By Valdinoci E.

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The lines of the derived geometry are the lines in Q through p that have been punctured at p. 3 Definitions of Frequently Used Terms 21 point p coincides with the derived geometry at this point. If the point set of Q is equipped with a parallelism, the derived plane of Q at the point p is the derived geometry at p whose line set has been augmented by the nontrivial parallel classes not containing p that have been punctured at all points parallel to p. A map from the point set of one geometry to the point set of a second geometry is collineation-preserving if it maps sets of collinear points to sets of collinear points.

This means that the associated flat projective plane still contains the solid black part of the configuration, but does not contain the grey segment. Since the three grey points are contained in exactly one Euclidean line or circle, Desargues' configuration does not close in the new flat projective plane. Hence this plane is not classical. 3 The Classical Point Mobius Strip Plane The surface S is the vertical cylinder over the unit circle S 1 in the xyplane. The points of the first geometry are the points of the cylinder.

For a proof of (iii) see Buchanan-Hahl-Lowen [1980]. • Topological ovals play an important role in the study of flat rank 3 circle planes. Consider, for example, the derived affine plane of a flat Mobius plane at a point p. Every circle in the Mobius plane not passing through this point induces a topological oval in the derived plane. This means that a flat Mobius plane has a representation in terms of a special flat affine plane plus a set of topological ovals in this affine plane. In the classical case this is the geometry of Euclidean lines and circles in R 2 .