[rsp1984]

If you take the Matrix logarithm of an SO(3) (3x3 rotation matrix) you get a 3-vector that represents the axis of rotation, scaled by the rotation amount (in radians). This is also a cheap operation using the inverse Rodrigues formula [1].

The 3-vector is not a bijective representation (starts repeating after length == 2*pi) but otherwise is the most elegant of them all, IMO. No need for rotors or quaternions. Plus you can simply use Rodrigues to get a rotation matrix back.

[1] https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula

[mecsred]

Thanks for the link to the Rodrigues form, that's quite interesting. Slightly confused by your comment though, shouldn't the matrix logarithm produce another matrix?

[rsp1984]

You're correct. The logarithm produces what's essentially a cross product matrix, 0 on the diagonal and symmetric off-diagonal. The off-diag elements are the 3-vector I was talking about. Thanks for pointing that out.

[nyrikki]

SO(3) is nonabelian, and isn't simply connected, which is why the surjective homomorphsin to SU(2) is valuable, particularly in 3D graphics.

[itishappy]

Composing axis-angle representations gets real weird real fast. You can convert them into 9 element rotation matrices, but then you lose the benefits of storing them using only 3 elements in the first place.