Marsaglia 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 (x_1, x_2) and (x_3, x_4), rejecting any points for which x_1^2 + x_2^2>=1 and x_3^2 + x_4^2>=1. Then the points x | = | x_1 y | = | x_2 z | = | x_3 sqrt((1 - x_1^2 - x_2^2)/(x_3^2 + x_4^2)) w | = | x_4 sqrt((1 - x_1^2 - x_2^2)/(x_3^2 + x_4^2))

hypersphere | sphere point picking