Cartesian oval

  

with foci F₁ (-1, 0) and F₂ (1,0).
This bipolar equation defines the Cartesian oval as the collection of points for which the distances to F₁ and F₂ are related linearly.
The curve is also called the oval of Descartes, or the Cartesian curve ¹⁾.
One can imagine that 

The curve is a quartic, in fact a bicircular quartic and a cyclic of a circle.
When working out the bipolar equation into the Cartesian form ²⁾, a second oval appears. In fact the bipolar equation extends to a r₁ ± b r₂ = ±1.

When the inside and outside of an Cartesian oval have refraction indices n₁ and n₂, respectively, with n₁/n₂ = b/a, then the refracted rays sent from one focal point, seem to be issued from the other focal point. This led to the name of aplanatic curve ³⁾.
For certain values the inner part of the curve has the form of an egg, the egg of Descartes.

Some special cases of the Cartesian oval are:

• a = b: ellipse
• a = -b: hyperbola
• a = 1: limaçon

It was Descartes (1637) who was the first to describe the curve; Newton studied the curve while classifying his cubic curves.

notes

1) In French: cartésienne
In Italian: ovale Cartesiana

2) Cartesian equation of the curve is, with c = a² - b² and d = a² + b²:
(c(x² +y²+1)-2dx))² = 2d(x²+y²+1)-4cx-1

This is said to be equivalent with the following:
(x²+y²)² + k(x²+y²) + lx + m = 0

3) in French: courbe aplanétique