# Homoeoid

Homoeoid in 3D

A homoeoid is a shell 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.

## 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.

## References

• Chandrasekhar, S.: Ellipsoidal Figures of Equilibrium, Yale Univ. Press. London (1969)
• Routh, E. J.: A Treatise on Analytical Statics, Vol II, Cambridge University Press, Cambridge (1882)