Variations on the Witch of Agnesi

Abstract

At some point in their mathematical training, most teachers encounter the Witch of Agnesi. The Witch, a plane curve named and studied in the mid-1700s by the Italian mathematician Maria Agnesi, is constructed as follows:

1. Center a circle with radius a at the point (0,a).
2. Choose any point A on the line y=2a and connect it to the origin O with a line segment.
3. Label the point where the line segment OA intersects the circle as B.
4. Let P be the point whose x-coordinate is the same as point A and whose y-coordinate is the same as point B.