# Question for geometers: How to describe this problems of this kind?

 P: 234 Wikipedia's article on "affine connection" has exactly this situation as a motivating example. But I don't think it's any less mathematically demanding than the one on holonomy, so that may not be helpful. I thought about it for a while, and I believe I can answer your second question in the affirmative: any rotation of the (assume unit) sphere can be realized by rolling along a closed path in the plane (and I think that $6\pi$ is an upper bound on the length of the path). The way to see it is to ask a related question: (1) which points in the plane can I reach if I require the sphere to end in the same orientation it began with? If the answer is "all of them", then we can reduce the original problem a simpler question: (2) can we achieve any orientation by rolling along an arbitrary path? For (1), just notice that I can roll a distance of $2\pi$ along a straight line in any direction I like, and end up back in the original orientation (I've just done a full rotation about the axis parallel to the plane and perpendicular to travel). I can combine two such motions to reach any point within a radius of $4\pi$, and obviously this means I can reach any other point in the plane by similar means. So now, if I can roll along a non-closed path to achieve the orientation I want, I can then close that path up with an orientation-preserving path, giving me a closed path that ends up with the desired orientation. So all that remains is (2) to show that we can always do that. Any rotation of the sphere has an axis-angle description, i.e. it is specified by rotation by a certain angle around a particular axis. By rolling in a straight line, I can rotate by an arbitrary angle, around any axis which is parallel to the plane. And if I want to rotate around an axis that isn't plane-parallel, it's obvious that there's another straight-line path P I can roll along to make it become so. Then I rotate the desired amount around it, then I roll along the reverse of P (translated to my new position), and I'm done. The upper bound comes from the fact that I have to roll at most $\pi/2$ to bring my rotation axis parallel to the plane (in the case where it's the vertical axis), and that a roll by $\pi$ one way or the other is sufficient for any rotation around that axis. Then I roll at most $\pi/2$ again to put my axis back where it started, and then $4\pi$ to go home. I'd be astonished if there's not a better upper bound that doesn't involve such ugly (albeit easy to think about) paths.