Axis-angle composition of orthogonal spin rotations
= Axis-angle composition of orthogonal spin rotations
{title2=$U=CI-i\mathbf v\cdot\boldsymbol\sigma$}
For $A=\Omega T/2$, $B=\Phi/2$, the ordered product of a $z$ <rotation> after an $x$ <rotation> has $C=\cos A\cos B$ and $\mathbf v=(\cos A\sin B,\sin A\sin B,\sin A\cos B)$. Since $C^2+|\mathbf v|^2=1$, its angle is $2\operatorname{atan2}(|\mathbf v|,C)$ and its axis is $\mathbf v/|\mathbf v|$. At $\mathbf v=0$ the group element is central and its axis is undetermined.