# 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}) + 2*m*^{ 2}*cx* + *a*^{ 2} – *m*^{ 2}*c*^{ 2})^{2} = 4*a*^{ 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.