The Minimum Number of Rotations About Two Axes for Constructing an Arbitrary Fixed Rotation

For any pair of three-dimensional real unit vectors $\widehat{m}$ and $\widehat{n}$ with $\mathrm{\mid }{\widehat{m}}^{\mathrm{T}}\widehat{n}\mathrm{\mid }<1$ and any rotation $U$, let $N_{\widehat{m},\widehat{n}}(U)$ denote the least value of a positive integer $k$ such that $U$ can be decomposed into a product of $k$ rotations about either $\widehat{m}$ or $\widehat{n}$. This work gives the number $N_{\widehat{m},\widehat{n}}(U)$ as a function of $U$. Here a rotation means an element $D$ of the special orthogonal group $\mathrm{S}\mathrm{O}(3)$ or an element of the special unitary group ${\rm SU}(2)$thatcorrespondsto$D$\mathrm{.}Decompositionsof$U$ attaining the minimum number $N_{\widehat{m},\widehat{n}}(U)$ are also given explicitly.