We consider the Earth is a spheroid and that the horizon is an intersection of the sky and Earth. From this we can conclude that the horizon is always flat, because it's a circle. This explains why we observe a flat horizon, given the geometrical properties described.

My question is, when we get pretty high in altitude over 10.6km, why can we start to see a curved horizon? Please correct me if the first assumptions are wrong as well.

  • 2
    $\begingroup$ I smell flat earth... If this is indeed the case believe me: curvature of the horizon is the least of your problem. You should focus on statistics.. $\endgroup$
    – Noumeno
    Dec 14, 2020 at 11:06
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    $\begingroup$ The horizon seems flat from the point of view of an observer who is very close to the surface (relative to the radius of the Earth). $\endgroup$ Dec 14, 2020 at 14:25
  • $\begingroup$ @Noumeno No, I'm just trying to figure out why the horizon curves as I was under the impression that since the horizon is a circle it would always look flat regardless of how high we were. If the Earth was flat, the concept of a horizon would likely not even exist. $\endgroup$
    – Kramen
    Dec 14, 2020 at 17:13

1 Answer 1


The horizon is pretty much flat when observed from pretty low. It is not a perfect straight line. In fact, what you observe is a circle with several kilometers of radius that has its centre only about four or five meters below your eyes (namely the tangent circle of a cone and the earth sphere where the cone apex is in your eye). That is almost as if you observed that circle from within its plane, in which case you would observe a straight line exactly. More importantly, if you rotate your head left or right, you will not follow some further downfall of the curved horizon line, but instead its central point directly ahead of you will again be highest at (pretty close to) eye level. Unless you adjust your direction of sight: In order to look exactly at the horizon in front of you, you'd have to nod forward by a very, very tiny angle; if you then rotate around the tilted, not quite vertical axis of your head, you'd look slightly above the horizon after half a round.

Eyes are unreliable instruments anyway, so let's build a camera obscura from a used shoe box. The horizon on the image plane on the back is very close to a straight line and is only approximately at eye level, i.e., with a properly aligned box, the horizon will be slightly above (inverted image!) the middle of the screen and the outer ends will be even further above. Now if you rotate the box around the vertical axis, the image does not really change, i.e., you still have the outer ends higher above the middle line than the centre. How is that even possible? Well, by rotating the box, the image of an outer point of the horizon moves not only from boundary to middle of screen, but also closer to the pinhole!

Some back-of-the-envelope numbers: Standing on the ground, the horizon is about 3 arc minutes below "eye level". So in a perfectly aligned shoe-box camera of length 35 cm, the horizon will be about 0.3 mm above the center line. And if the box is 25 cm wide, the outer edges of the screen will be just about 2 cm further from the pinhole, and the curvature of the horizon line will be very subtle: only about 20 µm difference at the outer ends!

If you make a digital photo (standing at the beach, say) with a decent camara and a short(!) lens, you can digitally distort the image by making it only a few percent of its width while keeping the height. You should be able to see the curvature in the result (a curvature that was there before, but was too subtle to notice). Make yourself clear that this curvature is not caused by fish-eye effects of the lens (which do not play a role for a line though the image centre). Indeed, it happens in the same manner if you start with an upside down photo.

  • $\begingroup$ Thank you for your answer, it was very enlightening. $\endgroup$
    – Kramen
    Dec 14, 2020 at 17:06

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