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Given a unit quaternion in hamiltonian notiation $q$ which describes some arbitrary rotation of $\alpha$ about axis $\beta$ as

$$ q =\begin{bmatrix} \cos(\alpha/2) \\ \sin(\alpha/2)\beta_x \\ \sin(\alpha/2)\beta_y \\ \sin(\alpha/2)\beta_z \\ \end{bmatrix} $$

how would I calculate the magnitude of rotation (in radians) that quaternion gives me about some arbitrary axis $v$?

As further clarification, what I mean by "magnitude of rotation about an arbitrary axis" would be the same as finding the angle $\alpha^\star$ which minimizes the angle $\tilde{\alpha}$ encoded by the "error" between the two quaternions $$ \tilde{q} = q^{-1} \otimes q^\star $$ where $$ q^\star =\begin{bmatrix} \cos(\alpha^\star/2) \\ \sin(\alpha^\star/2)v_x \\ \sin(\alpha^\star/2)v_y \\ \sin(\alpha^\star/2)v_z \\ \end{bmatrix} $$ and $$ \tilde q =\begin{bmatrix} \cos(\tilde\alpha /2) \\ \sin(\tilde\alpha /2)\tilde\beta_x \\ \sin(\tilde\alpha /2)\tilde\beta_y \\ \sin(\tilde\alpha /2)\tilde\beta_z \\ \end{bmatrix} $$

and $v=\begin{bmatrix} v_x & v_y & v_z \end{bmatrix}^\top$ is the arbitrary axis of rotation.

I believe that this calculation is similar to calculating euler angles from a quaternion, i.e. "yaw" is typically the amount of rotation about the $z$ axis, and is can be calculated with (link)

$$ \psi = \textrm{atan}\frac{2(q_w q_z + q_x q_y)}{ 1-2(q_y^2+q_z^2)} $$

and you can form a quaternion with rotation only about $z$ through an angle $\psi$ with

$$q_\psi = \begin{bmatrix} \cos\left(\frac{\psi}{2}\right) & 0 & 0 & \sin\left(\frac{\psi}{2}\right) \end{bmatrix}^\top. $$

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    $\begingroup$ How are you defining the rotation about an arbitrary axis? For instance, take a rotation of say $\frac\pi4$ radians about the $x$ axis. What is the magnitude of this rotation about the $z$ axis? $\endgroup$ – Steven Stadnicki Mar 13 '18 at 17:20
  • $\begingroup$ In that case, there would be no rotation about $z$, because it was purely about $x$. I believe that the calculation is somehow related to the dot product between the axis of rotation of the quaternion and the arbitrary axis in question. Whether it's a direct scaling or not is what I'm a little confused about. $\endgroup$ – superjax Mar 13 '18 at 17:23
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    $\begingroup$ The catch is that one usually thinks of a rotation based on its affect on points, and while it's true that there are points where the rotation about the $x$ axis has no '$z$ axis component', there are other points where that's not true; for instance, the transform of the point $(1,0,1)$ by that $\frac\pi4$ rotation would be $(1,\sqrt{2}/2, \sqrt{2}/2)$ (modulo some minus signs) and it's clear that some rotation around the $z$ axis has occurred here. I don't think there's any way of getting what you're after in a frame-independent fashion. $\endgroup$ – Steven Stadnicki Mar 13 '18 at 17:36
  • $\begingroup$ @superjax: your comment above suggests that you not only don't know the answer to your question, you don't actually have a completely clear idea of what your question itself means. That makes it very tough for us. Where did you get the notion of "magnitude of a rotation about some axis other than the axis of rotation"? If you can show us a clear presentation of that notion, we can probably help you compute it. Without that, we're all just shooting in the dark. $\endgroup$ – John Hughes Mar 13 '18 at 17:45
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    $\begingroup$ I think this is more clear, though it also makes the question relatively straightforward, I think - you want to maximize the real part of $\widetilde{q}$ (in your notation), and that's relatively easily expressed as a function of $\alpha$. Can you talk about what you're trying to use this for, by any chance? This smells a little bit like an XY problem... $\endgroup$ – Steven Stadnicki Mar 13 '18 at 18:50
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First of all, I want to note that your formulas are slightly wrong: you should have, e.g., $q=(\cos(\alpha/2), \sin(\alpha/2)\beta_x, \sin(\alpha/2)\beta_y, \sin(\alpha/2)\beta_z)$, where ${\bf \beta}=(\beta_x, \beta_y, \beta_z)$ is the unit vector representing the rotation axis; instead you write $\cos(\beta_x/2)$, etc., which doesn't really make mathematical sense. I'll assume that all of the quaternions are in this corrected form in what follows.

Let's break this down a little bit. For clarity's sake, I'll use $\bf{q}$ for the imaginary (i.e. vector) part of $q$, and similarly $\bf{\widetilde{q}}$, etc; $\mathbf{v}$ here is the unit vector that you're trying to find a 'comparable' rotation around. (Note that by this notation, ${\bf q^\star}=\sin(\alpha^\star/2){\bf v}$).

By your own formula, we have $\widetilde{q} = q^{-1}q^\star$, and we're interested in finding a $q^\star$ that maximizes the real part of $\widetilde{q}$. (This is equivalent to minimizing the angle $\tilde{\alpha}$, since $\cos$ is a decreasing function in the domain we're interested in.) But that part is $\widetilde{q}_w=(q^{-1})_wq^\star_w-{\bf q^{-1}}\cdot{\bf q^\star}$ $= \cos(\alpha/2)\cos(\alpha^\star/2) + \sin(\alpha/2)\sin(\alpha^\star/2){\bf \beta}\cdot{\bf v}$. (Sanity check: when ${\bf \beta\cdot v}=1$ — i.e., when they're the same vector — this expression is $\cos((\alpha-\alpha^\star)/2)$ and we maximize it by taking $\alpha^\star=\alpha$, which is what you'd expect.)

Now, in general, to maximize an expression of the form $a\cos(\theta)+b\sin(\theta)$, we can view it as the (2d) dot product of $(a,b)$ with $(\cos(\theta), \sin(\theta))$; seen this way, it's clear that the maximum is achieved when these two vectors are aligned — that is, when $\theta=\arctan(\frac ba)$. Written in this form, you can see that the optimal $\alpha^\star$ is the one with $\tan(\alpha^\star/2)={\bf \beta\cdot v}\tan(\alpha/2)$.

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