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 P whose distances from two foci F_1 and F_2 in two-center bipolar coordinates satisfy


where m,n are positive integers, k is a positive real, and r and r^' are the distances from F_1 and F_2 (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


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


Once again squaring both sides gives




gives the slightly simpler form


which corresponds to the form given by Lawrence (1972, p. 157) in the case a=1 and k=1.

If m=n, the oval becomes a central conic.

If c^' is the distance between F_1 and F_2, and the equation


is used instead, an alternate form is


See also

Bipolar Coordinates, Oval

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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.", J. "Cartesian Oval."

Cite this as:

Weisstein, Eric W. "Cartesian Ovals." From MathWorld--A Wolfram Web Resource.

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