The Witch of Agnesi

This page, and the Sketchpad document on which it is based, are from the Visual Dictionary of Special Plane Curves, copyright ©1995, 1996 by Xah Lee. This information can be freely distributed provided it remains 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 press Animate to move it around the circle. Press Show Trace to begin tracing point X as it moves; press Hide Trace to stop. You can also drag any other red point or, while the JavaSketch is animating, press the > or < keys to speed up and slow down the animation. Click the 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 the center at (0, a).
2. Let there be a horizontal line l passing through (0, 2a).
3. From any point M on the circle, draw a secant passing through the origin and M. Let the intersection of this secant and line l be N.
4. The Witch of Agnesi is the locus of intersections of a horizontal line passing through M and a vertical line passing through N.

Formulas

• Parametric: (2a tan(t), 2a cos)t)2}, -π/2 < t < π/2.
• Cartesian: y (x2 + 4 a2) == 8 a3

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

History

This was studied by Maria Gaetana Agnesi (1718–1799) in 1748 and 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."