On methods for generating uniform random points on the surface of a sphere

Some methods for generating random points uniformly distributed on the surface of ann-sphere have been proposed to simulate spherical processes on computer. A standard method is to normalize random points inside of the sphere, see M. Muller 5]. Improved methods were given by J. M. Cook 1] and G. Marsaglia 4] in three and four dimensions, and computational methods in higher dimensions by J. S. Hicks and R. F. Wheeling 3] and M. Sibuya 6]. In this paper we shall offer direct methods for generating uniform random points on the surface of a unitn-sphere, which can be easily combined with Marsaglia's idea for getting more improved methods. Our method in even dimensions was obtained by M. Sibuya 6], but a differential-geometric view-point will make analyses simpler, even in odd dimensions.