The Witch of Agnesi


Maria Gaetana Agnesi, for whom this curve is named, was an Italian mathematician. She called the curve versiera, from the Italian vertere, "to turn." Her translator, however, had just learned Italian, and confused the word with avversiera, which means "wife of the devil." Hence the curve's unusual name. (source)


The witch of Agnesi is defined by taking a circle of radius a, centered at the point (0, a), and drawing a tangent line at the point (0, 2a). The origin is then connected to a point A on the tangent line, and the point of intersection of this line with the circle is labeled Q. Taking the x-coordinate from A and the y-coordinate from Q, we have a new point P. As A moves along the tangent line, P traces out the witch of Agnesi. The curve is parameterized by the equations:

An implicit definition is given by:


Velocity and Acceleration:

The Tangent Vector:

Arc Length: