Abstract
An incidence structure (P, N, I) consisting of a set P of points, a set N of circles and an incidence relation I is called a locally affine circular space R, if the following holds: For any point P∈P the points and the circles of the internal structure R P =(P P , N P , I P ) with P P =P∖{P}, N P ={k∈N|PIk} and I p =I∩(P P ×N P ) are respectively the points and the lines of an affine space.
The purpose of this paper is at first to find conditions for representing a finite locally affine circular space R as a substrcture of a finite projective space B. In our representation (called S-representation) the points of R correspond to the elements of a spread L in B and the circles to certain lines of B not contained in elements of L. It is then shown that a finite inversive plane M of order q admits an S-representation if and only if q is even. As consequences we get characterizations of miquelian inversive planes of even order.
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Hölz, G. Zur S-Darstellbarkeit endlicher lokal-affiner Kreisräume. Geom Dedicata 9, 477–496 (1980). https://doi.org/10.1007/BF00181563
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DOI: https://doi.org/10.1007/BF00181563