Geodesic
Uniform distribution of points on hypersurfaces: simulation of random equiprobable rotations
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 1, pp. 29-35 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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: random points on a hypersphere
Mots-clés : uniform distribution of points on hypersurfaces, quaternions, random rotations. 
@article{VUU_2015_25_1_a3,
     author = {N. P. Kopytov and E. A. Mityushov},
     title = {Uniform distribution of points on hypersurfaces: simulation of random equiprobable rotations},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {29--35},
     year = {2015},
     volume = {25},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VUU_2015_25_1_a3/}
}
TY  - JOUR
AU  - N. P. Kopytov
AU  - E. A. Mityushov
TI  - Uniform distribution of points on hypersurfaces: simulation of random equiprobable rotations
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2015
SP  - 29
EP  - 35
VL  - 25
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VUU_2015_25_1_a3/
LA  - en
ID  - VUU_2015_25_1_a3
ER  - 
%0 Journal Article
%A N. P. Kopytov
%A E. A. Mityushov
%T Uniform distribution of points on hypersurfaces: simulation of random equiprobable rotations
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2015
%P 29-35
%V 25
%N 1
%U http://geodesic.mathdoc.fr/item/VUU_2015_25_1_a3/
%G en
%F VUU_2015_25_1_a3
N. P. Kopytov; E. A. Mityushov. Uniform distribution of points on hypersurfaces: simulation of random equiprobable rotations. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 1, pp. 29-35. http://geodesic.mathdoc.fr/item/VUU_2015_25_1_a3/

[1] Marsaglia G., “Choosing a point from the surface of a sphere”, Ann. Math. Stat., 43:2 (1972), 645–646 | DOI | Zbl

[2] Muller M. E., “A note on a method for generating points uniformly on $n$-dimensional spheres”, Communications of the ACM, 2:4 (1959), 19–20 | DOI | Zbl

[3] Weisstein E. W., Sphere point picking, From MathWorld – A Wolfram Web Resource http://mathworld.wolfram.com/SpherePointPicking.html

[4] Weisstein E. W., Hypersphere point picking, From MathWorld – A Wolfram Web Resource http://mathworld.wolfram.com/HyperspherePointPicking.html

[5] Rubinstein R. Y., Kroese D. P., Simulation and the Monte Carlo methods, Wiley-Interscience, New York, 2007, 345 pp. | MR

[6] Melfi G., Schoier G., “Simulation of random distributions on surfaces”, Atti della XLII Riunione Scientifica, Societa Italiana di Statistica (SIS), Bari, 2004, 173–176

[7] Petrillo S., Simulation de points aleatoires independants et non-independants sur surfaces non planes, Diplome postgrade en statistique. Travial de diplome, Universite de Neuchatel, 2005 http://gibonet.ch/pub/travail.pdf

[8] Gel'fand I. M., Shapiro Z. Ya., “Representations of the group of rotations in three-dimensional space and their applications”, Uspekhi Mat. Nauk, 7:1(47) (1952), 3–117 (in Russian) | MR | Zbl

[9] Dubrovin B. A., Novikov S. P., Fomenko A. T., Modern geometry: methods and applications, URSS, Moscow, 2013 | MR

[10] 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

[11] 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

[12] 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, 55–66 (in Russian)

[13] 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), 2263–2264 (in Russian)

[14] 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, 618–622 (in Russian)

[15] Volkov S. D., Klinskikh N. A., “On the distribution of elastic constants in the quasi-isotropic polycrystals”, Dokl. Akad. Nauk SSSR, 146:3 (1962), 565–568 (in Russian) | Zbl

[16] Miles R. E., “On random rotations in $\mathbb R^3$”, Biometrika, 52:3–4 (1965), 636–639 | MR | Zbl

[17] Borisov A. V., Mamaev I. S., Rigid body dynamics, Regular and Chaotic Dynamics, Izhevsk, 2001, 384 pp. | MR

[18] Golubev Yu. F., Quaternion algebra in rigid body kinematics, Preprint no. 39, Keldysh Institute of Applied Mathematics, Moscow, 2013, 23 pp. (in Russian) http://library.keldysh.ru/preprint.asp?id=2013-39

[19] Roberts P. H., Winch D. E., “On random rotations”, Adv. Appl. Prob., 16 (1984), 638–655 | DOI | MR | Zbl

[20] Borovkov M. V., Savelova T. I., Normal distributions on $SO(3)$, Moscow Engineering Physics Institute, Moscow, 2002