A Modified Schwarz-christoffel Transformation for Elongated

The numerical computation of a conformal map from a disk or a half plane onto an elongated region is frequently difficult, or impossible, because of the so-called crowding phenomenon. This paper shows that this problem can often be avoided by using another elongated region, an infinite strip, as the standard domain. A transformation similar to the Schwarz-Christoffel formula maps this strip onto an arbitrary polygonal channel, and a slightly modified transformation maps an elongated rectangle onto an arbitrary closed polygon. By using robust and efficient software for numerical integration and solution of the parameter problem, high-accuracy maps of distorted regions with aspect ratios as high as thousands to one are constructed. The modified mapping method has natural applications in fluid mechanics and electrical engineering.