Disc to upper half-plane, cross-ratio

Idea: rotate the upper hemisphere \((\HH)\) of \(\CP^1\) to make the equator \(\bd \DD\), “zoom” by placing \(i\) at the center so \(0\to i\mapsto -1\to 0\) and \(i\to \infty\mapsto 0\to 1\). Accomplish this by sending

• \(\infty\to 1\)
• \(i\to 0\)
• \(-i\to \infty\)

Use the cross-ratio \[ R(z) \da (z, \infty, i, -i) = {z-i \over z-(-i)} {\infty - (-i) \over \infty - i} = {z-i\over z+i} .\]

Checking that this works:

• If \(z\in \RR\) then \(\abs{z-i} = \abs{z+i}\) so \(\abs{F(z)} = 1\).
• If \(z\in \HH\) then \(\abs{z-i}\leq \abs{z+i}\) so \(\abs{F(z)}< 1\).