Well, that question just about states my issue. We have a body and we rotate about, say, the 3-axis of its body frame. Then, we must do the next rotation about the 1 or 2 axis. Let me say we choose the 1-axis Then we have a choice: continue on to the 2 axis or repeat the 3 axis. One set is called the Euler angles: precession, nutation, spin The other set is called the Tait-Bryan angles: pitch, yaw, roll So now my question is HOW do you KNOW that you have covered all rotations? I mean, I can see by a geometric argument. But how do you KNOW it other than by geometry? Because it seems rather odd, from a distance, that that final choice of either going to the last axis (2) or repeating the first axis (3) SHOULD cover all orientations. (I understand the proof of how the space of orientations is a 3 dimensional sub manifold of R9. I get that... I follow the proof. I just don't get how one can be so sure, without testing it or without geometry, that the two sequences Euler or Tait can define the orientation (and actually be the three variables) '