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Is here u just a 3-D direction vector or a 4-d vector including translation? I thought if u is translated, then the position of new point will also be changed. Is it right?


Sorry I guess it will be a 6-d vector if it includes translation.


I understand why quaternions are useful. Can you please explain the fatal shortcomings of spherical coordinates in describing rotations? I feel like you would have good insight on this.


Is $\overline{q}xq$ equivalent to a change of basis transformation?


I'm not certain I concretely see how quaternions map 3D rotations. I see on a high level that a unit 4D quaternion gets mapped to a sphere in 3D, and we can describe them with axis and angle like polar coordinates -- but what's u exactly in this instance? Would it be possible to work out an example?


If rotations are the main use for quarternions, what are the other applications of them, in and out of graphics?


I think robotics is another good application of quaternions because robots also have to be designed to work in 3D and have good state estimation. This is probably accomplished with quaternions because I can definitely see using euler angles would have some bad consequences in some situations as we talked about in class.