This textbook is a comprehensive and yet accessible introduction to
non-Euclidean Laguerre geometry, for which there exists no previous
systematic presentation in the literature. Moreover, we present new
results by demonstrating all essential features of Laguerre
geometry on the example of checkerboard incircular nets. Classical
(Euclidean) Laguerre geometry studies oriented hyperplanes,
oriented hyperspheres, and their oriented contact in Euclidean
space. We describe how this can be generalized to arbitrary
Cayley-Klein spaces, in particular hyperbolic and elliptic space,
and study the corresponding groups of Laguerre transformations. We
give an introduction to Lie geometry and describe how these
Laguerre geometries can be obtained as subgeometries. As an
application of two-dimensional Lie and Laguerre geometry we study
the properties of checkerboard incircular nets.
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