title: Homoeoid
text: # Homoeoid
  
  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
  
      ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1$
  
  with semiaxes $a,b,c$ the inner shell is given for $0\leq m\leq 1$ by
  
      ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=m^{2}$.
  
  The thin homoeoid is then given by the limit $m\to 1$
  
  ## 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.
  
  