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!!
One thing to remember is that bilinear transformations preserve angles. The set has a couple of right-angles corners in its boundary, whereas does not. So you can be sure that a bilinear transformation will not be sufficient on its own to transform into . The best you can do with a bilinear transformation is to take the ends of the diameter of the unit circle to the points 0 and . Then the operation of squaring has the effect of doubling angles, so it will convert the right angle at 0 into a straight line.
That may sound a bit vague, but it is the way that I used to find the map from to .