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.

About the Author

Alexander Bobenko is a professor at the Technische Universit�t Berlin. He is an author with Yuri Suris of the book „Discrete Differential Geometry“, and editor of several books in geometry and mathematical physics. He is the Coordinator of the DFG Collaboration Research Center „Discretization in Geometry and Dynamics“.

Carl Lutz is a doctoral student at Technische Universit�t Berlin. He wrote his master thesis under the supervision of Alexander Bobenko on the topic “Laguerre Geometry in Space Forms”.

Helmut Pottmann is a professor at King Abdullah University of Science and Technology in Saudi Arabia and at Technische Universit�t Wien. He has co-authored two books (“Computational Line Geometry” and “Architectural Geometry”) and has been founding director of the Visual Computing Center at KAUST and the Center for Geometry and Computational Design at TU Wien.

Jan Techter is a postdoc at Technische Universit�t Berlin. He wrote his doctoral thesis under the supervision of Alexander Bobenko on the topic “Discrete Confocal Quadrics and Checkerboard Incircular Nets”.

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