Post by Gary on Nov 2, 2016 21:27:21 GMT
If you look at paragraph two on page 312 where Needham starts to describe the construction for [32], you will notice that he starts with $\mathcal{M}$ as a direct motion. It seems to me that if we want to actually construct something like [32], we need to have an explicit function for $\mathcal{M}$ in order to be able to calculate $w$ and $dw$. The angle $\theta$ would then emerge as part of this calculation.
Vasco
Vasco,
I've gone round with this. Could you say that $M = e^{i \lambda} z$ and apply this to z on L? Then one could plot $d\tilde{z}$ using $M'(z) = e^{i \lambda}$ to get the angle right (it looks right on the map). M provides a direct motion on an h-line from z to w. It is equivalent to $\mathcal{R}_{B} \circ \mathcal{R}_{A}$. You obtain z and w first by inverting z in A to get w. (And you get z by rotating an arbitrary M and then w by inverting z in A.) But then I think you need to work backwords by assigning an arbitrary value to $\theta$, say Pi/4, which is needed to plot C. Once C is plotted, you can get dw from $\mathcal{R}_{C} \circ \mathcal{R}_{B}$. When you apply this to $\mathcal{R}_{B} \circ \mathcal{R}_{A}$, you have an M composed of four reflections that sends z to w AND dz to dw (not the same M as as we used to send z to w). However, we can show that this M equals $\mathcal{R}_{C} \circ \mathcal{R}_{A}$.
Gary
