What would the horizon look like on an infinite plane? I can imagine the growth of a horizon to be similar to a decreasing exponential function. Where the growth of horizon decreases more as the distance away from it increases. See graph below (assume y is horizon growth and x is plane length).

enter image description here

But what if the length was infinite? Clearly the horizon can't just keep going up and up, that would be mathmatically impossible. But in the real world if there is a limit, where is it?


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The diagram shows how the angle your line of site to the horizon makes depends on how far away the horizon is. On a flat infinite plane the horizon would appear at your eye level.

  • $\begingroup$ Dear John, please let me know what happens if two persons stand next to each other, one is taller then the other, will they both see the horizon at their own eye level? So they will not agree where the horizon is? $\endgroup$ – Árpád Szendrei Apr 6 '18 at 9:16
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    $\begingroup$ @ÁrpádSzendrei correct. But that's always true even when the horizon is not at infinity. For example it's true for observers standing on the Earth's surface. $\endgroup$ – John Rennie Apr 6 '18 at 9:19
  • $\begingroup$ so no matter how far I go up (to infinity), I will see the horizon in front of me at my eye level (this is not true in case of the Earth)? $\endgroup$ – Árpád Szendrei Apr 6 '18 at 9:22
  • $\begingroup$ @ÁrpádSzendrei correct $\endgroup$ – John Rennie Apr 6 '18 at 9:29

Consider the following triangle:

$h$ is your height, $l$ is the distance to the object you want to see.

Let's calculate $\varphi$:



Let $x=\dfrac{l}{h}$. Plotting $\varphi=arctan(x)$, you get:

The plot means that for $x\to\infty$ (or, in other words, for $l>>h$, i.e. for the distance being a lot of larger than your height), the angle equals $90^\circ$ and you'll the horizont right atthe height $h$, i.e. the level of your eyes. The higher person is, the more they will see (this is also true for Earth, actually).


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