Uniform random point on the n-dimensional unit sphere

Returns a single point drawn uniformly from the surface of the \(n\)-dimensional unit sphere via the Box-Muller / Marsaglia method: draw \(n\) independent standard Gaussians and normalize the resulting vector to unit Euclidean norm.

Usage

generate_point_on_sphere(n)

Arguments

n

Integer. Dimension of the sphere.

Value

Numeric vector of length n representing a point on the unit sphere.

Details

This function uses the Box-Muller transform: if \(Z_1, Z_2, \ldots, Z_n\) are standard normal random variables and \(\hat{e}_1, \hat{e}_2, \ldots, \hat{e}_n\) are canonical basis vectors, then the uniformly distributed vector \(\hat{s}\) on the sphere surface is: $$\hat{s} = \frac{Z_1\hat{e}_1 + Z_2\hat{e}_2 + \cdots + Z_n\hat{e}_n} {\sqrt{Z_1^2 + Z_2^2 + \cdots + Z_n^2}}$$

References

Box, G. E. P., & Muller, M. E. (1958). A note on the generation of random normal deviates. Annals of Mathematical Statistics, 29(2), 610-611. doi:10.1214/aoms/1177706645

Marsaglia, G. (1972). Choosing a point from the surface of a sphere. Annals of Mathematical Statistics, 43(2), 645-646. doi:10.1214/aoms/1177692644

See also

generate_cone_sample, generate_cone_samples for cone-restricted sampling that builds on this primitive; rotate_from_canonical for the cheap O(n) rotation used to align cone samples with a target axis; verify_cone_uniformity for goodness-of-fit testing of generated samples.

Examples

point <- generate_point_on_sphere(3)
sqrt(sum(point^2))   # equals 1 up to floating-point error
#> [1] 1