Geodesic
Sub-Riemannian distance in the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$
Matematičeskie trudy, Tome 18 (2015) no. 2, pp. 3-21 
V. N. Berestovskii; I. A. Zubareva. Sub-Riemannian distance in the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$. Matematičeskie trudy, Tome 18 (2015) no. 2, pp. 3-21. http://geodesic.mathdoc.fr/item/MT_2015_18_2_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

We calculate distances between arbitrary elements of the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$ for special left-invariant sub-Riemannian metrics $\rho$ and $d$. In computing distances for the second metric, we substantially use the fact that the canonical two-sheeted covering epimorphism $\Omega$ of $\mathrm{SU(2)}$ onto $\mathrm{SO(3)}$ is a submetry and a local isometry in the metrics $\rho$ and $d$. Despite the fact that the proof uses previously known formulas for geodesics starting at the unity, F. Klein's formula for $\Omega$, trigonometric functions, and the conventional differential calculus of functions of one real variable, we focus attention on a careful application of these simple tools in order to avoid the mistakes made in previously published mathematical works in this area.

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