A Möbius transformation is given by
with parameters $a$, $b$, $c$, and $d$. The Wikipedia article provides rules for finding these parameters based on three points $z_1$, $z_2$, and $z_3$ and their images $w_1$, $w_2$, and $w_3$. It is my goal to understand how we can derive the equations which yield the parameters.
Möbius transformations preserve the cross-ratio, so I assume we start with the cross-ratios of the original points and their images: $$(z,z_1;z_2,z_3)=(f(z),w_1;w_2,w_3)$$ which can be reformulated as
I imagine the solution is obtained by reformulating this equation above somehow to solve for $f(z)$. But how is this done? I could not find a proper tutorial for this online - most tutorials I find plug in specific points at this stage, but I would like to learn how the general approach is derived.