Marsaglia (1972) has given a simple method for selecting points with a uniform distribution on the surface of a 4-sphere. This is accomplished by picking two pairs of points
and 
,
rejecting any points for which 
and 
. Then the points
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(1)
|
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(2)
|
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(3)
|
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(4)
|
have a uniform distribution on the surface of the hypersphere. This extends the method of Marsaglia (1972) for sphere point picking, but unfortunately does not generalize to higher dimensions.
An easy way to pick a random point on a hypersphere of arbitrary dimension is to generate 
Gaussian random variables 
, 
, ..., 
. Then the distribution of the vectors
![1/(sqrt(x_1^2+x_2^2+...+x_n^2))[x_1; x_2; |; x_n]](/images/equations/HyperspherePointPicking/NumberedEquation1.svg) |
(5)
|
is uniform over the surface 
(Muller 1959, Marsaglia 1972).
-Dimensional Sphere." Comm. Assoc. Comput. Mach. 2,
13-15, 1959.Marsaglia, G. "Choosing a Point from the Surface of
a Sphere." Ann. Math. Stat. 43, 645-646, 1972.Muller,
M. E. "A Note on a Method for Generating Points Uniformly on 
-Dimensional Spheres." Comm. Assoc. Comput. Mach. 2,
19-20, Apr. 1959.