Cartesian oval


last updated: 20041204 
with foci F_{1} (1, 0) and F_{2} (1,0).
This bipolar equation defines the Cartesian oval as the collection of points
for which the distances to F_{1} and F_{2}
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 r_{1}
± b r_{2} = ±1.
When the inside and outside of an Cartesian oval have refraction indices n_{1
}and n_{2}, respectively, with n_{1}/n_{2} = 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:
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 = a^{2} 
b^{2} and d = a^{2} + b^{2}:
(c(x^{2} +y^{2}+1)2dx))^{2} = 2d(x^{2}+y^{2}+1)4cx1
This is said to be equivalent with the following:
(x^{2}+y^{2})^{2} + k(x^{2}+y^{2}) + lx
+ m = 0
3) in French: courbe aplanétique
