# Homoeoid

Cut view of a homoeoid in 3D

A homoeoid is a shell (a bounded region) bounded by two concentric, similar ellipses (in 2D) or ellipsoids (in 3D).[1][2] 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.[3]

## Mathematical definition

If the outer shell is given by

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

with semiaxes $\displaystyle{ a,b,c }$ the inner shell is given for $\displaystyle{ 0 \leq m \leq 1 }$ by

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

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

## 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.[4]