# Mean value of the rotation angle is 126.5°

In the paper "Applications of Quaternions to Computation with Rotations" by Eugene Salamin, 1979 (click here), they get 126.5 degrees as the mean value of the rotation angle of a random rotation (by integrating quaternions over the 3-sphere).

How can I make sense of this result? If rotation angle around a given axis can be 0..360°, should not the mean be 180? or 0 if it can be −180..180°? or 90° if you take absolute values?

• I think the angles they're talking about must be in the range from $0^\circ$ to $180^\circ$. If you rotate a circle $181^\circ$ clockwise, that's the same as $179^\circ$ counterclockwise, and I think the latter is the angle they're talking about. But the rotations above are rotations in $3$-dimensional Euclidean space, so just plus or minus signs do not suffice to express the direction in which the sphere was rotated. Aug 10, 2013 at 16:50
• The key phrase is "Random rotations are to be chosen uniformly distributed over the 3-sphere" Aug 10, 2013 at 16:54