Consider the Mobius transformation that maps onto the circle . Note that maps onto the circle .

Now choose an , where such that or equivalently .

Note that is the inverse of with respect to the real axis. So and must be inverses of each other with respect to . Hence or equivalently .

When . Also notice that . Therefore .

By converting into polar coordinates, the above equality yields .

Therefore .

So notice that only two points were chosen to map the upper-half plane onto the unit disk and certain constraints were given to a third point but no third point was chosen explicitly, therefore it makes sense that there is not a unique solution to do this.

I hope this answers your question.

-Chip