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I'm reading a magazine article on 3d orientation and want to know what the mathematical issue is and where to read about it, preferably a theorem title that I can google or a topic and textbook:

It’s possible to prove that no three-scalar parameterization of 3D orientation exists that doesn’t suck, for some suitably mathematically rigorous definition of “suck.” I haven’t done this proof (I think it uses some pretty high-end group theory, which I haven’t learned yet), so I can’t tell you exactly how it works, but I believe the gist of the proof is that no minimal parameterization exists that doesn’t contain singularities. These singularities can take different forms — depending on how you allocate the three degrees of freedom — but according to the math, it’s impossible to get rid of them.

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    $\begingroup$ This is a nice question, but it seems like it's pure math. (It is addressed in Math.SE, under a slightly different context, here.) $\endgroup$
    – knzhou
    May 9, 2020 at 7:23

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