Focaloid

In

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 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.

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).

External links

Media related to Focaloid at Wikimedia Commons

Mathematical definition (3D)

Confocal

Physical significance

See also

References

External links