The Ahlfors Iteration for Confromal Mapping

Abstract

The Riemann Mapping Theorem states that for any proper, simply connected planar domain there exists a conformal mapping from the disk onto the domain. But can this map be explicitly described? For general domains, there is no obvious answer. However, if the domain is the interior of a simple polygon, a convenient formula for the Riemann map was discovered independently by Schwarz and Christoffel. In this dissertation, we present a local quadratically convergent algorithm, the Ahlfors Iteration, based on the theory of quasiconformal maps in the plane, to compute the Schwarz-Christoffel mapping. This algorithm will also apply to a larger collection of simply connected Riemann surfaces. The Ahlfors Iteration improves upon current algorithms that compute the Schwarz-Christoffel map, in that, it is proven to converge, has a simple iterative form, and is easy to implement.