This article provides, over any field, infinitely many algebraic embeddings of the affine spaces A(1) and A(2) into smooth quadrics of dimension two and three, respectively, which are pairwise non-equivalent under automorphisms of the smooth quadric. Our main tools are the study of the birational morphism SL2 -> A(3) and the fibration SL2 -> A(3) -> A(1) obtained by projections, as well as degenerations of variables of polynomial rings, and families of A(1)-fibrations.