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Focaloid

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Focaloid

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.

Contents

Mathematical definition (3D)

If one boundary surface is given by

x 2 a 2 + y 2 b 2 + z 2 c 2 = 1

with semiaxes a, b, c the second surface is given by

x 2 a 2 + λ + y 2 b 2 + λ + z 2 c 2 + λ = 1.

The thin focaloid is then given by the limit λ → 0 .

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

f 1 2 = a 2 − b 2 = ( a 2 + λ ) − ( b 2 + λ ) , f 2 2 = a 2 − c 2 = ( a 2 + λ ) − ( c 2 + λ ) , f 3 2 = b 2 − c 2 = ( b 2 + λ ) − ( c 2 + λ ) .

Physical meaning

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.

References

Focaloid Wikipedia
(Text) CC BY-SA