Note on planar functions over the reals

The following construction was used in a paper of Kárteszi [7] illustrating the role of Cremona transformations for secondary school students.This is a typical construction in the theory of flat affine planes, see Salzmann [9], Groh [4] and due to Dembowski and Ostrom [3] for the case of finite ground fields. Let $R2$ be the classical euclidean affine plane and $\tilde{f}$ be the graph of a real function $f : R → R$ (R denotes the field of real numbers).Define a new incidence structure $A = A(f)$ on the points of $R2$ in which the new lines are the vertical lines of $R2$ and the translates of $\tilde{f}$.The incidence is the set-theoretical element of relation. (For the definition of incidence structure,affine plane etc. we refer to Dembowski [2])