This article studies a particular process that approximates solutions of the
Beltrami equation (straightening of ellipse fields, a.k.a. measurable Riemann
mapping theorem) on C. It passes through the introduction of a
sequence of similarity surfaces constructed by gluing polygons, and we explain
the relation between their conformal uniformization and the Schwarz-Christoffel
formula. Numerical aspects, in particular the efficiency of the process, are
not studied, but we draw interesting theoretical consequences. First, we give
an independent proof of the analytic dependence, on the data (the Beltrami
form), of the solution of the Beltrami equation (Ahlfors-Bers theorem). For
this we prove, without using the Ahlfors-Bers theorem, the holomorphic
dependence, with respect to the polygons, of the Christoffel symbol appearing
in the Schwarz-Christoffel formula. Second, these Christoffel symbols define a
sequence of parallel transports on the range, and in the case of a data that is
C2 with compact support, we prove that it converges to the parallel
transport associated to a particular affine connection, which we identify.Comment: 95 pages, 17 figure