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Homoeoid

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Geometric shell bounded by two concentric, similar ellipses or ellipsoids
Cut view of a 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. The name homoeoid was coined by Lord Kelvin and Peter Tait.

Mathematical definition

If the outer shell is given by

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

with semiaxes a , b , c {\displaystyle a,b,c} the inner shell is given for 0 ≤ m ≤ 1 {\displaystyle 0\leq m\leq 1} by

x 2 a 2 + y 2 b 2 + z 2 c 2 = m 2 {\displaystyle {\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 → 1 {\displaystyle 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.

See also

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. 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

  • Media related to Homoeoid at Wikimedia Commons


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