notes

Gary

Presumably, the question you are referring to is the one in the edited version of your original post above, is that right?
Looking at your document, one thing struck me immediately. In your diagram (b) of the map on page 1 you have drawn the h-line incorrectly. It should "sit" on the horizon, not on the line $y=1$.

I will think about your question above and get back - In the meantime you can see what the geodesics on the pseudosphere are supposed to look if you explore the links in my post in the Useful Links board.

Update: 7th Feb 2017 10:45 GMT
Note: When I use the word Tractrix I mean the curve shown in figure 18a on page 294 and when I use the word pseudosphere I mean the surface in figure 20a on page 297.

I have looked at your function mapToTracTrix (which in fact is intended to map to the pseudosphere) and it doesn't seem correct to me for the following reasons:

1. The IF at the beginning should be checking for Im$(z)\geq1$ because points on the pseudosphere are only defined when $y$ on the map $\geq1$.
Just out of interest, what values does the current code return for $(X,Y,Z)$ if Im$(z)=0$?

2. The value of $Z$ should be calculated from the equation you found in Wikipedia (which you have copied incorrectly into your document - the last term should not be inside the logarithm). This means that in your function you should write:

$Z=\ln\bigg[\frac{1+\sqrt{(1-r^2)}}{{r^2}}\bigg]-\sqrt{1-r^2}$ where $r=1/$Im$(z)$ is the $r$ in the current version of your function.

This is because your $Z$ is the value of $Y$ in 18a on page 294.

Because of 1. above, if you start moving along on the map in a direction orthogonal to $y=1$, then you will continue straight up the tractrix generator on the pseudosphere. The only way to travel along a geodesic on the pseudosphere which is not a tractrix generator is to start moving on the map in a direction which is not orthogonal to $y=1$.
Recall exercise 16 part (ii).

3. Your diagram of the pseudosphere on page 1 of your document looks wrong because the slope of the tractrix generators at $X=1$ should be zero and yours clearly isn't. This is because (using the notation in the book), $\frac{dY}{dX}=-\sqrt{\frac{1-X^2}{X^2}}$, which is zero when $X=1$.

Vasco

Gary

Yes, sorry, I copied it wrongly from Wikipedia. Your diagram of the pseudosphere looks really excellent! You can see that the maximum vertical distance of the geodesic on the pseudosphere corresponds with the maximum $y$ point of the semicircular h-line in the map, as expected.

Vasco