Abstract

In the first paragraph of this paper we start with a geometry consisting of points, circles, and an equivalencerelation on the set of points. There is just one circle through any three pairwise non-parallel points and any four non-concyclic points generate a Laguerre-plane or an inversive plane. If dimension d>2, we show that we must have the geometry induced on a cone of a projective space by plane-cuts. In the following part this result is used to give a characterization of the geometry of paraboloids in affine spaces.

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