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I stand up and I look at two parallel railroad tracks. I find that converge away from me. Why? Can someone explain me why parallel lines seem to converge, please?

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    $\begingroup$ Because we live in a three dimensional world. $\endgroup$
    – Horus
    Aug 24, 2015 at 15:30
  • $\begingroup$ A more detailed explanation? $\endgroup$
    – Stefan
    Aug 24, 2015 at 15:32
  • $\begingroup$ Added an answer in the exact same question here: physics.stackexchange.com/a/292342/78842 $\endgroup$ Nov 13, 2016 at 15:00
  • $\begingroup$ I'm sure there's a proper answer but perhaps one could say - "parallel lines intersect at infinity". $\endgroup$
    – Prahar
    Jan 1, 2021 at 13:21

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The distance between the two is assumed to be constant. However, when you look at an object that is far away, it "seems smaller". What that means mathematically is that the angle from one end to the other, as seen at your eye, is smaller.

enter image description here

Now you can't tell the difference between something that is "small and close", and something that is "big and far". So the illusion presented by the railway track is that the distance between the rails gets smaller as you look at a point that is further in the distance.

And because your eye is a little bit above the ground, points on the railroad closer to the horizon are actually further away from you.

In art, the point where the railroad lines appear to converge is called the "vanishing point". Google that term if you want more information.

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Keep in mind that your eyes perceive a fixed field of view. That is, you see a certain angle in front of you. They do not see a fixed distance across your line of sight. This means that the farther away for you you look, the more distance there is from one side of your FOV to the other. When you look at parallel railroad tracks that maintain a certain distance apart, they seem to converge as they go off into the distance. This is because the proportion of your field of view that the tracks take up diminishes.

Let's say the tracks are $x$ meters apart and close to you, your FOV encompasses $6x$ meters across. That means $1/6$ of your field of view is taken up by the tracks' separation. But far away from you, your field of view might encompass $600x$ meters across. This means the tracks would have converged to only take up $1/600$ of your FOV. They're still the same distance apart, but the distance between them takes up less of your field of view. The brain interprets this as converging because if an image did this that was at the same distance in front of you throughout, it would actually be converging.

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Things in the distance seem smaller due to the viewing angle from one end to the other, A 100 feet tall tree viewed from 50 feet away has a viewing angle of 76 degrees. The same height tree viewed from 1,000 feet away has a viewing angle of 5.26 degrees

Railroad tracks at only 4 feet 8.5 inches wide have a viewing angle of 5.39 degrees at 50 feet, and 0.27 degrees at 1,000 feet 5.39 degrees is about 2 inches at arms length 0.29 degrees is 0.1 inches (1/10th of an inch) at arms length It makes no difference if you use your eyes or a camera to view the tracks or the trees. The result is the sameDiagram showing 100 foot high trees

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There is a well-known episode of Father Ted where he explains to Dougal that the small plastic cow in his hand is near but the real cows in the field outside the window are far away. Poor Dougal remains utterly baffled.

Ultimately, the phenomenon is grounded in the principles of projective geometry, especially perspective projections, but I doubt that would have helped Dougal. It might help here, though.

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