Also check out:
and
and especially
http://dspace.mit.edu/handle/1721.1/8031
where in Ch 3 there is an nice explanation of how you get a 3-ball (solid ball of values for axis-angle vectors) as the locally flat tangent space of the 4-sphere (hollow 4D shell of values for quaternions).
@vperetroukhin’s explanation is exactly right: unit vectors pointing in the viewing direction, scaled by the in-plane rotation. Unfortunately understanding this encoding of rotation is much easier than understanding how they all relate. The conversion functions in pyem.geom cover all the representations and are mutually consistent with the cryoSPARC/Relion conventions if you want to see. I admit I found most of them through trial and error.
Oh, also it’s pretty intuitive that the “viewing direction” unit vector is just the first two Euler angles as spherical coordinates with radius = 1.