Remark on Continued Fractions, Moebius Transformations and Cycles

Abstract

In a recent paper A.Beardon and I.Short proposed to use chains of tangent horocycles as an extended tool describing continued fractions. We review the origin of such construction from the Moebius transformations point of view. Related descriptions with chains of orthogonal horocycles emerged as a result. The approach is extended to several dimensions in a way which is compatible to the early proposition of A.Beardon based on Clifford algebras.