Assume that I want to find the conformal mapping from a square onto the unit disk. In the regular case, where the four edge points are positioned along cardinal directions, this mapping seems to have a relatively simple solution. I have plotted this in the figure below (a).
The scenario I am interested in is a bit more complicated. I still want to map from a square (or rectangle) onto a disk, but I want to place the edge points of the square irregularly on the target circle (figure b). Assume that I know the positions of these target points.
Is it possible to find a mapping like this? If so, how would I go about solving this issue?
Your question is equivalent to asking if, given two quadruples $(z_1, z_2, z_3, z_4)$ and $(w_1, w_2, w_3, w_4)$ of pairwise distinct points on the unit circle, there is a conformal map from the unit disk onto itself which extends continuously to the closed disk, and maps $z_k$ to $w_k$ for $k=1, 2, 3, 4$.
Conformal automorphisms of the unit disk are Möbius transformations, they are uniquely determined by the images of three distinct points, preserve the cross-ratio and the orientation. Therefore such a mapping exists exactly if the cross-ratios $(z_1, z_2; z_3, z_4)$ and $(w_1, w_2; w_3, w_4)$ are equal, and if the two quadruples have the same orientation with respect to the unit disk. In that case the mapping $T$ is given by $$ (T(z), w_2; w_3, w_4) = (z, z_2; z_3, z_4) \, . $$