that characteristic of a one-parameter subgroup, i.e. exponential, and that the formulas match for infinitesimal . For an alternative derivation based on this exponential relationship, see exponential map from to . For the inverse mapping, see log map from to . The Hodge dual of the rotation is just which enables the extraction of both the axis of rotation and the sine of the angle of the rotation from the rotation matrix itself, with the usual ambiguity, where . The above simple expression results from the fact that the Hodge duals of and are zero, and . When applying the Rodrigues' formula, however, the usual ambiguity could be removed with an extended form of the formula. See also Axis angle Rotation (mathematics) SO(3) and SO(4) Euler–Rodrigues formula References Leonhard Euler, "Problema algebraicum ob affectiones prorsus singulares memorabile", Commentatio 407 Indicis Enestoemiani, Novi Comm. Acad. Sci. Petropolitanae 15 (1770), 75–106. Olinde Rodrigues, "Des lois géométriques qui régissent les déplacements d'un système solide dans l'espace, et de la variation des coordonnées provenant de ces déplacements considérés indépendants des causes qui peuvent les produire", Journal de Mathématiques Pures et Appliquées 5 (1840), 380–440. online. Friedberg, Richard (2022). "Rodrigues, Olinde: "Des lois géométriques qui régissent les déplacements d'un systéme solide...", translation and commentary". arXiv:2211.07787. Don Koks, (2006) Explorations in Mathematical Physics, Springer Science+Business Media,LLC. . Ch.4, pps 147 et seq. A Roundabout Route to Geometric Algebra External links Johan E. Mebius, Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations., arXiv General Mathematics 2007. For another descriptive example see: http://chrishecker.com/Rigid_Body_Dynamics#Physics_Articles, Chris Hecker, physics section, part 4. "The Third Dimension" – on page 3, section ``Axis and Angle'', http://chrishecker.com/images/b/bb/Gdmphys4.pdf Rotation in three dimensions Euclidean geometry Orientation (geometry) fr:Rotation vectorielle#Cas général