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I have a simple paper with parallel grid lines and I know the height of the observer from the ground level.

Now when I snap the picture with my camera, the effect i get is what is seen in the image below. Even if the lines are parallel, the perception is that they converge in the middle.

When i draw lines on the lines of the page, i get angles.

enter image description here

Is it possible that given the information below:

  1. The height of the observer from the ground
  2. The angles generated

That I can calculate the distance from the observer to this "convergance" point? Is there already some formula for this?

The goal is this: If there is some generic formula to calculate the distance from the observer to the convergence point, then I can use this formula in simulation program involving the calculation of small distances.

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There is indeed a generic formula: the convergence point is infinitely far away.

Why is this? Assume it is not. Then there is some finite distance, $d$, where the lines converge. There are then two possibilities:

  1. the lines must appear curved;
  2. at distances greater than $d$ they must appear to have crossed over each other (this is basic Euclidean geometry).

It should be easy to convince yourself that projected straight lines appear straight, and that very distant objects do not appear as mirror images. The remaining option is that there is no distance greater than $d$, in other words, $d$ is not finite.

Another way to see this is to think about making two marks, the same distance from the camera, on two of the lines. You can then ask, from the point of view of the observer (the camera), what is the angle between those marks, or equivalently, how far apart do they appear to be? At the point where the lines converge the answer obviously is that the angle is zero: everything appears to be a single point.

If the separation between the points is $s$ and the distance from the observer is $d$, then if $d \gg s$, $s/d \approx \sin\theta \approx \theta$, where $\theta$ is the angle between them (note I’ve ignored the camera being above the plane &c: all of this washes out as $d/s$ becomes large). Well, we want $\theta \to 0$, so $s/d \to 0$, so $d \to \infty$ since $s$ is constant.

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  • $\begingroup$ As an aside, this point is also known as the vanishing point, and the concept appears in a variety of disciplines that involve projections, from mathematics using homogeneous coordinates to art and photography. $\endgroup$ – Muphrid Oct 1 '17 at 0:45
  • $\begingroup$ man.. you guys speak math that i am not good at. What if i used the red lines as reference and simply use that convergence point as the "vanishing point"? Or i'm i asking something stupid.. I just want to know if it is possible to know where the red lines converge. $\endgroup$ – iOS Calendar patchthecode.com Oct 1 '17 at 0:55
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    $\begingroup$ @iOSCalendarViewOnMyProfile And I have answered that: the lines appear to converge infinitely far away if they are parallel lines. If they are not parallel lines then it's up to you where they converge. $\endgroup$ – tfb Oct 1 '17 at 10:57

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