Is there a rotation representation that can also represent "turns", instead of collapsing coincident rotations into the same representation?


In 2D, a simple angle satisfies this, as it can have additional multiples of $2\pi$. For example, rotating by a turn and a half would be $3\pi$.


Is there something similar for 3D rotations? Does the concept even make sense there? Quaternions don't work for this since they only have two representations of any given rotation. Rotation vectors ($\theta\hat{e}$) seem to work, though they are very hard to work with.


EDIT: My objective with this is to extend quaternion spherical interpolation to rotations of more than 180° in terms of beginning and end "orientation with turns" objects, so you could, for example, interpolate over an entire revolution using the same machinery as you would with normal small rotation interpolation.



3D Rotation Representation for Multiple Turns
Since you haven't described what you plan to do with this representation, I'm not sure what method would work well.


One of the problems with representing "turns" in more than two dimensions is that you don't have much in the way of discrete invariants. This is because the fundamental group of $SO(n)$ only has two elements when $n \gt 2$. Indeed, a double rotation in any direction can be continuously deformed to the identity rotation. This means that two turns are indistinguishable from zero turns, if you consider paths that can be deformed to each other to be equivalent.


If you want to keep track of how your rotation came about, here are two suggestions:


1. You can use based paths in the group of rotations, i.e., continuous maps from a real interval into rotation matrices.
2. If your paths are smooth, you can take the derivative, and use paths in velocity space, i.e., the Lie algebra of the rotation group.


Both methods give you points in an infinite dimensional path space.


