Quaternions can be defined from the axis angle of rotation by
[ q 1 q 2 q 3 q 4 ] = [ e 1 s i n ( α / 2 ) e 2 s i n ( α / 2 ) e 3 s i n ( α / 2 ) c o s ( α / 2 ) ] {\displaystyle \left[{\begin{matrix}q_{1}\\q_{2}\\q_{3}\\q_{4}\end{matrix}}\right]=\left[{\begin{matrix}e_{1}sin(\alpha /2)\\e_{2}sin(\alpha /2)\\e_{3}sin(\alpha /2)\\cos(\alpha /2)\end{matrix}}\right]}
![{\displaystyle \left[{\begin{matrix}q_{1}\\q_{2}\\q_{3}\\q_{4}\end{matrix}}\right]=\left[{\begin{matrix}e_{1}sin(\alpha /2)\\e_{2}sin(\alpha /2)\\e_{3}sin(\alpha /2)\\cos(\alpha /2)\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6251c3fa1b08b444cb313f4c4a61d91f5ae4b689)