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.

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.