Correspondence between rotation representations

Question

I was wondering if there is a bijection between unit quaternions and other rotation representations such as vector of rotation, Euler angles or rotation matrices.

It seems to me this is not the case but I cannot find a theoretical arguments to prove this point.

Thanks.

Answer 1

No there isn't, since the unit quaternions $1$ and $-1$ both give the identity rotation.

Answer 2

In any method of representation, if you have chosen unique representations in terms of quaternions (or Euler angles, or rotation matrices,) then they correspond in a one-to-one way with rotations, and hence with each other. That is the meaning of "uniquely represented".

Of couse in some of these systems, there is more than one way to write a single rotation. That would break a correspondence, if you allowed multiple representations of the same rotation. So that is why I emphasize choosing unique representations.

I'm pretty sure the English Wiki page on the topic contains more that you would like to know.