When you look at a distant object in hyperbolic geometry, the main problem is that it will be tiny. You're using HyperRogue as a base, which, in its standard tiling, can be considered to have absolute unit of 3 meters. A circle with diameter of 15 meters will then have circumference around 1400 m. In Euclidean space, a circle with this circumference would have radius 222 m, so in other words: you would see an object 15 m away just as small as you would see an object 222 m away in our world. Since the circumference of the circle is the same, the angular size of the object (which is basically how big part of the circumference it takes) is also the same.
Let's say you can see an object (perhaps a mountain) that is 50 km away on Earth. A circle with radius of 50 km has circumference 314.16 km. In HyperRogue world, this circumference would correspond to radius of only around 31.3 m; in other words, if you could only notice something as big as a mountain at THIS distance, there's not much chance of seeing anything that is farther.
And this assumes that you could see the whole mountain, which you actually can't -- and this brings me to the next point.
In Euclidean world, we take it for granted that when we look at a distant 3D object, we see a half of it. You look up to the Moon and you see half of the whole Moon. If we ignore the pesky fact that we have two eyes for a moment, then a single eye can actually never see a full half of the Moon, but the difference is negligible.
In hyperbolic world, it's not negligible. If you have a sphere in this world, it turns out that you will always see just a small part of it -- the bulk of it will be so diminished by the insane hyperbolic perspective that it will actually hide behind the nearest bit.
When you calculate it, you will find that there is an upper limit on an AREA of a sphere that is visible. And that area is pi -- that's what you'd get with infinite distance and infinitely large sphere (horosphere). In HyperRogue, an area of pi would mean around 28 square metres. Let that sink in. If there was, say, Jupiter near you (no matter whether we're talking Jupiter's radius, circumference, mass or whatever -- in this case there's no real difference), you couldn't actually see more than 28 square metres of it! This also means that you couldn't find a meaningful difference between a small sphere and a large one -- as the radius grows, the curvature approaches horosphere, and it would be very hard to distinguish how big the sphere is, exactly -- or whether it's even finite.
That means you couldn't actually see a mountain at the distance of over 30 m as previously mentioned -- the bulk of that mountain would be hidden and you could only see a very small front of it, which could, of course, be seen at smaller distance.
Not only would be distant objects small, but they would also move insanely fast when you move. Hyperbolic parallax doesn't work the same way the Euclidean one does; distant objects would not seem to move more slowly. If you walked under a sky full of stars (and you could see them for some reasons), the stars would seem to move -- regardless of their true distance.
So, given all this, how would you navigate in such a world? The best bet probably is to put together a set of local coordinate systems, but... not even that wouldn't work in HyperRogue. The curvature of that world is simply so big that I have to question whether things like cities or civilization are even possible in there.
One idea I came up with was to have square map plates, each corresponding to a square from a hyperbolic tiling like {4,5} or {4,6}. You could then put the plates together to get a limited view. But even a {4,6} square -- square with inner angle of 60 degrees -- would have an edge of only around 5 meters in HyperRogue. What's worse, no square at all could cover an area bigger than 56 square meters (2 pi square units) because that's the maximum area of a square in hyperbolic geometry. If we used hexagonal map plates, we could get to double of that, but that's all.
Tricky.