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 R<sup>2</sup> 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 R<sup>2</sup> in which the new lines are the vertical lines of R<sup>2</sup> 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]).

DOI Code: 10.1285/i15900932v10n1p59

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