quaternionsrotations
I have two quaternions, $q_1$ and $q_2$. I want to rotate the axis of rotation of $q_1$ by $q_2$. Is there any way of doing this directly, without extracting the axis of rotation from $q_1$, rotating it and re-inserting it again?
Please note that I am not trying to perform a combined rotation by quaternion multiplication ($q' = q_2q_1$). Instead I want to rotate the axis of rotation of $q_1$.
The formula is given in this excerpt from a journal paper. It was discovered by the French mathematician Olinde Rodrigues in 1840, which was before the invention of vectors or even quaternions (which were invented before vectors).
The composition of $\alpha\hat{l}$ and $\beta\hat{m}$ (where the second rotation is applied and then the first is applied) is given by $\gamma\hat{n}$, where $\cos\frac{\gamma}{2} = \cos\frac{\alpha}{2}\cos\frac{\beta}{2} - \sin\frac{\alpha}{2}\sin\frac{\beta}{2}\hat{l}\cdot\hat{m}$ and $\sin\frac{\gamma}{2}\hat{n}=\sin\frac{\alpha}{2}\cos\frac{\beta}{2}\hat{l}+\cos\frac{\alpha}{2}\sin\frac{\beta}{2}\hat{m}+\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\hat{l}\times\hat{m}$.
As a sanity check, it's easy to see that when $\hat{l}=\hat{m}$, then it's easy to see that $\gamma=\alpha+\beta$ and $\hat{n}= \hat{l}=\hat{m}$.
In any case, these formulas are proven in detail in this chapter of Simon Altman's book "Rotations, Quaternions, and Double Groups", but it basically boils down to this spherical triangle:
See also this related result proven by William Rowan Hamilton after he invented quaternions.
If you have unit quaternion $q$ then formula $u \to qu\bar q$ define rotation of imaginary quaternion $u$ over axis $Im(q)$ and angle having cosine of half equal to $Re(q)$. The oposite rotation is obtained by conjugate quaternion $\bar q$ giving formula $u \to \bar quq$. If you combine both transformations you obtain $\bar q qu\bar q q$ which is equal to $u$ because $q$ is unit quaternion and multiplication of quaternions is associative.
If you do not obtain the same point after applying rotation $q$ and $\bar q$ then you must have done some mistake. It is hard to know where might be the mistake without seeing your code.
Best Answer
The whole thing can be done with quaternion arithmetic.
The axis of $q_1$ is $a=\frac{q_1-\overline{q_1}}{2}$, and by computing $q_2aq_2^{-1}$ you will be rotating the vector $a$ into a new direction.