I'm currently writing the report on my master thesis project, where I use Android sensors to perform inertial navigation in a heavy industrial environment. In my application, I make use of quaternions to represent the rotation of the device, and I feel it necessary to include a short description of what quaternions are and why they are suitable to represent rotations.
When defining a rotation quaternion (with vector notation) representing a rotation of $\theta$ around a vector $u=(u_x, u_y, u_z)$ as:
$$
q=\left(\cos\left(\frac{\theta}{2}\right), u_x\sin\left(\frac{\theta}{2}\right), u_y\sin\left(\frac{\theta}{2}\right), u_z\sin\left(\frac{\theta}{2}\right)\right),
$$
I think it is necessary to explain why there is a $\frac{1}{2}$ tied to the angle everywhere. I found some suggestions in the answer to this question, but I feel the answers given are either too simple:
The actual rotation is defined by the map $x↦qxq^*$. You get a $θ/2$ from $q$ on the left, and another $θ/2$ from $q^∗$ on the right, which adds up to a $θ$.
and
If it were $\cosθ+a\sinθ$ instead of $\cos(θ/2)+a\sin(θ/2)$, then rotation of $\pi$ about any axis would give you the same result.
… or way too in depth (especially the accepted answer). I feel like both of the quotes above are relevant to what I want to convey, but not quite there.
Related to the first quote, one answer to the question linked above suggests that the reason for using $\theta/2$ is to, given a vector $p$ to rotate expressed as a pure imaginary quaternion, keep the resulting rotated vector after a conjugation operation $qpq^*$ in the pure imaginary 3D space (essentially, the multiplication from the left rotates it $\theta/2$, but also moves it out of the imaginary 3D space – then the multiplication from the right moves it back into the imaginary 3D space and rotates it $\theta/2$ again for a full rotation of $\theta$).
There is also this proof on Wikipedia that shows that using $\theta/2$ makes the conjugation operation equivalent to Rodrigues's rotation formula. However, I feel like this proof is too much to include in my report.
Is there a concise (at most a few lines of text) way of conveying why one has to use $\theta/2$ in rotation quaternions?
It could be something along the lines of:
- This is necessary to prevent overlap of positive and negative rotations around the same axis.
- This is because the rotation is, in effect, applied twice during the conjugation operation described in […].
- …
Best Answer
I think the simpler way to explain the half angle is the analogy with rotations in a plane represented by complex numbers.
If we identify a vector $\vec v=(a,b)^T$ with a complex number $z=a+bi$, then a rotation about the origin of $\vec v$ by an angle $\theta$ is given by $R_{\theta}(z)= e^{\theta i } z$ . The idea of Hamilton was to find some generalization of this formula for three-dimensional rotations.
The quaternions can do such a generalization identifying a $3D$-vector with a pure imaginary quaternion $\mathbf{v}$ and using a pure imaginary versor $\mathbf{u}$ to identify the axis of rotation. But when we calculate the product $e^{\theta \mathbf{u}}\mathbf{v}$ we see that the result is not a pure imaginary quaternion , so it's not a vector.
We can find a pure imaginary quaternion only if we perform the multiplication $e^{\theta \mathbf{u}}\mathbf{v}e^{-\theta \mathbf{u}}$ , and this really represents a rotation about the axis $ u$, but the angle of rotation is now $2 \theta$.