It's a bit of a general question, what is the process of constructing a Möbius transformation?
For example how can I find $a,b,c,d$ that make the transformation $\frac{az+b}{cz+d}$ send some disk to a half of plane.
It seems to me like trial and error, one needs to make a guess to the transformation expression and check if it works for 3 edge points and one inner point.
I saw these examples but I don't understand how to get the transformation from scratch.
For a given disk with centre $a$ and radius $r$, first use a translation $f_1:z \rightarrow z-a$ to map the disk to a disk centred at the origin. Then use an expansion/contraction $f_2:z \rightarrow \frac{z}{r}$ to map this disk to the unit disk. Then use the Mobius transformation
$f_3: z\rightarrow \frac{1-iz}{z-i}$
to map the unit disk to the upper half plane.
Now $f_1$, $f_2$ and $f_3$ are all Mobius transformations and the composition of two Mobius transformations is another Mobius transformation. So you can compose the three maps to get a Mobius transformation $f_3 \circ f_2 \circ f_1$ that maps the original disk to the upper half plane.