# Find rotation axis in 3D space with quaternions

Rotation of $u_1$ around $q_1$(unknown) axis is $u_2$, and rotation of $u_2$ around $q_2$(unknown) axis is $u_3$. What is the value of axis $q$,in case $q$=$q_1$=$q_2$?

Note that possible values of $q_1$ makes a circle,$c_1$, in 3d space,and $q_2$ another circle,$c_2$, in 3d space. Question is to find the intersection($q$,unit vector) of these circles, $c_1$ and $c_2$.

related to question in find quaternion scalar from end points of the rotation

• @jyrki-lahtonen wondering if you know the answer for this problem? – user818117 Apr 15 '17 at 17:41

Axis of rotation of $q_1$ is $c_{1} = u_1 \times u_2$.
Axis of rotation of $q_2$ is $c_{2} = u_2 \times u_3$.
The two circles intersect at $\pm (c_{1} \times c_{2})$ normalized to unit length.
• in your case $c_1$ is a possible vector from the solution set,as well as $c_2$,imagine there are 2 circles in 3D space having the same center,if you pick random vectors from each as $c_1$,$c_2$, and consider a case $c_1$,and $c_2$ are very close to to the intersection point then $c_1$x$c_2$ will be close to perpendicular of intersection point, which also sates the intersectin may not be at the vectors ±($c_1$x$c_2$) – user818117 Apr 15 '17 at 17:39
• yes, if $\pm (c_{1} \times c_{2})$ is nearly zero, then the three vectors are yearly in the same plane. – Tpofofn Apr 15 '17 at 21:29