## Cartesian Ovals

A curve consisting of two ovals which was first studied by Descartes in 1637. 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 . If , the oval becomes an Ellipse. In Cartesian Coordinates, the Cartesian ovals can be written
 (2)

 (3)

 (4)
Now define
 (5) (6)

and set . Then
 (7)

If is the distance between and , and the equation
 (8)

is used instead, an alternate form is
 (9)

The curves possess three Foci. If , 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

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.