Abstract
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.
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References
Cook, J. M. (1957) Rational formulae for the production of a spherically symmetric probability distribution,Math. Comp.,11, 81–82.
Eisenhart, L. P. (1966).Riemannian Geometry, sixth print, Princeton Univ. Press.
Hicks, J. S. and Wheeling, R. F. (1959) An efficient method for generating uniformly distributed points on the surface of ann-dimensional sphere,Commun. Ass. Comput. Math.,2, 17–19.
Marsaglia, G. (1972). Choosing a point from the surface of a sphere,Ann. Math. Statist.,43, 645–646.
Muller, M. E. (1959). A note on a method for generating points uniformly onN-dimensional spheres,Commun. Ass. Comput. Math. 2, 19–20.
Sibuya, M. (1964). A method for generating uniformly distributed points onN-dimensional spheres,Ann. Inst. Statist. Math.,14, 81–85.
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Tashiro, Y. On methods for generating uniform random points on the surface of a sphere. Ann Inst Stat Math 29, 295–300 (1977). https://doi.org/10.1007/BF02532791
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DOI: https://doi.org/10.1007/BF02532791