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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  and  in two-center bipolar
coordinates satisfy
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(1) |
where m, n are positive
integers, k is a positive real,
and r and  are the distances from and . If m = n, the oval becomes an ellipse. In Cartesian
coordinates, the Cartesian ovals can be written
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(2) |
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(3) |
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|
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(4) | Now define
and set a = 1. Then
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(7) |
If  is the distance between and , and the equation
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(8) | is used instead, an alternate form is
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(9) |
The curves possess three foci. If m
= 1, one Cartesian oval is a central conic, while
if  , then the curve is a Limaçon
and the inside oval touches the outside one. Cartesian ovals are anallagmatic
curves.
References
Baudoin, P. Les ovales de Descartes et le limaçon de
Pascal. Paris: Vuibert, 1938.
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." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cartesian.html.
Author: Eric W. Weisstein
© 1999 CRC Press
LLC, © 1999-2002 Wolfram Research,
Inc.
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