General Great-Circle-Arrangement


Question:

 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.

Conjecture:

 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).

Remarks:

  • 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)