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While using unit quaternions to represent rotations, does a quaternion (and its inverse) represent uniquely a certain rotation? Do quaternions suffer from aliasing problems like euler angles? (E.g. 0 degrees = 360*k degrees)

I know the same rotation can be achieved from a unit quaternion and its inverse (same as the conjugate in this case), but I was wondering whether other unit quaterions that can represent the same rotation exist.

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Unit quaternions represent rotations uniquely up to a sign; $q$ and $-q$ represent the same rotation. You can see this in the half-angle formulas (e.g. here); if you add $2\pi$ to the angle (thus not changing the rotation), the resulting unit quaternion is inverted.

There are two different "inverses" at play here. $-q$ is the additive inverse of $q$. This is not the conjugate. For unit quaternions, the conjugate is the multiplicative inverse.

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