As you stare at this, try to imagine yourself as one of the fish. You are exactly the same size and shape as every other fish, and you can swim in a straight line forever without ever seeing any change in your surroundings or in your fellow fish. But looking into the map from outside, the compression of distances makes you appear to be swerving along a circular path and to be shrinking as you go. Pg 63 of Tristan Needham's VDG
I don't get it, how would it be that I see all the fish are of same size if I was in the disc but not as an outside observer? Could we explain the above situation using principles from geometric optics?
I think the I understood it a bit more now. In the real world, if we have a ruler, and we graduate it by defining a unit marking on it, then wherever we travel on the earth, the unit won't change.
Here, the issue is that, as we move from point to point on the beltrami poincare' disc, the rule shrinks and expands and so does one unit. It's sort of like how currencies work between countries, as we move from country to country, what is the purchasing power of one unit currency is different( eg: 1$ is certainly not the same value on as 1 euro).
Now, the metric formula $ds' = \frac{2ds}{1-r^2}$ tells us how to convert the unit distance at some point (measured by an outsider) with how that amount of distance is actually valued nearby in that point. Sort of like converting all foreign currency into the US dollar (example), and seeing how much it would be worth on that scale.
This causes what we think of as straight-line to be different because due to our unit not being fixed point to point, there are better ways to travel between two points than to take a straight line.