The "Cartesian ovals," sometimes also known as the Cartesian curve or oval of Descartes, are the quartic curve consisting of
two ovals. They were first studied by Descartes in 1637 and by Newton while classifying
cubic curves. It is the locus of a point 
whose distances from two foci 
and 
in two-center bipolar
coordinates satisfy
 |
(1)
|
where 
are positive integers, 
is a positive real, and 
and 
are the distances from 
and 
(Lockwood 1967, p. 188).
Cartesian ovals are anallagmatic curves. Unlike the Cartesian ovals, these curves possess three foci.
In Cartesian coordinates, the Cartesian ovals can be written
 |
(2)
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Moving the quantity involving 
to the right-hand side, squaring both sides, simplifying,
and rearranging gives
 |
(3)
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Once again squaring both sides gives
![[(m^2-n^2)(x^2+y^2+a^2)-2ax(m^2+n^2)-k^2]^2
=4k^2n^2[(a+x)^2+y^2].](/images/equations/CartesianOvals/NumberedEquation4.svg) |
(4)
|
Defining
 |  |  |
(5)
|
 |  |  |
(6)
|
gives the slightly simpler form
![[a^2b-k^2-2acx+b(x^2+y^2)]^2=2(c-b)k^2[(a+x)^2+y^2],](/images/equations/CartesianOvals/NumberedEquation5.svg) |
(7)
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which corresponds to the form given by Lawrence (1972, p. 157) in the case 
and 
.
If 
,
the oval becomes a central conic.
If 
is the distance between 
and 
, and the equation
 |
(8)
|
is used instead, an alternate form is
![[(1-m^2)(x^2+y^2)+2m^2c^'x+a^('2)-m^2c^('2)]^2=4a^('2)(x^2+y^2).](/images/equations/CartesianOvals/NumberedEquation7.svg) |
(9)
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