 TOPICS # Cartesian Ovals 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)

Moving the quantity involving to the right-hand side, squaring both sides, simplifying, and rearranging gives (3)

Once again squaring both sides gives (4)

Defining   (5)   (6)

gives the slightly simpler form (7)

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 (9)

Bipolar Coordinates, Oval

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## References

Baudoin, P. Les ovales de Descartes et le limaçon de Pascal. Paris: Vuibert, 1938.Cundy, H. and Rollett, A. "The Cartesian Ovals." §2.4.3 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 35, 1989.Lawrence, J. D. "Cartesian Oval." §5.17 in A Catalog of Special Plane Curves. New York: Dover, pp. 155-157, 1972.Lockwood, E. H. "The Ovals of Descartes." A Book of Curves. Cambridge, England: Cambridge University Press, p. 188, 1967.MacTutor History of Mathematics Archive. "Cartesian Oval." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cartesian.html.Wassenaar, J. "Cartesian Oval." http://www.2dcurves.com/quartic/quarticct.html.

## Cite this as:

Weisstein, Eric W. "Cartesian Ovals." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CartesianOvals.html