The classical Laguerre plane is the geometry of oriented lines and oriented circles in the Euclidean plane with oriented contact as incidence relation. The same construction can be applied to the classical elliptic and hyperbolic plane, yielding two-dimensional elliptic and hyperbolic Laguerre geometry. Each of these geometries admits a quadric model, and these quadrics can all be naturally viewed as subquadrics of the Lie quadric. These constructions can all be generalized to Euclidean and non-Euclidean spaces of higher dimensions. The authors give a comprehensive modern presentation of these constructions and connections.
In the second part of the book, the authors study the more modern concept of circular nets and Lie checkerboard incircular nets. They use classical configuration theorems to show that incircular nets and Lie checkerboard incircular nets really do exist, and they investigate how these two notions are related. Moreover, they give parametrizations of checkerboard incircular nets using Jacobi elliptic functions.
The book is very geometric in flavour and contains lots of instructive illustrations.