Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?
Hint: Use $z\overline{z}=|z|^2$ and compare $|z-a|^2$ to $|1-\overline{a}z|^2$.
Have a look at the comments to see why these transformations are special.
Hint: Use $z\overline{z}=|z|^2$ and compare $|z-a|^2$ to $|1-\overline{a}z|^2$.
Have a look at the comments to see why these transformations are special.