Poincare Extension of Moebius Transformations

Given sphere preserving (M\"obius) transformations in $n$-dimensional Euclidean space one can use the Poincar\'e extension to obtain sphere preserving transformations in a half space of $n+1$ dimensions. The Poincar\'e 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 M\"obius transformations acting on triples of quadratic forms. To focus on the concepts, this paper deals with the M\"obius transformations of the real line only.