I've been reading up on affine geometry. An affine space (correct me if I'm wrong) is a set of "points" along with a set of translations on those points such that for any two points $P, Q$ there exists a unique translation $T$ such that $T(P)=Q$.
This has me wondering if a different notion than translation can be used as the "fundamental transformation" on the set of points. For instance, would it be possible to consider a space which is a set of points along with the set of rotations on those points? Or with a set of reflections? If I'm just thinking of the Euclidean plane (which includes both the notion of translation and rotation so it should be both an affine space and the hypothetical space whose fundamental transformation is rotation) then there should always be a (non-unique) rotation $R$ such that $R(P) = Q$ for any two points $P,Q$. And if you specify the "center of the circle along which you rotate" (again just thinking about Euclidean space at the moment) then you should have a unique rotation (up to traveling one way or the other along the circle) that maps a point $P$ to a point $Q$.
Is there any literature on this? Is there some giant flaw I'm not considering?