Mapping from Poincare's disk model to UHP

Question

I have a question that :

How can I map any point in Poincare's disk model to Upper-half-plane model? I know the function

$$f(z) = \frac{z + i}{iz+1}$$

But I want to know the geometric constructive method. Would you help me?

Answer 1

It seems a little easier to construct the equivalent

$$f(z)=\frac{z+i}{i(z-i)}$$

Given point $z$ in the complex plane, draw the points $z+i$ and $z-i$ one unit above and below $z$. You could use the obvious parallelograms with $z$, $0$, $i$, and each of those two points. Then rotate the line segment from $0$ to $z-i$ by $90°$ counterclockwise to get point $i(z-i)$.

We now have the numerator and denominator of our fraction. We now use the facts that (1) the modulus of $\frac uv$ is the ratio of the moduli of $u$ and $v$, and (2) the argument (angle) of $\frac uv$ is the difference of the arguments of $u$ and $v$.

To get the correct modulus, draw a vertical line from the intersection (point $A$ in my diagram) of the ray from $0$ to $z-i$ with the unit circle. Mark the intersection of this line with the ray from $0$ to $z+i$ (point $B$ in my diagram). By similar triangles, point $B$ has the desired ratio of moduli.

To get the correct angle, copy the signed angle of $i(z-i)$ to $0$ to the positive real axis (shown in red) to the ray from $0$ to $z+i$. The argument of points on that ray have the desired difference of arguments in our fraction.

The desired $f(z)$ is then the intersection of that last ray with the circle that goes through point $B$ and is centered at the origin. That intersection has both the correct modulus and the correct argument.

The only problem I see with this construction is that point $B$ and thus point $f(z)$ is undefined when $z$ is on the imaginary axis. This is easily handled by rotating point $A$ off the imaginary axis then continuing.