Focaloid
Introduction Focaloid in 3DIn geometry, a '''focaloid ''' is a shell bounded by two concentric, confocal ellipses (in 2D) or ellipsoids (in 3D). When the thickness of the shell becomes negligible, it is called a '''thin focaloid'''.
Mathematical definition (3D) If one boundary surface is given by: with semiaxes ''a'', ''b'', ''c'' the second surface is given by: The '''thin focaloid''' is then given by the limit .In general, a focaloid could be understood as a shell consisting out of two closed coordinate surfaces of a confocal ellipsoidal coordinate system.
Confocal Confocal ellipsoids share the same foci, which are given for the example above by: : :
Physical significance A focaloid can be used as a construction element of a matter or charge distribution. The particular importance of focaloids lies in the fact that two different but confocal focaloids of the same mass or charge produce the same action on a test mass or charge in the exterior region.
See also * Homoeoid* Confocal conic sections
References *Subrahmanyan Chandrasekhar (1969): ''Ellipsoidal Figures of Equilibrium.'' Yale University Press, London, Connecticut*Routh, E. J.: ''A Treatise on Analytical Statics, Vol II'', Cambridge University Press, Cambridge (1882).
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