Descartes oval

A plane curve for which the distances and between any point of the curve and two fixed points and (the foci) are related by the non-homogeneous linear equation

A Descartes oval may be defined by means of the homogeneous linear equation

where is the distance to the third focus located on the straight line . In the general case, a Descartes oval consists of two closed curves, one enclosing the other (see Fig.). The equation of a Descartes oval in Cartesian coordinates is

where is the length of the segment . If and , the Descartes oval is an ellipse; if and , it is a hyperbola; if , it is a Pascal limaçon. First studied by R. Descartes in the context of problems of optics [1].

Figure: d031340a

References

[1]	R. Descartes, "Géométrie" , Leyden (1637)
[2]	A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)

Comments

A Descartes oval is also called a Cartesian oval, or simply a Cartesian.

The caustic of a circle with respect to a point light source in the plane of the circle is the evolute of a Descartes oval.

There are several ways in which a Cartesian oval can arise. One is as follows. Let and be two cones with axes parallel to the -axis and with circular intersections with the -plane. Then the vertical projection onto the -plane of their intersection is a Cartesian oval. This is a result of A. Quetelet. Suppose the apexes of and are at and , respectively, and the intersections with the -plane are circles of radius 1 and (see below). Then in the case , the projection of the intersection is a cardioid.

In polar coordinates the equation for a Cartesian oval is . Hence when the equation becomes (plus ) and one obtains the Pascal limaçon; when in addition , the cardioid results, whose equation can also be written as .

References

[a1]	J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
[a2]	K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)