Focaloid

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)

If one boundary surface is given by

{\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1

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

{\frac {x^{2}}{a^{2}+\lambda }}+{\frac {y^{2}}{b^{2}+\lambda }}+{\frac {z^{2}}{c^{2}+\lambda }}=1.

The thin focaloid is then given by the limit $\lambda \to 0$ .

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}+\lambda )-(b^{2}+\lambda ),\,

f_{2}^{2}=a^{2}-c^{2}=(a^{2}+\lambda )-(c^{2}+\lambda ),\,

f_{3}^{2}=b^{2}-c^{2}=(b^{2}+\lambda )-(c^{2}+\lambda ).

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

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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Focaloid

Mathematical definition (3D)

Confocal

Physical meaning

See also

References