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Original post on worldbuilding.stackexchange by Dennis
I am imagining a fantasy world where the horizon is a real place. If you travel to where the dome of the sky appears to meet the ground, you can even find intrepid human settlements on the edge of — something.
At first I thought of the horizon ending in a solid sky-painted wall. But then I realized there’s another, more terrifying option.
First, consider a couple models of possible geometry: An infinite sky above an infinite flat plane would have a vanishing point, but would not appear domed. The sky of the real Earth appears domed because the world is round and sky and ground curve downward together. Lastly there is the medieval view of a flat Earth covered by a domed heavens.
What I realized is there’s a fourth model which has the infinite extent of the former, but the dome appearance of the latter two. Take each line of sight from an observer and project it through the spot where the “dome” would be, onto the infinite-plane sky beyond it. This gives you the difference from where the point (in the sky) “appears to be” versus where how far away it actually lies (where the ray intersects the plane of the sky). (Essentially a sphere projected onto a plane.)
From angles of straight up (90 degrees) through 45 degrees, the difference would not be that great. However as the angle of your line of sight to the sky approaches zero degrees, the place where the ray intersects the plane tends to infinity.
As to why this would create a physical place where you can “reach” the horizon, imagine that this visual effect is due to a change in the nature of space itself. There is a radius on the surface of the flat Earth past which a motion that would normally cause 1 inch of displacement causes 2 inches of displacement, whereupon another inch (3 total) causes 9 inches of displacement, and so on.
Rather than the horizon being a barrier that acts like a wall with negative effect on movement, it would be a boundary with an amplifying or positive feedback effect on outward movement. This would be a dangerous place because a misstep could send you flying outward to eternity.
It would act somewhat like an event horizon, but differs from a black hole event horizon in important ways. It would not block light or information from crossing. A ship passing the event horizon of a large black hole might not notice an difference in space from its local point of view. For the hyperbolic horizon I’m describing, the observer would most definitely notice a problem; it would feel like space had acquired an inherent negative viscosity in the outward direction.
As for why clouds on the horizon don’t become infinitely tiny as they become infinitely distant, perhaps the increase in displacement-per-velocity as a function of distance causes them to puff up to enormous size. Since stars are point sources of light, their apparent size need not change.