A classification of finite \(\{ 0,1,2\}\)-inversive planes
A circular space is a geometry such that any three points determine a unique circle, every circle contains at least three points, and there are at least two circles. A circular space is called an \(\{0,1,2\}\)-inversive plane if the residue space in every point is a \(\{0,1,2\}\)-semi-affine plane, i.e., a linear space (or incidence space) such that for every non-incident pair of a point \(P\) and a line \(\ell,\) the number of lines through \(P\) not intersecting \(\ell\) is in the set \(\{0,1,2\}.\) The authors give a complete classification of all finite \(\{0,1,2\}\)-inversive planes. Their list contains more than two dozen items most of which only have a small number of points. Indeed, there are only three infinite classes, namely inversive planes, punctured inversive planes, and one such that the residue space is obtained from a projective plane of order~\(n\) by deleting a \(k\)-arc, where \(k(k-1)(k-2)=12n(n+1).\)
scientific article; zbMATH DE number 1369781