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March 26

Distribution of points on a sphere

I've been using this algorithm for coming up with a uniform distribution of points on an n-sphere (Python notation):

X = [random.gauss(0, 1) for i in range(n)]
t = math.sqrt(sum([i**2 for i in X]))
Y = [X[i] / t for i in range(n)]
return Y

Does the resulting Y distribution have a name, or can a closed form be derived for it? 70.190.182.236 (talk) 20:39, 26 March 2014 (UTC)[reply]

This is the unique rotationally-invariant probability measure on the sphere. I'm not sure it has a name as such, but it can be obtained as a symmetry reduction of the Haar measure on the group of rotations SO(n). The fact that your probability distribution has this property follows from the fact that if

X_{1},\dots ,X_{n}

are iid (standard) Gaussian random variables, then the n-dimensional random variable

X=(X_{1},\dots ,X_{n})

is normally distributed with pdf

{\frac {1}{(2\pi \sigma ^{2})^{n/2}}}\exp(-x^{T}x/2\sigma ^{2})

, which is rotationally invariant. Sławomir Biały (talk) 21:02, 26 March 2014 (UTC)[reply]

Note that this topic was discussed here before (although not specifically what to name the method you describe): See Wikipedia:Reference_desk/Archives/Mathematics/2007_August_29#Distributing_points_on_a_sphere and Wikipedia:Reference_desk/Archives/Mathematics/2010_April_30#Even_distribution_on_a_sphere. StuRat (talk) 21:16, 26 March 2014 (UTC)[reply]

I think those are about optimally distributing multiple points on a sphere, whereas the OP is asking about the probability distribution of a single point chosen "uniformly at random" on a sphere. Sławomir Biały (talk) 21:20, 26 March 2014 (UTC)[reply]

That's a very nice algorithm, thanks very much 70.190.182.236 and Sławomir. Dmcq (talk) 15:28, 27 March 2014 (UTC)[reply]

There might be a slight bias to the output due to limits of machine floating point operations. I'm not really an expert on such things, but I think it would be prudent to drop the cases where the norm t is either very small or very large. Sławomir Biały (talk) 17:08, 27 March 2014 (UTC)[reply]

I formatted the code for easier reading; I hope you don't mind. By the way I'd use

Y = [x/t for x in X];

no need for range there. —Tamfang (talk) 07:27, 28 March 2014 (UTC)[reply]

See also b:Mathematica/Uniform_Spherical_Distribution, which seems as sensible a name as any; it also gives explicit formulas in terms of θ and φ. --Tardis (talk) 08:46, 30 March 2014 (UTC)[reply]

phi := 2 ArcSin[Sqrt[Random[]]] (for colatitude) is weird, with an unnecessary sqrt; I don't understand the voodoo that got there (being unacquainted with Jacobians), and haven't worked out whether it matches the formula that I've always used: arcsin(1-2*random()), if the output of random() is between 0 and 1 – an application of Archimedes' theorem that the area between two planes both perpendicular to the axis of a cylinder is the same on the cylinder as on an inscribed sphere (if each plane cuts the sphere in at least one point). —Tamfang (talk) 04:50, 31 March 2014 (UTC)[reply]