Let \(T\) be a lattice triangulation of the complex plane \(\mathbb C\) with congruent triangles. The sets of vertices and edges of \(T\) are denoted by \(V\) and \(E\), respectively. A discrete conformal \(PL\)-mapping \(g\) is a continuous and orientation-preserving map of a subcomplex \(T_S\) of a triangle lattice \(T\) to \(\mathbb C\) which is locally a homeomorphism in a neighborhood of each inner point and whose restriction to every triangle is a linear map onto the corresponding image triangle. Furthermore, there exists a function \(u:V_S\to\mathbb R\) on the vertices, called associated scale factors, such that, for all edges \(e=[v,w]\in E_S\), one has \[ |g(v)-g(w)|=|v-w|e^{(u(v)+u(w))/2}. \] In the present article, the authors find a sequence of discrete conformal \(PL\)-maps which approximate a given conformal map.
Theorem 1.2: Let \(f:D\to\mathbb C\) be a conformal map. Let \(K\subset D\) be a compact set which is the closure of its simply connected interior \(\text{int}(K)\), \(0\in\text{int}(K)\). Let \(T\) be a triangular lattice with acute angles. For each \(\epsilon>0\), let \(T_K^{\epsilon}\) be a subcomplex of \(\epsilon T\) whose support is contained in \(K\) and is homeomorphic to a closed disk. Assume that 0 is an interior vertex of \(T_K^{\epsilon}\). Let \(e_0=[0,\hat v_o]\in E_K^{\epsilon}\) be one of its incident edges. Then, if \(\epsilon>0\) is small enough (depending on \(K\), \(f\) and \(T\)), there exists a unique discrete conformal \(PL\)-map \(f^{\epsilon}\) on \(T_K^{\epsilon}\) which satisfies the following two conditions: the associated scale factors \(u^{\epsilon}:V_K^{\epsilon}\to\mathbb R\) satisfy \(u^{\epsilon}(v)=\log|f'(v)|\) for all boundary vertices \(v\) of \(V_K^{\epsilon}\); and the discrete conformal \(PL\)-map is normalized according to \(f^{\epsilon}(0)=f(0)\) and \(\arg(f^{\epsilon}(\hat v_0)-f^{\epsilon}(0))=\arg(\hat v_0)+\arg(f'(\hat v_0/2))\;(\text{mod}\;2\pi).\) Furthermore, the following estimates for \(u^{\epsilon}\) and \(f^{\epsilon}\) hold for all vertices \(v\in V_K^{\epsilon}\) and points \(x\) in the support of \(T_K^{\epsilon}\), respectively, with \(C_1\), \(C_2\) and \(C_3\) depending only on \(K\), \(f\) and \(T\):

(i)

The scale factors \(u^{\epsilon}\) approximate \(\log|f'|\) uniformly with error of order \(\epsilon^2\): \(\big|u^{\epsilon}(v)-\log|f'(v)|\big|\leq C_1\epsilon^2\);

(ii)

The discrete conformal \(PL\)-mappings \(f^{\epsilon}\) converge to \(f\) for \(\epsilon\to0\) uniformly with error of order \(\epsilon\): \(\big|f^{\epsilon}(x)-f(x)\big|\leq C_2\epsilon\);

(iii)

The derivatives of \(f^{\epsilon}\) converge to \(f'\) uniformly for \(\epsilon\to0\) with error of order \(\epsilon\): \(\big|\partial_zf^{\epsilon}(x)-f'(x)\big|\leq C_3\epsilon\) and \(\big|\partial_{\overline z}f^{\epsilon}(x)\big|\leq C_3\epsilon\) for all points \(x\) in the interior of a triangle \(\Delta\) of \(T_K^{\epsilon}\).

For the entire collection see [Zbl 1354.53005].