Geometries on surfaces

"The projective, Mobius, Laguerre, and Minkowski planes over the real numbers are just a few examples of a host of fundamental classical topological geometries on surfaces that satisfy an axiom of joining. This book summarises all known major results and open problems related to these classical geometries and their close (non-classical) relatives." "Topics covered include: classical geometries; methods for constructing non-classical geometries; classifications and characterisations of geometries. This work is related to a host of other fields including interpolation theory, convexity, differential geometry, topology, the theory of Lie groups and many more. The authors detail these connections, some of which are well-known, but many much less so." "Acting both as a referee for experts and as an accessible introduction for beginners, this book will interest anyone wishing to know more about incidence geometries and the way they interact."--Jacket. Geometries for Pedestrians -- Geometries of Points and Lines -- Geometries on Surfaces -- Flat Linear Spaces -- Models of the Classical Flat Projective Plane -- Convexity Theory -- Continuity of Geometric Operations and the Line Space -- Isomorphisms, Automorphism Groups, and Polarities -- Topological Planes and Flat Linear Spaces -- Classification with Respect to the Group Dimension -- Constructions -- Planes with Special Properties -- Other Invariants and Characterizations -- Related Geometries -- Spherical Circle Planes -- Models of the Classical Flat Mobius Plane -- Derived Planes and Topological Properties -- Constructions -- Groups of Automorphisms and Groups of Projectivities -- The Hering Types -- Characterizations of the Classical Plane -- Planes with Special Properties -- Subgeometries and Lie Geometries -- Toroidal Circle Planes -- Models of the Classical Flat Minkowski Plane -- Derived Planes and Topological Properties -- Constructions -- Automorphism Groups and Groups of Projectivities -- The Klein-Kroll Types -- Characterizations of the Classical Plane -- Planes with Special Properties -- Subgeometries and Lie Geometries -- Cylindrical Circle Planes -- Models of the Classical Flat Laguerre Plane -- Derived Planes and Topological Properties -- Constructions -- Automorphism Groups and Groups of Projectivities -- The Kleinewillinghofer Types -- Characterizations of the Classical Plane -- Planes with Special Properties -- Subgeometries and Lie Geometries -- Generalized Quadrangles