I want to take the upper half of the unit disc,

and send it to the upper half of the plane

. My instructor hinted that if I can specify three distinct points that go to 1, 0, and the point at infinity under the transformation that I can use the cross-ratio to determine a bilinear transformation that will "almost work".

I'm totally lost in this course right now, so I wasn't even sure what the hint meant, but I tried *something* (even if ignorantly).

Looking at the

, I was hoping that if I sent

and left 1 and 0 fixed, that the boundary would "unfurl" to the real axis and the interior of

would map to

as a consequence. In terms of the cross-ratio, this translates to

.

But I don't even know how to tell if this works. And what might my instructor mean by "almost work"? Thanks ahead of time for any advice!!