Cartesian oval

A Cartesian oval is a curve that actually consists of two ovals, one inside the other. It is the locus of a point whose distances s and t from two fixed points S and T satisfy the equation s + mt = a. When c is the distance between S and T then the curve can be expressed in the form:


    ((1 – m 2)(x 2 + y 2) + 2m 2cx + a 2m 2c 2)2 = 4a 2(x 2 + y 2)


The curves were first studied by René Descartes in 1637 and are sometimes called the ovals of Descartes. They were also investigated by Isaac Newton in his classification of cubic curves.


If m = +1 then the Cartesian oval is a central conic. If m = a/c then it becomes a limacon of Pascal, in which case the inside oval touches the outside one. Cartesian ovals are anallagmatic curves.