Poincare Extension of Moebius Transformations

Abstract

Given sphere preserving (Möbius) transformations in $n$-dimensional Euclidean space one can use the Poincaré extension to obtain sphere preserving transformations in a half space of $n+1$ dimensions. The Poincaré extension is usually provided either by an explicit formula or by some geometric construction. We investigate its algebraic background and describe all available options. The solution is given in terms of one-parameter subgroups of Möbius transformations acting on triples of quadratic forms. To focus on the concepts, this paper deals with the Möbius transformations of the real line only.

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References in corpus (2)

• Erlangen Program at Large-0: Starting with the group SL(2,R)
• An Extension of Moebius--Lie Geometry with Conformal Ensembles of Cycles and Its Implementation in a GiNaC Library

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