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haojun

Should it be q x ( q^(-1) ) ? I find in some places, the left side of x is the original quaternion, while on the right side is its inverse... eg https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation. I know once I know how to prove it in math, I will know the exact answer. Still trying hard to find and understand the proof on the Internet.....

keenan

Good catch: both $\bar{q}xq$ and $qx\bar{q}$ are rotations, but you have to be careful about the sign in the axis-angle expression. For the former it should be $\cos(\theta/2) - \sin(\theta/2)u$, whereas for the latter it should be $\cos(\theta/2) + \sin(\theta/2)u$. So the sign is wrong in the slide (or equivalently, the conjugation is wrong). Geometrically this just means the axis is flipped, so that the direction of rotation is reversed. So you could also negate the angle.