Motivated by recent results of Tao–Ziegler Discrete Anal. 2016 and Greenfeld–Tao (2022 preprint) on concatenating affine-linear functions along subgroups of an abelian group, we show three results on recovering affine linearity of functions f : V → W from their restrictions to affine lines, where V ,  W are F -vector spaces and dim V ⩾ 2 . First, if dim V < | F | and f : V → F is affine-linear when restricted to affine lines parallel to a basis and to certain “generic” lines through 0, then f is affine-linear on V . (This extends to all modules M over unital commutative rings R with large enough characteristic.) Second, we explain how a classical result attributed to von Staudt (1850 s) extends beyond bijections: If f : V → W preserves affine lines ℓ , and if f ( v ) ∉ f ( ℓ ) whenever v ∉ ℓ , then this also suffices to recover affine linearity on V , but up to a field automorphism. In particular, if F is a prime field Z / p Z ( p > 2 ) or Q , or a completion Q p or R , then f is affine-linear on V . We then quantitatively refine our first result above, via a weak multiplicative variant of the additive B h -sets initially explored by Singer Trans. Amer. Math. Soc. 1938, Erdös–Turán J. London Math. Soc. 1941, and Bose–Chowla Comment. Math. Helv. 1962. Weak multiplicative B h -sets occur inside all rings with large enough characteristic, and in all infinite or large enough finite integral domains/fields. We show that if R is among any of these classes of rings, and M = R n for some n ⩾ 3 , then one requires affine linearity on at least n ⌈ n / 2 ⌉ -many generic lines to deduce the global affine linearity of f on R n . Moreover, this bound is sharp.