General Great-Circle-Arrangement


Let us place n great circles on the sphere so that the largest distance
from any spherical point to the next great circle shall be as small as possible.


The largest incircle of an arrangement of n great circles is always >= pi/2n.
Equality holds only at the regular mosaik {2,2n}, i.e. when all great circles intersect
at two antipodal points and when they are "uniformly" distributed then.

other formulation:

The radius of the (smallest) circle containing n points in the elliptic plane is always
<= π/2 * ( 1 - 1/n ); equality holds only if the points are equidistant on a line.
(The elliptic plane is given by the set of antipodal point-pairs on the sphere).


  • n great circles separate the sphere at least into n facets
  • the incircle radius of a spherical triangle is given by sin r = S/(2*sin s )
  • the incircle radius of a spherical diangle is given by the angle α of the diangle
  • the diangle's area is 2*α
  • Literature:

  • Johann Linhart: "Eine extremale Verteilung von Grosskreisen", Elemente der Mathematik 29 (1974), S. 57-80
  • Vera Rosta: "An Extremal Arrangement of three Great Circles on the Sphere", Mat. Lapok 24 (1973)