Witch of Agnesi

The "witch of Agnesi" is a cubic curve studied by Maria Agnesi in her book Instituzioni analitiche ad uso della gioventù italiana, published in 1748 (MacTutor History of Mathematics Archive). The curve is also known as cubique d'Agnesi or agnésienne and was studied earlier by Fermat (MacTutor History of Mathematics Archive) and by Guido Grandi, whose treatment appeared in 1703 (O'Connor and Robertson 2014).

The name "witch" derives from a mistranslation of the term averisera ("versed sine curve," from the Latin vertere, "to turn") in the original work as avversiera ("witch" or "wife of the devil") in an 1801 translation of the work by Cambridge Lucasian Professor of Mathematics John Colson (Gray).

The curve is obtained by drawing a line from the origin through the circle of radius and center , then picking the point with the coordinate of the intersection with the circle and the coordinate of the intersection of the extension of line with the line .

The parametric equations are

			(1)
			(2)

for , or

			(3)
			(4)

for .

The Cartesian equation can be obtained by eliminating in the parametric equations, giving

(5)

which is equivalent in functional form to the Lorentzian function.

The area between the curve and the x-axis is

(6)

The curve has inflection points at . The line is an asymptote to the curve.

The curvature and tangential angle in the second parameterization are given by

			(7)
			(8)