Conformal mappings of stretched polyominoes onto half-plane

TL;DR

This paper presents an algorithm to compute conformal mappings of stretched polyominoes, specifically P-pentominoes and L-tetrominoes, onto the upper half-plane, using advanced complex analysis techniques.

Contribution

It introduces a novel algorithm for conformal mapping of specific polygonal shapes derived from polyominoes, expanding computational methods in geometric function theory.

Findings

Algorithm successfully maps polyominoes onto the upper half-plane

Conformal modules of the polygons are computed effectively

Method leverages Riemann-Schwarz reflection and uniformization techniques

Abstract

We give an algorithm for finding conformal mappings onto the upper half-plane and conformal modules of some types of polygons. The polygons are obtained by stretching along the real axis polyominoes i.e., polygons which are connected unions of unit squares with vertices from the integer lattice. We consider the polyominoes of two types, so-called the $P$-pentomino and the $L$-tetromino. The proofs are based on the Riemann-Schwarz reflection principle and uniformization of compact simply-connected Riemann surfaces by rational functions.