The right way to do this is to apply the Cauchy Integral Formula. With this you can map any polygon to any other polygon. If the polygons don't self-intersect, it will send the boundary to the boundary and the interior to the interior. The mapping will also have the excellent property that it is conformal, meaning that angles are preserved. By that I mean that if a pair of curves intersect in your region, then they will be mapped to a pair of curves that intersect at the same angle
