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 - m2)(x2 + y2) + 2m2cx + a2 - m2c2)2 = 4a2(x2 + y2)
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.