  In this paper, we show that the analogue of Thurston's asymmetric metric on
the Teichm{\"u}ller space of flat structures on the torus is weak Finsler and
we give a geometric description of its unit sphere at each point in the tangent
space to Teichm{\"u}ller space. We then introduce a family of weak Finsler
metrics which interpolate between Thurston's asymmetric metric and the
Teichm{\"u}ller metric of the torus (which coincides with the the hyperbolic
metric). We describe the infinitesimal unit spheres of the metrics in this
family.The final version of this paper will appear in Annales Academi\ae \
Scientiarum Fennic\ae \ Mathematica.
