If a conformal mapping, e.g., a Schwarz-Christoffel mapping, maps the real line (from left to right) to a polygon, which is traced out from left to right, why is the upper half plane mapped to the interior of the polygon?
Is it a preservation of orientation argument? I.e., since the upper half plane was to the "left" of the moving point in the z-plane (moving along the real axis), then the image of the upper half plane will again be to the left of the curve, which is also going from left to right, until it closes at the final vertex to form the polygon.
I know that one can use a test point and then make a connectedness argument, but if the integral (or any tricky conformal mapping) were hard to evaluate, such as an elliptic integral, I don't feel that this is a wise move to make, and that there could possibly be a better way to argue why the image of the UHP is inside the polygon.
Thanks,