Oval

Oval (Lat. ovum, egg), in geometry, a closed curve, generally more or less egg-like in form. The simplest oval is the ellipse; more complicated forms are represented in the notation of analytical geometry by equations of the 4th, 6th, 8th. .. degrees. Those of the 4th degree, known as bicircular quartics, are the most important, and of these the special forms named after Descartes and Cassini are of most interest. The Cartesian ovals presented themselves in an investigation of the section of a surface which would refract rays proceeding from a point in a medium of one refractive index into a point in a medium of a different refractive index. The most convenient equation is lr? mr' = n, where r,r are the distances of a point on the curve from two fixed and given points, termed the foci, and 1, n are constants. The curve is obviously symmetrical about the line joining the foci, and has the important property that the normal at any point divides the angle between the radii into segments whose sines are in the ratio 1: m. The Cassinian oval has the equation y r' = a 2 , where r,r are the radii of a point on the curve from two given foci, and a is a constant. This curve is symmetrical about two perpendicular axes. It may consist of a single closed curve or of two curves, according to the value of a;  the transition between the two types being a figure of 8, better known as Bernoulli's lemniscate.

See CURVE; also Salmon, Higher Plane Curves.