Cartesian Oval

Cartesian equation:
((1m2)(x2+y2)+2m2cx+a2m2c2)2=4a2(x2+y2)((1 - m^{2})(x^{2} + y^{2}) + 2m^{2}cx + a^{2} - m^{2}c^{2})^{2} = 4a^{2}(x^{2} + y^{2})


This curve CC consists of two ovals so it should really be called Cartesian Ovals. It is the locus of a point PP whose distances ss and tt from two fixed points SS and TT satisfy s+mt=as + mt = a. When cc is the distance between SS and TT then the curve can be expressed in the form given above.

The curves were first studied by Descartes in 1637 and are sometimes called the 'Ovals of Descartes'.

The curve was also studied by Newton in his classification of cubic curves.

The Cartesian Oval has bipolar equation r+mr=ar + mr' =a.

If m=±1m = ±1 then the Cartesian Oval CC is a central conic while if m=a/cm = a/c then the curve is a Limacon of Pascal (Étienne Pascal). In this case the inside oval touches the outside one.

Cartesian Ovals are anallagmatic curves.