Homoeoid

Homoeoid in 3D

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.

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)

See also

• Focaloid