Homoeoid and focaloid
{{Short description|Geometric shell bounded by two concentric, similar ellipses or ellipsoids}}
A homoeoid or homeoid is a shell (a bounded region) bounded by two concentric, similar ellipses (in 2D) or ellipsoids (in 3D).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)
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.Harry Bateman. "Partial differential equations of mathematical physics.", Cambridge, UK: Cambridge University Press, 1932 (1932). Closely related is the focaloid, a shell between concentric, confocal ellipses or ellipsoids.{{cite journal |last1=Rodrigues |first1=Hilário |title=On determining the kinetic content of ellipsoidal configurations |journal=Monthly Notices of the Royal Astronomical Society |date=11 May 2014 |volume=440 |issue=2 |pages=1519–1526 |doi=10.1093/mnras/stu353|doi-access=free |arxiv=1402.6541 }}
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 , the inner shell of a homoeoid is given for by
:
\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=m^2,
and a focaloid is defined for by
:
\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 , and the thin focaloid is the limit .
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.Michel Chasles, [http://sites.mathdoc.fr/JMPA/PDF/JMPA_1840_1_5_A41_0.pdf Solution nouvelle du problème de l’attraction d’un ellipsoïde hétérogène sur un point exterieur], Jour. Liouville 5, 465–488 (1840) Meanwhile, two uniform, concentric focaloids with the same mass or charge exert the same potential on a test particle outside of both.
See also
References
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