in terms of the special orthogonal group. A splitting determines a subgroup This is the subgroup that leaves each of the two factors fixed. Choosing a set of coset representatives for the quotient is the same as choosing representative angles for this step of the polyspherical coordinate decomposition. In polyspherical coordinates, the volume measure on and the area measure on are products. There is one factor for each angle, and the volume measure on also has a factor for the radial coordinate. The area measure has the form: where the factors are determined by the tree. Similarly, the volume measure is Suppose we have a node of the tree that corresponds to the decomposition and that has angular coordinate . The corresponding factor depends on the values of and . When the area measure is normalized so that the area of the sphere is 1, these factors are as follows. If , then If and , and if denotes the beta function, then If and , then Finally, if both and are greater than one, then Stereographic projection Just as a two-dimensional sphere embedded in three dimensions can be mapped onto a two-dimensional plane by a stereographic projection, an -sphere can be mapped onto an -dimensional hyperplane by the -dimensional version of the stereographic projection. For example, the point on a two-dimensional sphere of radius 1 maps to the point on the -plane. In other words, Likewise, the stereographic projection of an -sphere of radius 1 will map to the -dimensional hyperplane perpendicular to the -axis as Generating random points Uniformly at random on the -sphere To generate uniformly distributed random points on the unit -sphere (that is, the surface of the unit -ball), gives the following algorithm. Generate an -dimensional vector of normal deviates (it suffices to use , although in fact the choice of the variance is arbitrary), . Now calculate the "radius" of this point: The vector is uniformly distributed over the surface of the unit -ball. An alternative given by Marsaglia is to uniformly randomly select a point in the unit -cube by sampling each independently from the uniform distribution over , computing as above, and rejecting the point and resampling if (i.e., if the point is not in the -ball), and when a point in the ball is obtained scaling it up to the spherical surface by the factor ; then again is uniformly distributed