Homoeoid and focaloid

A homoeoid or homeoid 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] Closely related is the focaloid, a shell between concentric, confocal ellipses or ellipsoids.^[4]

Cut view of a homoeoid in 3D

Cut view of a focaloid in 3D

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 of a homoeoid is given for ${\displaystyle 0\leq m\leq 1}$  by

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

and a focaloid is defined for ${\displaystyle \lambda \geq 0}$  by

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

The thin homoeoid is then given by the limit ${\displaystyle m\to 1}$ , and the thin focaloid is the limit ${\displaystyle \lambda \to 0}$ .^[3]

Physical properties

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.^[5] Meanwhile, two uniform, concentric focaloids with the same mass or charge exert the same potential on a test particle outside of both.^[4][1]

See also

• Focaloid

References

1. Chandrasekhar, S.: Ellipsoidal Figures of Equilibrium, Yale Univ. Press. London (1969)
2. Routh, E. J.: A Treatise on Analytical Statics, Vol II, Cambridge University Press, Cambridge (1882)
3. Harry Bateman. "Partial differential equations of mathematical physics.", Cambridge, UK: Cambridge University Press, 1932 (1932).
4. Rodrigues, Hilário (11 May 2014). "On determining the kinetic content of ellipsoidal configurations". Monthly Notices of the Royal Astronomical Society. 440 (2): 1519–1526. arXiv:1402.6541. doi:10.1093/mnras/stu353.
5. Michel Chasles, Solution nouvelle du problème de l’attraction d’un ellipsoïde hétérogène sur un point exterieur, Jour. Liouville 5, 465–488 (1840)

External links