Hypersphere point picking is the selection of points from a uniform distribution on a hypersphere. Marsaglia (1972)
gives a simple method for the unit hypersphere 
in four-dimensional Euclidean space. Draw two
independent pairs 
and 
uniformly from the square ![[-1,1]^2](/images/equations/HyperspherePointPicking/Inline4.svg)
.
Resample either pair if its squared distance from the
origin is at least 1, and resample the second pair if
.
Then the points
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
have a uniform distribution on 
. This extends Marsaglia's (1972) method for sphere
point picking.
For arbitrary dimension, generate independent and identically distributed random variables 
,
,
..., 
with a standard normal distribution. The normalized
vector
![1/(sqrt(x_1^2+x_2^2+...+x_n^2))[x_1; x_2; |; x_n]](/images/equations/HyperspherePointPicking/NumberedEquation1.svg) |
(5)
|
has a uniform distribution on 
(Muller 1959, Marsaglia 1972).
Schnabel and Janke (2022, p. 2) give a generalization based on ordered disk samples. For 
with 
,
draw 
independent pairs 
uniformly from the unit disk,
rejecting zero radii and boundary
points. Sort the whole pairs so that 
, where 
, and set 
. The coordinates
 |
(6)
|
for 
,
..., 
give a uniform distribution on 
. For odd 
, generate a point in 
dimensions and normalize its first 
coordinates.
-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.Schnabel, S. and Janke, W. "A Simple Algorithm
for Uniform Sampling on the Surface of a Hypersphere." 5 Apr 2022.