SCHWARZ-CHRISTOFFEL MAPPINGS FOR NONPOLYGONAL REGIONS

An approximate conformal mapping for an arbitrary region Omega bounded by a smooth curve Gamma is constructed using the Schwarz-Christoffel mapping for a polygonal region in which Omega is embedded. An algorithm for finding this so-called outer polygon is presented. The resulting function is a conformal mapping from the upper half-plane or the unit disk to a region R, approximately equal to Omega. R is bounded by a C-infinity curve, and since the mapping function originates from the Schwarz-Christoffel mapping and tangent polygons are used to determine it, important properties of Gamma such as direction, linear asymptotes, and inflexion points are preserved in the boundary of R. The method makes extensive use of existing Schwarz-Christoffel software in both the determination of outer polygons and the calculation of function values. By the use suggested here, the capabilities of such well-written software are extended.