Negative Pedal Curves of an Ellipse

An ellipse can be described parametrically by two equations, each of which contains a constant; in this Demonstration these constants have been labeled and : , . The same two constants appear in the parametric equations of the negative pedal curves. The set of negative pedal curves with respect to the origin includes a curve known as "Talbot's curve" which has four cusps and two ordinary double points. The set of curves with respect to one of the foci is called "Burleigh's ovals" and includes a fish-like curve that inspired a paper by H. Martyn Cundy (Mathematical Gazette, 85(504), pp. 439-445).