Assume that I have a (possibly unit) square in some $z$-plane which I want to distort into the $w$-plane through a conformal mapping. My question has two successive parts:
- If I know the images of (i) the edge points and (ii) circular arcs which are orthogonal to each other, does a conformal map such as the one shown below exist? If so, how could I find it?
- Is it possible to find a unique conformal map similar to the one below if I only have the edgepoints in the $z$-plane and their images in the $w$-plane?
