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Focaloid in 3D

In 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)[edit]

If one boundary surface is given by

with semiaxes abc 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 ellipsoids share the same foci, which are given for the example above by

Physical meaning[edit]

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[edit]