Witch of Agnesi - Equations

Equations

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=1/2, then this equation becomes rather simple:

Parametrically, if is the angle between OM and OA, measured clockwise, then the curve is defined by the equations

Another parameterization, with being the angle between OA and the x-axis, increasing anti-clockwise is