info prev up next book cdrom email home

Cartesian Ovals

\begin{figure}\begin{center}\BoxedEPSF{CartesianOvals.epsf scaled 600}\end{center}\end{figure}

A curve consisting of two ovals which was first studied by Descartes in 1637. It is the locus of a point $P$ whose distances from two Foci $F_1$ and $F_2$ in two-center Bipolar Coordinates satisfy

mr\pm nr'=k,
\end{displaymath} (1)

where $m, n$ are Positive Integers, $k$ is a Positive real, and $r$ and $r'$ are the distances from $F_1$ and $F_2$. If $m=n$, the oval becomes an Ellipse. In Cartesian Coordinates, the Cartesian ovals can be written
\end{displaymath} (2)

\end{displaymath} (3)

$ = 2(m^2+n^2)(n^2+y^2+a^2)-4ax(m^2-n^2)-k^2.\quad$ (4)
Now define
$\displaystyle b$ $\textstyle \equiv$ $\displaystyle m^2-n^2$ (5)
$\displaystyle c$ $\textstyle \equiv$ $\displaystyle m^2+n^2,$ (6)

and set $a=1$. Then
\end{displaymath} (7)

If $c'$ is the distance between $F_1$ and $F_2$, and the equation
\end{displaymath} (8)

is used instead, an alternate form is
\end{displaymath} (9)

The curves possess three Foci. If $m = 1$, one Cartesian oval is a central Conic, while if $m = a/c$, then the curve is a Limaçon and the inside oval touches the outside one. Cartesian ovals are Anallagmatic Curves.


Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 35, 1989.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 155-157, 1972.

Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 188, 1967.

MacTutor History of Mathematics Archive. ``Cartesian Oval.''

info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein