Harmonic Shears and Numerical Conformal Mappings

Abstract

In this article we introduce a numerical algorithm for finding harmonic mappings by using the {\it shear construction} introduced by Clunie and Sheil-Small in 1984. The MATLAB implementation of the algorithm is based on the numerical conformal mapping package, the Schwarz-Christoffel toolbox, by T.~Driscoll. Several numerical examples are given. In addition, we discuss briefly the minimal surfaces associated with harmonic mappings and give a numerical example of minimal surfaces.