  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.
