How to Distribute N Points on the Surface of a Sphere ?
Volume of the Convex Hull
When has the convex hull of N points on the sphere volume as large as possible ?
For evaluation we have to use:
m: | sum (l1)
|
D: |
volume of spherical ABC,
i.e. the volume of the tetrahedron defined by A,B,C and the sphere's center O. |
The best polyhedron of a given combinatorical type fullfills a local criterion:
p = q/| q | with
q = p1 × p2 +
p2 × p3 + ... +
pk × p1
The problem is completely solved for N = 4,..,8,12.
About the combinatorical type of N = 9,10,11,13,14 is known:
- it contains no vertex of degree 3
- it is determined uniquely for N = 9 and N = 14.
- for N = 10,11,13 the number of possible types had been reduced to two.
Literatur:
- [1] Fejes-Tóth,Lászlo:
-
Lagerungen in der Ebene, auf der Kugel und im Raum
Springer, Berlin-Heidelberg, 1956, 1972
- [2] Berman, J.D.; Hanes, K.:
-
Volumes of polyhedra inscribed in the unit sphere in E ³
Math.Ann 188 (1970), 78-84
- [3] Hardin, R.H.; Sloane, N.J.A.; Smith, W.D.
-
www.research.att.com/~njas/maxvolumes/
- [4] Wimmer, Lienhard:
-
Über das maximale Volumen der konvexen Hülle
von Punkten auf der Einheitskugel
Dissertation, Salzburg, 1997
Lienhard Wimmer
2004-01-09