observations and plots re three types of direct motion

Gary

Here is some feedback on your document:

In your first paragraph I think I understand what you mean by the $z$-plane and $w$-plane, but referring to the LHS and RHS as the $z$-plane and $w$-plane could potentially be confusing as in figure 29 on page 163 $z$ and $w$ are both on the LHS and $\widetilde{z}$ and $\widetilde{w}$ are both on the RHS.
In figure 1a you have written $w$ and $\widetilde{M}(w)$ and in 1b you have written $z=F^{-1}(w)$, but in the book Needham has $w=M(z)$ in LHS of figure 29 and $\widetilde{w}=\widetilde{M}(\widetilde{z})$ in RHS of figure 29.
The word hyperbolic on page 153 is not related to the ideas of the hyperbolic plane, as I understand it - see footnote on page 308.

Discussion of h-rotation on pages 2-3
Since the fixed points $\xi_+$ and $\xi_-$ can be any two points then we can choose $\xi_+=\alpha i$ and $\xi_-=-\alpha i$ where $\alpha$ is real and positive and then the picture will be like your figure 2 on page 2.
As I understand things the Möbius transformation $F$ is not the one we are interested in here, but the one called $M$ which transforms $z$ to $w$ on the LHS of figures 29, 34, 35 and 36 of chapter 3. I see the RHS pictures as being there to help us understand what is happening on the LHS.
On page 3 you talk about the figures 1a and 1b having fixed points, but it's the Möbius transformation $M$ which has fixed points at $\xi_+$ and $\xi_-$, and $\widetilde{M}$ which has fixed points at 0 and $\infty$.
For me the fact that $M$ is elliptic is not determined by $F$, but is a characteristic of $M$ itself.

If you look at (46) on page 173, you will see that the Möbius transformation $M$ is the composition of reflections in any two circles which pass through the fixed points. It then follows from (38) and (37) on page 303 that the h-rotation about $a$ is a perfect rotation (to a Poincarite), as described on pages 308-9.

I think your paragraph entitled Limit-translation should be H-translation from looking at figure 3 on page 3 and comparing it with figure 30 on page 311.

Vasco