Optimizing Arrangements of Points on the Sphere


Since my Master Thesis
Gleichmässige Verteilung von Punkten auf der Einheitskugel (pdf,html)

I am interested in the question how to distribute "equally" n points on the unit sphere.
There are many different possibilities what "equally distribution" of n points on the sphere could be.
E.g. one can ask to
  • maximize the largest volume of the polyhedron inscribed the sphere
  • minimize the smalles volume of the polyhedron circumscribed the sphere
  • maximize the radius of n congruent circles packed on the sphere
  • minimize the radius of n congruent circles covering the sphere
  • optimize weighted sums of the points' distances
  • optimize functionals related to stochastic uniformify theory
    •  In my PhD-Thesis
    Das Maximale Volumen der Konvexen Hülle von Punkten auf der Einheitkugel

    I found the polyhedron with maximal volume and 8 points on the sphere and I found good ones 9, 10, 11, 13 and 14 points.

    Meanwhile I developed some applets for numerical research on this subject and placed them on www.sphopt.com
    They take a randomly choosen polyhedron and try to find the best polyhedron for a select "uniformity measure" from the list above.

    Let me add some remarks on the so called covering problem which can sometimes be diguised:
    Answer to a Question of Marek Teichmann


    Lienhard Wimmer ( Impressum )
    2008-01-26