  We give a parametrization to the asymptotic Teichmuller space of the open
unit disk through equivalent classes of shear functions induced by
quasisymmetric homeomorphisms on the Farey tesselation of the unit disk. Then
using the parametrization, we define a new metric on the asymptotic Teichmuller
space. Two other related metrics are also introduced on the asymptotic
Teichmuller space by using cross-ratio distortions or quadrilateral dilatations
under the boundary maps on degenerating sequences of quadruples or
quadrilaterals. We show that the topologies induced by the three metrics are
equivalent to the one induced by the Teichmuller metric on the asymptotic
Teichmuller space. Before proving our main results, we revisit and rectify a
mistake in the proof in the reference [21] on the characterization of
quasisymmetric homeomorphisms in terms of shear functions.
