Non-Euclidean Laguerre geometry and incircular nets

TL;DR

This paper generalizes classical Laguerre geometry from Euclidean space to Cayley-Klein spaces, exploring the associated transformation groups and applying the theory to analyze properties of incircular nets.

Contribution

It extends Laguerre geometry to hyperbolic and elliptic spaces and connects it with Lie geometry, providing new insights into geometric transformations and incircular nets.

Findings

Generalization of Laguerre geometry to Cayley-Klein spaces

Description of Laguerre transformation groups in these spaces

Analysis of incircular nets within the generalized framework

Abstract

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 incircular nets.