Agnesi is a versed sine curve. (A versed sine, or versine curve, is an outdated expression for 1 - cos x, which was commonly used to describe curves in early mathematics.) It is alternatively known as Cubique d'Agnesi (Agnesi's cubic), Agnésienne, or the Agnesi curve, and was studied by both Pierre de Fermat and Guido Grandi as early as 1703. Termed "versiera" by Agnesi, the name of the curve was mistranslated "wife of the devil" or "witch" and hence the name "Witch of Agnesi" stuck. The curve is often most described algebraically, parametrically, or geometrically.

To construct the Witch of Agnesi geometrically, first draw a circle of diameter d centered at (0,d/2)--that is, the circle should be symmetric about the y-axis with a tangent line of y = 0 (the x-axis). Draw a second tangent line to the circle of y = d. To draw the Witch of Agnesi curve, draw a line segment from the origin (0,0) to any point along the line y = d (x,d); call this point Q. The line segment OQ will cross the circle at exactly two points--the origin (0,0) and one other point (a,y)--call it point B--along the circumference of the circle. The intersection of all points (x,y) made by creating a vertical line (that is, parallel to the y-axis) through the point (x,d) and a horizontal line (that is, parallel to the x-axis) through the point (a,y) is Agnesi's curve. Agnesi's curve extends infinitely in both the positive x and negative x directions and is generally bell-shaped. The peak, or highest point, of Agnesi's curve will always occur at the point (0,d), and its lower bound is the asymptote y = 0. The maximum height of the Agnesi curve is exactly the magnitude chosen for the diameter d.

An algebraic equation for Agnesi's curve, in Cartesian coordinates, is y = d³/(x² + d²). The power of the numerator--3--is the reason that the curve is sometimes referred to as "Agensi's cubic." It has inflection points at the line y = 3a/2. Using Cartesian coordinates, one parametric form is x = dt, y = d/(1 + t²), while a parametric form of the curve in polar coordinates is x = 2dcot(, y = d(1 - cos(2()). Agnesi's curve can also be approximated using Taylor polynomials, with x₀ = 0. However, since the denominator x² + d² = 0 when x = +di and -di, the Taylor approximation will hold only on the interval (-d, d). Outside that interval, Taylor's polynomial will not converge to the curve.

Agnesi's curve and others that are similar were studied carefully by mathematicians such as Agnesi and Fermat in order to better explore the relationship between plane geometry and algebraic expressions, as well as the relationships between plane geometry and differential calculus. During this period, mathematicians and even educated elite were especially interested in simplifying mathematical expressions and concepts. Among the educated elite, even hosts would sometimes pose the simplification of a mathematical problem as a parlor game to their guests. Today, Agnesi's curve is used primarily as a modeling and statistical tool. Some computer models for weather and atmospheric conditions, for example, use Agnesi's curve to model mountains and other topographic peaks of terrain. In one application, for example, the mountain may be incorporated into the model using Agnesi's curve. Then scientists use the ideas of slope and velocity to determine wind speed and direction on the downslope. Although less common, the Agnesi curve is also used as a distribution model (akin to the normal curve) for statistics. Since the algebraic expression for the curve is relatively straight-forward to integrate over a specified interval, it holds advantages over using the standard normal curve in statistics.