Conformal mapping of circular arc polygons

An algorithm is described which computes the conformal mapping from the unit disk onto an arbitrary polygon having circular arcs as sides. This generalizes the Schwarz-Christoffel program of Trefethen (SIAM J. Sci. Stat. Comp., 1 (1980), pp. 82–102). Our algorithm must also determine certain parameters by solving a nonlinear least squares problem. Instead of using Gauss-Jacobi quadrature to evaluate the Schwarz-Christoffel integral, however, an ordinary differential equation solver is applied to a non-singular formulation of the Schwarzian differential equation. The construction of a conformal mapping reduces simple elliptic partial differential equations on an irregular region to similar problems on a disk, for which existing programs can compute solutions very efficiently. Typical examples arise in the modeling of conductivity past an array of conducting cylinders and electrical fields inside a waveguide.