Homoeoid

A homoeoid is a shell (a bounded region) bounded by two concentric, similar ellipses (in 2D) or ellipsoids (in 3D). When the thickness of the shell becomes negligible, it is called a thin homoeoid. The name homoeoid was coined by Lord Kelvin and Peter Tait.

Mathematical definition
If the outer shell is given by



\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 $$ with semiaxes $$a,b,c$$ the inner shell is given for $$0 \leq m \leq 1 $$ by



\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=m^2 $$.

The thin homoeoid is then given by the limit $$ m \to 1 $$

Physical meaning
A homoeoid can be used as a construction element of a matter or charge distribution. The gravitational or electromagnetic potential of a homoeoid homogeneously filled with matter or charge is constant inside the shell. This means that a test mass or charge will not feel any force inside the shell.