A homoeoid or homeoid 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. Closely related is the focaloid, a shell between concentric, confocal ellipses or ellipsoids.
If the outer shell is given by
with semiaxes , the inner shell of a homoeoid is given for by
and a focaloid is defined for by
The thin homoeoid is then given by the limit , and the thin focaloid is the limit .
Thin focaloids and homoeoids can be used as elements of an ellipsoidal matter or charge distribution that generalize the shell theorem for spherical shells. The gravitational or electromagnetic potential of a homoeoid homogeneously filled with matter or charge is constant inside the shell, so there is no force on a test particle inside of it. Meanwhile, two uniform, concentric focaloids with the same mass or charge exert the same potential on a test particle outside of both.