Two rotations generate all Euclidean motions and all elliptic motions (Q1060421)

By \(A^{\alpha}\) the author denotes the rotation by the angle \(\alpha\) about the point A of the plane or, according to context, about the diameter through the point A on the sphere. Let A, B be two points of the Euclidean plane. Then any rotation as well as any translation of the Euclidean plane can be presented as a finite chain of \(A^{\alpha_{k+1}}B^{\beta_ k}A^{\alpha_ k}...B^{\beta_ 1}A^{\alpha_ 1}\); there is no bound for k. For any distinct axes A, B of the sphere, every rotation \(C^{\gamma}\) of it can be presented as \(C^{\gamma}=A^{\alpha_{k+1}}B^{\beta_ k}A^{\alpha_ k}...B^{\beta_ 1}A^{\alpha_ 1}\) where k is bounded by a function of \(| AB|\), in particular, \(k=1\) for \(| AB| =\pi /2\).