2-dimensional Minkowski planes and desarguesian derived affine planes (Q756701)

Let \({\mathcal M}\) be a Minkowski plane and let \({\mathcal D}\) denote the set of those points of \({\mathcal M}\) for which the derived affine plane is desarguesian. It is well known that if \({\mathcal M}\) is finite and \({\mathcal D}\) is nonempty, then \({\mathcal M}\) is miquelian. If \({\mathcal M}\) has odd order this follows from Segre's theorem and nonmiquelian Minkowski planes of even order do not exist at all by Heise's theorem. A similar theorem holds for \({\mathcal M}\) locally compact with a 4-dimensional point space. If \({\mathcal D}\) is nonempty, then Buchanan's theorem implies that \({\mathcal M}\) is miquelian. Assume now that \({\mathcal M}\) is locally compact with a 2-dimensional point space. Then the author proves the following theorems: If \({\mathcal D}\) contains one parallel class of points and at least one further point, then \({\mathcal M}\) belongs to a 1-parameter family of Minkowski planes discovered by \textit{E. Hartmann} [Geom. Dedicata 10, 155-159 (1981; Zbl 0454.51004)]. If \({\mathcal D}\) contains one parallel class of points and at least three pairwise nonparallel points, then \({\mathcal M}\) is isomorphic to the classical real Minkowski plane. The proof is achieved by comparing the description of circles of \({\mathcal M}\) in derived affine planes with respect to different points of \({\mathcal D}\).