2

I would like to know how to formally define a finite (but large) set of points uniformly distributed on a $d$-dimensional unit sphere $S^{d-1}$.

For $d=2$, the answer is trivial - we can arrange the points as a regular polygon on a circle. However, the answer for $d=3$ becomes less obvious.

My goal for ''uniform'' is that for some regular subset of the sphere, it contains points approximately proportional to the area of the subset.

Any thoughts or comments are highly appreciated!

andy
  • 318
  • 1
  • 5
  • 4
    You can pick a random set of points which will be uniform in this sense with high probability (explicitly you can do this by generating samples from a multivariate Gaussian, then normalizing them). For the question of how to deterministically specify such a set of points there has been work done on the question of how to find the "most uniform" set of points on a sphere, discussed e.g. here: https://arxiv.org/abs/2407.01503v1 – Qiaochu Yuan Aug 05 '24 at 01:34
  • 4
    If you just need such points, use the Mathematica call RandomPoint[Sphere[n],k] to get $k$ points uniformly distributed on an $n$-dimensional hypersphere, e.g., RandomPoint[Sphere[6],1000]. – David G. Stork Aug 05 '24 at 01:51
  • 5
    This question is similar to: Algorithm to generate an uniform distribution of points in the volume of an hypersphere/on the surface of an hypersphere.. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. – Semiclassical Aug 05 '24 at 03:00
  • For $d=3$ you could use the vertices of a geodesic sphere. – Karl Aug 05 '24 at 03:32
  • 1
    To fully appreciate the difficulties of higher dimensional geometry, the nuances of placing points on spheres as far apart and as uniformly as possible in particular, you may want to begin by studying Conway & Sloane. Close to 600 pages of condensed math. Warning, an applicable PhD in algebra/combinatorics/geometry does not guarantee the ability to absorb more than, say, a quarter of the material. – Jyrki Lahtonen Aug 05 '24 at 04:13
  • 1
    For $d=3$, the Fibonacci sphere algorithm gives a pretty good distribution. – Mike Earnest Aug 05 '24 at 13:14
  • This question was closed as a duplicate, but I repoened because it was not a duplicate. This question is about finding finite sets of points that are "uniformly" distributed on $S_n$, but the proposed dupe was about generating the continuous uniform distribution on $S_n$. – Mike Earnest Aug 14 '24 at 18:56

0 Answers0