Optimizing Arrangements of Points on the Sphere
Since my Master Thesis
Gleichmässige Verteilung von Punkten auf der Einheitskugel
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 )