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When we look at parallel rail tracks, it appears to converge at a certain angle. Let us say the width of the tracks is d and the height of the eye is h. Assuming one stands in the middle of the tracks, what is the angle at which the tracks appear to converge at infinity?

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  • $\begingroup$ This is an interesting problem that I've not really thought about before. Have you tried working out any of the geometry yourself? $\endgroup$ – Kyle Kanos Aug 17 '15 at 2:19
  • $\begingroup$ Yes I did using a simplistic model. It turned out to be 2 arctan(d/2h) where d is the width of the tracks and h is the height of observation. Try it out :-) $\endgroup$ – Pratik Rath Aug 28 '15 at 14:20
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When your eyesight is exactly horizontal; i.e. the axis of the convex lens of the eye is parallel to the ground, for given distance $x$ sufficiently far away from the eye, the point of one of the rail tracks $(x,h,d/2)$ forms an image at the point $ (\frac{ld}{2x}, \frac{lh}{x})$ on the image plane at the back of the eye. Here, $l$ is the distance between the image plane and the lens of the eye. (For the given fixed distance $x$, you are always straining your eye to adjust the focal length so that the image distance is $l$.) Thus $x$ parametrises a straight line which makes an angle $\tan^{-1}(d/2h)$, with the vertical axis on the image plane, and thus the total angle is $2 \tan^{-1}(d/2h)$. The configuration on the image plane is inverted, when the signal goes to the brain but this subtended angle, of course, remains the same.

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