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

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

## 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).