Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $B$.

share|improve this question
    
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
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.