Say I have two disks $A$ and $B$, with two points $a\in A$ and $b\in B$. Is there a way to explicitly construct a linear fractional transformation from $A$ onto $B$ that sends $a$ to $b$?
I know a linear fractional transformation is determined by its image on 3 distinct points, and that they sends circles and lines to circles and lines. Would it then be enough to map $a\mapsto b$, and then choose two boundary points on $A$ to map to two arbitrary boundary points on $B$, or does more care need to be taken? My worry is that the boundary points may map onto a circle which isn't the boundary of $B$.