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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 $B$.

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Try mapping through a half-plane and then combining the results. Also, what is the image of the line that passes through the circle and the orgin of transformation? – dtldarek Mar 15 '12 at 4:44
up vote 0 down vote accepted

See Is there a Moebius transformation that scales disks to the unit disk?

Define the two LFT's that take your two circles, with distinguished interior points $a,b$ each to the unit disk and $0.$ Then take the inverse of one of the maps (given by the LFT of the inverse matrix) and compose it with the other map, the result takes $a$ to $b$ or $b$ to $a$ depending on order.

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