## Distributing points on the sphere. I.(English)Zbl 1087.52009

The authors study four different methods for distributing points on the sphere and numerically analyze their relative merits with respect to certain metrics. In particular, given points $$z_1, \dots, z_N$$ on $$S^2$$, then the metrics the authors consider are the following: $\theta _{s}=\theta _{s}(z_{1},\dots ,z_{N})=\sum_{1\leq j<k\leq N} {1\over | z_{j}-z_{k}| ^s}\quad\text{for } 0<s<\infty,$
$\theta _{\infty }=\min_{1\leq j<k\leq N} | z_{j}-z_{k}| ,\quad \text{and\;}$
$\theta _{0}=\sum \log\left[{1\over | z_{j}-z_{k}| }\right] .$
The four methods studied are $$\varepsilon$$-good sets introduced by P. Sarnak [Some applications of modular forms (Cambridge Tracts in Mathematics, 99. Cambridge: Cambridge University Press) (1990; Zbl 0721.11015)]; lattice points introduced by E. W. Altschuler et al. [Phys. Rev. Lett. 78, No. 14, 2681–2685 (1997)], subdivision of the icosahedron by L. Kobbelt [Proceedings of SIGGRAPH 2000, Computer Graphics Proceedings, Annual Conference Series, Reading, MA: Addison-Wesley, 103–112 (2000)], and polar coordinates subdivision.
The numerical experiments performed show that in most cases the polar coordinates subdivison method is better than the other three methods.
There are some results on the degrees of the vertices of a triangulation of the sphere induced by the $$N$$ points.

### MSC:

 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 31-04 Software, source code, etc. for problems pertaining to potential theory

Zbl 0721.11015
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### References:

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