Numerical Computation of the Schwarz–Christoffel Transformation

A program is described which computes Schwarz-Christoffel transformations that map the unit disk conformally onto the interior of a bounded or unbouded polygon in the complex plane. The inverse map is also computed. The computational problem is approached by setting up a nonlinear system of equations whose unknowns are essentially the "accessory parameters" $z_k$. This system is then solved with a packaged subroutine. New features of this work include the evaluation of integrals within the disk rather than along the boundary, making possible the treatment of unbounded polygons; the use of a compound form of Gauss-Jacobi quadrature to evaluate the Schwarz-Christoffel integral, making possible high accuracy at reasonable cost; and the elimination of constraints in the nonlinear system by a simple change of variables. Schwarz-Christoffel transformations may be applied to solve the Laplace and Poisson equations and related problems in two-dimensional domains with irregular or unbounded (but not curved or multiply connected) geometries. Computational examples are presented. The time required to solve the mapping problem is roughly proportional to $N^3$, where N is the number of vertices of the polygon. A typical set of computations to 8-place accuracy with $N \leq 10$ takes 1 to 10 seconds on an IBM 370/168.