  Two triangle meshes are conformally equivalent if for any pair of incident
triangles the absolute values of the corresponding cross-ratios of the four
vertices agree. Such a pair can be considered as preimage and image of a
discrete conformal map. In this article we study discrete conformal maps which
are defined on parts of a triangular lattice $T$ with strictly acute angles.
That is, $T$ is an infinite triangulation of the plane with congruent strictly
acute triangles. A smooth conformal map $f$ can be approximated on a compact
subset by such discrete conformal maps $f^\varepsilon$, defined on a part of
$\varepsilon T$ for $\varepsilon>0$ small enough, see [U. B\"ucking,
Approximation of conformal mappings using conformally equivalent triangular
lattices, in "Advances in Discrete Differential Geometry" (A.I. Bobenko ed.),
Springer (2016), 133--149]. We improve this result and show that the
convergence is in fact in $C^\infty$. Furthermore, we describe how the
cross-ratios of the four vertices for pairs of incident triangles are related
to the Schwarzian derivative of $f$.
