How do you define a uniform distribution? I've seen a little bit about the different definitions of potential energy, integration, maximizing the minimum distance between points. But do they really get to the heart of the problem? So what I want to say here is, can the problem be equated to how do you make a combination of n points look more like a ball? I know the phrase "more like a ball" doesn't sound very mathematical, but I think that's the essence of the problem. So can we find an algorithm or a definition that describes how similar a geometry is to a sphere? I love talking about math so feel free to speak your mind!
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1If you're asking about picking uniformly distributed points on the surface of a sphere, see here and here – ydd Oct 14 '24 at 15:59
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Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Oct 14 '24 at 15:59
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1I’m voting to close this question because stackexchange is a site for questions and answers, not for discussions, As for your curiousity - if you sample from a uniform distribution on the sphere you will find points that fill each region in proportion to its area. – Ethan Bolker Oct 14 '24 at 15:59
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3Your question is unclear: do you just want points on a sphere? And what is absolutely uniformly ?? – Oct 14 '24 at 16:00
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1Yeah, words mean very specific things in math, and I think the word "absolutely" in your title doesn't actually mean anything, and only confuses the rest of us, who do expect it to mean something. – JonathanZ Oct 14 '24 at 16:13
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1The answer depends on how you phrase the question. This MathOverflow answer describes two versions: The Thompson problem and the Tammes problem. – Dennis Oct 14 '24 at 16:27
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1Maybe this is what you are looking for. – Raskolnikov Oct 14 '24 at 16:35
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Perhaps this is not the sort of thing the questioner wants, but it might at least give some perspective:
For context, any point on the sphere can be said to be randomly chosen with uniform distribution. With two points, chosen with uniform distribution, being very close is less likely, etc.
A different framing, that can be made both rigorous and sort-of algorithmic (if desired), is asking about sequences of finite point sets $F_n$ on the sphere, so that, for any continuous function $f$ on the sphere, $$ \lim_{n\to \infty} {1\over \# F_n}\sum_{x\in F_n} f(x) \;\;=\;\; \int f $$ where the latter integral is over the sphere, with rotation-invariant measure. (Then Weyl's criterion proves that, in effect, this property can be certified by just checking that conclusion for $f$ among spherical harmonics.)
For example, using some basic modular form stuff, it is a classical result that the sequence of projections-to-the-sphere of lattice points in $\mathbb R^{8n}$ at distance $\sqrt{\ell}$ for integer $\ell$ form such a sequence.
paul garrett
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I do not understand what 'absolutely uniform on $S_{n-1}$' means.
To generate one point $\theta$ uniformly on the sphere $S_{n-1}\subset R^n$ generate $Z_1,\ldots, Z_n$ iid rv with distribution $N(0,1).$ Then $$\theta=(Z_1,\ldots, Z_n)\times\frac{1}{\sqrt{Z_1^2+\cdots+ Z_n^2}}$$ fits. For generating $N$ points, repeat $N$ times the operation. If $N$ is large, the law of large numbers garantees that the number of points $\theta_i$ with $i=1,\ldots,N$ falling into a small disk of $S_{n-1}$ is asymptotically proportional to the area of the disk.
Letac Gérard
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1This answer seems badly aligned with the level (and intent) of the question. – John Hughes Oct 14 '24 at 16:13