## The Witch of Agnesi

Credits. This page, and the Geometer's Sketchpad document on which it is based, is from the WWW project A Visual Dictionary of Special Plane Curves, Copyright 1995, 1996 by Xah Lee (xah@best.com). It can be freely distributed provided this information is intact.

The Witch of Agnesi is defined as the curve traced by X as Drag Me moves around the circle. You can drag Drag Me with the mouse, or click Animate to move it around the circle. Click Show Trace to begin tracing point X as it moves; click Hide Trace to stop. You can also drag any other red point, or while the sketch is animating, press '>' or '<' keys to speed up and slow down the animation. Click the red X in the lower-right corner to clear any visible traces.

Step by step description

1. Let there be a circle of radius a with center at {0,a}.
2. Let there be a horizontal line L passing through {0, 2 a}.
3. From any point M on the circle, draw a secant passing the Origin and M. Let the intersection of this secant and line L be N.
4. Witch of Agnesi is the locus of intersections of a horizontal line passing through M and a vertical line passing through N.

Formulas

1. Parametric: 2 a {Tan[t], Cos[t]^2}, -Pi/2 < t < Pi/2.
2. Cartesian: y (x^2 + 4 a^2) == 8 a^3

(a is the scaling factor. Geometrically, it's the radius of the circle on which the Witch is constructed.)

History

Studied by Maria Gaetana Agnesi (1718-1799) in 1748. Also studied by Fermat (1666), and Guido Grandi (1703). The name of this curve has a colorful history. "Versaria" is the name given by Grandi, meaning "turning in every direction". In the course of time the word versaria took on another meaning. The Latin words adversaria, and by aphaeresis, versaria, signify a female that is contrary to God. Thus gradually the curve versaria came to be known in English as "the witch."