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## Archive of Issues

Russia Yekaterinburg
Year
2015
Volume
25
Issue
1
Pages
29-35
 Section Mathematics Title Uniform distribution of points on hypersurfaces: simulation of random equiprobable rotations Author(-s) Kopytov N.P.a, Mityushov E.A.a Affiliations Ural Federal Universitya Abstract The paper describes a universal method for simulation of uniform distributions of points on smooth regular surfaces in Euclidean spaces of various dimensions. The authors give an interpretation of a set of possible values of Rodrigues-Hamilton parameters used to describe a rigid rotation as a set of points of a three-dimensional hypersphere in four-dimensional Euclidean space. The relationship between random equiprobable rotations of a rigid body and a uniform distribution of points on the surface of a three-dimensional hypersphere in four-dimensional Euclidean space is established. Keywords uniform distribution of points on hypersurfaces, random points on a hypersphere, quaternions, random rotations UDC 519.21 MSC 60D05 DOI 10.20537/vm150104 Received 27 December 2014 Language English Citation Kopytov N.P., Mityushov E.A. Uniform distribution of points on hypersurfaces: simulation of random equiprobable rotations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 1, pp. 29-35. References Marsaglia G. Choosing a point from the surface of a sphere, Ann. Math. Stat., 1972, vol. 43, no. 2, pp. 645-646. Muller M.E. A note on a method for generating points uniformly on $n$-dimensional spheres, Communications of the ACM, 1959, vol. 2, issue 4, pp. 19-20. Weisstein E.W. Sphere point picking, From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/SpherePointPicking.html Weisstein E.W. Hypersphere point picking, From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/HyperspherePointPicking.html Rubinstein R.Y., Kroese D.P. Simulation and the Monte Carlo methods, New York: Wiley-Interscience, 2007. Melfi G., Schoier G. Simulation of random distributions on surfaces, Societa Italiana di Statistica (SIS). Atti della XLII Riunione Scientifica, Bari, 2004, pp. 173-176. Petrillo S. Simulation de points aleatoires independants et non-independants sur surfaces non planes, Universite de Neuchatel. Diplome postgrade en statistique. Travial de diplome, 2005. http://gibonet.ch/pub/travail.pdf Gel'fand I.M., Shapiro Z.Ya. Representations of the group of rotations in three-dimensional space and their applications, Uspekhi Mat. Nauk, 1952, vol. 7, no. 1 (47), pp. 3-117 (in Russian). Dubrovin B.A., Novikov S.P., Fomenko A.T. Sovremennaya geometriya: metody i prilozheniya (Modern geometry: methods and applications), Moscow: URSS, 2013. Kopytov N.P., Mityushov E.A. The method for uniform distribution of points on surfaces in multi-dimensional Euclidean space, Preprint. http://www.intellectualarchive.com/?link=item&id=1170 Kopytov N.P., Mityushov E.A. Universal algorithm of uniform distribution of points on arbitrary analitic surfaces in three-dimensional space, Preprint. http://www.intellectualarchive.com/?link=item&id=473 Kopytov N.P., Mityushov E.A. A mathematical model of shells reinforcement made of fibrous composite materials and the problem of points uniform distribution on surfaces, Vestn. Perm. Gos. Tekh. Univ., Ser. Mekh., 2010, no. 4, pp. 55-66 (in Russian). Kopytov N.P., Mityushov E.A. Uniform distribution of points on surfaces to create structures of composite shells with transversely isotropic properties, Vestn. Nizhegorod. Univ., 2011, no. 4 (5), pp. 2263-2264 (in Russian). Kopytov N.P., Mityushov E.A. The universal algorithm of uniform distribution of points on arbitrary analitic surfaces in three-dimensional space, Fundamental Research, 2013, no. 4, part 3, pp. 618-622 (in Russian). Volkov S.D., Klinskikh N.A. On the distribution of elastic constants in the quasi-isotropic polycrystals, Dokl. Akad. Nauk SSSR, 1962, vol. 146, no. 3, pp. 565-568 (in Russian). Miles R.E. On random rotations in $\mathbb{R}$$3$ , Biometrika, 1965, vol. 52, issue 3-4, pp. 636-639. Borisov A.V., Mamaev I.S. Dinamika tverdogo tela (Rigid body dynamics), Izhevsk: Regular and Chaotic Dynamics, 2001. Golubev Yu.F. Quaternion algebra in rigid body kinematics, Keldysh Institute of Applied Mathematics Preprint, Moscow, 2013, no. 39, pp. 1-23 (in Russian). http://library.keldysh.ru/preprint.asp?id=2013-39 Roberts P.H., Winch D.E. On random rotations, Adv. Appl. Prob., 1984, vol. 16, pp. 638-655. Borovkov M.V., Savelova T.I. Normal'nye raspredeleniya na $SO(3)$ (Normal distributions on $SO(3)$), Moscow: Moscow Engineering Physics Institute, 2002. Full text