Some alternatives of the Rodrigues axis-angle formula (Q2822188)

In this paper an alternative formula of the classical Rodrigues representation is obtained. In the geometrical interpretation, the rotation is generated by two angles presenting the spherical coordinates on the unit sphere and a positive real parameter, namely \(\mathbf{R}(\alpha, \beta, \gamma)\). The authors generalized it and the rotations can be viewed as normal vectors \(\mathcal{A} \) to some surface in \(\mathbb R^3\) with a fixed length. In Theorem 1 the parametrization and the corresponding composition laws are obtained via Cayley map. Moreover, the relationships between the novel parametrization, the classical Rodrigues representation on the extended \(so(3,\mathbb R)\) vector-parametrization are obtained.