A classification of finite \(\{ 0,1,2\}\)-inversive planes
type: Item
instance of: scholarly article
title: A classification of finite \(\{ 0,1,2\}\)-inversive planes
zbMATH Open document ID: 0955.51008
DOI: 10.1016/S0012-365X(99)00146-6
author: Nicola Durante
author: Ursula Hübsch
author: V. Napolitano
author: Johannes Ueberberg
published in: Discrete Mathematics
publication date: 2000-02-01T00:00:00Z
review text: 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 \(k(k-1)(k-2)=12n(n+1).\)
reviewed by: Hubert Kiechle
Mathematics Subject Classification ID: 51E25
Mathematics Subject Classification ID: 51B10
zbMATH DE Number: 1369781
zbMATH Keywords: inversive plane
zbMATH Keywords: semi-affine plane
MaRDI profile type: Publication
full work available at URL: s0012-365x(99)00146-6
OpenAlex ID: W2080220411
description: scientific article; zbMATH DE number 1369781