# Numerical Conformal Mappings for Regions Bounded by Smooth Curves

Abstract: In many applications, conformal mappings are used to transform twodimensional regions into simpler ones. One such region for which conformal mappings are needed is a channel bounded by continuously diﬀerentiable curves. In the applications that have motivated this work, it is important that the region an approximate conformal mapping produces, has this property, but also that the direction of the curve can be controlled, especially in the ends of the channel. This thesis treats three diﬀerent methods for numerically constructing conformal mappings between the upper half–plane or unit circle and a region bounded by a continuously diﬀerentiable curve, where the direction of the curve in a number of control points is controlled, exact or approximately. The ﬁrst method is built on an idea by Peter Henrici, where a modiﬁed Schwarz–Christoﬀel mapping maps the upper half–plane conformally on a polygon with rounded corners. His idea is used in an algorithm by which mappings for arbitrary regions, bounded by smooth curves are constructed. The second method uses the fact that a Schwarz–Christoﬀel mapping from the upper half–plane or unit circle to a polygon maps a region Q inside the half–plane or circle, for example a circle with radius less than 1 or a sector in the half–plane, on a region Ω inside the polygon bounded by a smooth curve. Given such a region Ω, we develop methods to ﬁnd a suitable outer polygon and corresponding Schwarz–Christoﬀel mapping that gives a mapping from Q to Ω. Both these methods use the concept of tangent polygons to numerically determine the coeﬃcients in the mappings. Finally, we use one of Don Marshall’s zipper algorithms to construct conformal mappings from the upper half–plane to channels bounded by arbitrary smooth curves, with the additional property that they are parallel straight lines when approaching inﬁnity.

This dissertation MIGHT be available in PDF-format. **Check this page to see if it is available for download.**