The Upper Half Disk (Parametrizations 2)

One of the text book examples of conformal maps is given by the complex function f(z) = z+1/z which maps the upper half disk bijectively to the lower half plane. This is the main ingredient for the following problem:

We would like to have a conformal map from a rectangle to a half disk (say of radius r) so that the left and right sides of the rectangle are mapped to half circular arcs around two points a and b.

The inverse of this map can be obtained by composing the text book map above with a Möbius transformation that sends the images of a and b to 0 and infinity, and then use polar coordinates on the upper half plane. Details are in this notebook.

What is this good for? Examples are 4-ended surfaces like the ones above that have sufficient symmetry that one can place the ends as four punctures on the real line, symmetric with respect to a circle.

This can also be useful more generally when existing symmetries cut the surface into pieces with four ends along the boundary of the pieces, as above.