Numerical Conformal Mappings for Regions Bounded by Smooth Curves

Sammanfattning: 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 di?erentiable 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 di?erent methods for numerically constructing conformal mappings between the upper half–plane or unit circle and a region bounded by a continuously di?erentiable curve, where the direction of the curve in a number of control points is controlled, exact or approximately.The ?rst method is built on an idea by Peter Henrici, where a modi?ed Schwarz–Christo?el 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–Christo?el 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 ? inside the polygon bounded by a smooth curve. Given such a region ?, we develop methods to ?nd a suitable outer polygon and corresponding Schwarz–Christo?el mapping that gives a mapping from Q to ?.Both these methods use the concept of tangent polygons to numerically determine the coe?cients 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 in?nity.