Editing N-sphere (section)

=== Uniformly at random on the {{math|(''n'' − 1)}}-sphere ===
[[File:2sphere-uniform.png|thumbnail|A set of points drawn from a uniform distribution on the surface of a unit {{math|2}}-sphere, generated using Marsaglia's algorithm.]]

To generate uniformly distributed random points on the unit {{tmath|(n-1)}}-sphere (that is, the surface of the unit {{tmath|n}}-ball), {{harvtxt|Marsaglia|1972}} gives the following algorithm.

Generate an {{tmath|n}}-dimensional vector of [[normal distribution|normal deviates]] (it suffices to use {{tmath|N(0, 1)}}, although in fact the choice of the variance is arbitrary), {{tmath|\mathbf x {{=}} (x_1, x_2, \ldots, x_n)}}. Now calculate the "radius" of this point:

:<math>r=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}.</math>

The vector {{tmath|\tfrac 1r \mathbf x}} is uniformly distributed over the surface of the unit {{tmath|n}}-ball.

An alternative given by Marsaglia is to uniformly randomly select a point {{tmath|\mathbf x {{=}} (x_1, x_2, \ldots, x_n)}} in the unit [[hypercube|{{mvar|n}}-cube]] by sampling each {{tmath|x_i}} independently from the [[continuous uniform distribution|uniform distribution]] over {{tmath|(-1, 1)}}, computing {{tmath|r}} as above, and rejecting the point and resampling if {{tmath|r \geq 1}} (i.e., if the point is not in the {{tmath|n}}-ball), and when a point in the ball is obtained scaling it up to the spherical surface by the factor {{tmath|\tfrac 1r }}; then again {{tmath|\tfrac 1r \mathbf x}} is uniformly distributed over the surface of the unit {{tmath|n}}-ball. This method becomes very inefficient for higher dimensions, as a vanishingly small fraction of the [[unit cube]] is contained in the sphere. In ten dimensions, less than 2% of the cube is filled by the sphere, so that typically more than 50 attempts will be needed. In seventy dimensions, less than <math>10^{-24}</math> of the cube is filled, meaning typically a trillion quadrillion trials will be needed, far more than a computer could ever carry out.