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Encyclopedia Britannica



OVAL (Lat. ovum, egg)

This article appears in Volume V20, Page 382 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: ORC-PAI
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, ate 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 lrtmr' =n, where r,r' are the distances of a point on the curve from two fixed and given points, termed the foci, and 1, m, 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 rr' = a2, where r,r' are the radii of a point on the curve from two given foci, and a is a
constant
 . This curve issymmetrical 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 ler lniscate (q.v.).
See CURVE; also Salmon, Higher Plane Curves.


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