{{Short description|Geometric shell bounded by two concentric, similar ellipses or ellipsoids}} thumb|Cut view of a homoeoid in 3D thumb|right|Cut view of a focaloid in 3D A '''homoeoid''' or '''homeoid''' is a shell (a bounded region) bounded by two concentric, similar ellipses (in 2D) or ellipsoids (in 3D).<ref name="chandrasekhar">Chandrasekhar, S.: ''Ellipsoidal Figures of Equilibrium'', Yale Univ. Press. London (1969)</ref><ref>Routh, E. J.: ''A Treatise on Analytical Statics, Vol II'', Cambridge University Press, Cambridge (1882)</ref> 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.<ref name="bateman">Harry Bateman. "Partial differential equations of mathematical physics.", Cambridge, UK: Cambridge University Press, 1932 (1932).</ref> Closely related is the '''focaloid''', a shell between concentric, confocal ellipses or ellipsoids.<ref name="rodrigues">{{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 }}</ref>
==Mathematical definition== If the outer shell is given by :<math>\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1</math> with semiaxes <math>a,b,c</math>, the inner shell of a homoeoid is given for <math>0 \leq m \leq 1</math> by :<math>\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = m^2,\quad {\displaystyle {\frac{x^2}{a^2 + \lambda }}+{\frac{y^2}{b^2 + \lambda }}+{\frac{z^2}{c^2 + \lambda }}=1.}</math> and a focaloid is defined for <math>\lambda \geq 0</math> by :<math>\frac{x^2}{a^2 + \lambda} + \frac{y^2}{b^2 + \lambda} + \frac{z^2}{c^2 + \lambda} = 1.</math>
The thin homoeoid is then given by the limit <math>m \to 1</math>, and the thin focaloid is the limit <math>\lambda \to 0</math>.<ref name="bateman"/>
==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.<ref>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)</ref> Meanwhile, two uniform, concentric focaloids with the same mass or charge exert the same potential on a test particle outside of both.<ref name="rodrigues"/><ref name="chandrasekhar"/>
==References== {{Reflist}} Category:Ellipsoids Category:Physics theorems Category:Potential theory Category:Gravity Category:Electrostatics
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