Witch of Agnesi
Starting with a fixed circle, a point O on the circle is chosen. For any other point A on the circle, the secant line OA is drawn. The point M is diametrically opposite to O. The line OA intersects the tangent of M at the point N. The line parallel to OM through N, and the line perpendicular to OM through A intersect at P. As the point A is varied, the path of P is the Witch of Agnesi.
The curve is
Suppose the point O is the origin, and that M is on the positive y-axis. Suppose the radius of the circle is a.
Then the curve has Cartesian equation
Note that if a = 1/, then this equation becomes rather simple:
This is the derivative of the
Parametrically, if θ is the angle between OM and OA, measured clockwise, then the curve is defined by the equations
Another parameterization, with θ being the angle between OA and the x-axis, increasing anti-clockwise is