If I have two parallel lines…say railroad tracks…that have a very long length they will appear to intersect at a distance called the vanishing point. We’ve all see it.
Would it be possible to make the railroad tracks diverge, that is, be placed further apart, as a function of distance so that the rails would appear to be parallel past the typical vanishing point?
The question was posed to me by my sons in the form of a hole dug to the center of the earth. The initial response from one son was that you couldn’t see far enough down the hole to see the center of the earth. So, says the other, what if you made the hole wider as you went deeper…so here I am, appealing to the teeming millions for a practical solution to neutralizing the vanishing point.
If I could end up with a f(x) where x is the depth of the hole, or percentage of the distance tot he vanishing point, that would be great.
If the rails diverge such that they would intersect at your location, then they both appear as points; that is, you see each rail in profile, so the end of the rail closest to you hides the rest of that rail.
The typical vanishing point? There isn’t one. Parallel rails appear to meet at a point, but it doesn’t have an apparent distance. What would it mean to be “past the typical vanishing point”.
If you dug a hole straight through the earth, and it had perfectly cylindrical walls, you could see all the way to the center of the earth. The far end of the hole would be visually very small, and I guess we need a telescope to see it clearly, or else its end must be flooded with very bright light. But optically and with regard to perspective, we could see it.
If it diverged as a cone whose apex was in front of your eye (which means below you in the case of a hole), you would not be able to see the walls. They would be blocked by the entrance of the hole. Imagine looking at the small end of a funnel that you hold several feet in front of you. You see its outside surfaces, not the inside surfaces. If on the other hand the apex of the funnel was behind you, you would see a sizeable end of the hole, and the walls would get ever more distant until they met the far edge.
Expanding on what Napier said, the vanishing point is obviously not at some finite distance from you: some finite distance from you is still on the rails, where they don’t intersect (visibly so, if your resolution is good enough). The vanishing point is nominally infinitely far out (that it should appear that the rails actually reach such a vanishing point is only because the resolution of your eye isn’t perfect; if you had perfect resolution, the intersection would only appear to occur with infinitely long parallel rails).
To put it another way: the vanishing point is a point in your visual field which both parallel rails approach, but never reach. So it doesn’t make sense to speak of what goes on in your visual field on the portion of the rails beyond the vanishing point.
That having been said, if one wants to reify vanishing points as actual legitimate points in space, one is led to projective geometry. In projective geometry, what happens to the rails after they reach their infinitely far off intersection is… they start approaching you from behind. Every line wraps around the universe like a “straight” path across the Earth. But that’s something of an abstract tangent…
But in my dumbninity, can I just verify what I think the question is?
I can? Cool, thanks…
So you want to place two rails, angling out and away from your field of vision in such a way that they “appear” to be parallel?
Couldn’t you hold up a protractor to the parallel rails and note the angle at which they appear to be converging, and then just angle them out at the same degrees?
K…I don’t think I’ll be able to understand your responses, considering my head is already starting to hurt, but I’ll try
Let me rephrase what I think the OP wants. If two rail are 4 ft apart where you are standing, how far do they have to be apart at 50 ft so that they still appear parallel. Like if you took a picture, and just checked the imaged lines to see if they were parallel.
How far apart at 50 ft from you?
how about 300 ft?
then 1 mile
Is there some mathematical relationship that would describe the distance apart they have to be (like the distance away from you is the cube root of the divergent distance)
If I’m not having some kind of brain fart, they just have to follow the rule that their distance from each other is proportional to their distance from you.
After the lines reach the intersection, do they cross? When they approach you from behind, are they a mirror image of how they left you? And if so, what happens when they meet, at your position?
Or . . . You are standing on the railroad tracks and looking forward toward an infinite vanishing point. But if you turn around, there’s another one behind you. Are these the same point? And don’t the tracks have to optically bend or curve at your position, to accommodate both the convergence and divergence?
It is possible to construct things in such a way as to defeat the effects of normal perspective (from the eye’s point of view, at least) - the technique is called forced perspective and is probably most commonly known in the form of those rooms that appear normal, but make a person appear to grow when they are seen to walk into it (an example exists in the moview Willy Wonka And The Chocolate Factory).
They typically only work when viewed from one specific point.
In the case of railway tracks - yes, of course it’s possible - demonstrable by a thought experiment:
You’re in charge of the construction of the track and it will be assembled from short segments - you have two assistants placing the segments, while you direct them from the viewing point by walkie-talkie. You want to see the tracks disappearing into the distance without converging, so you direct them to place each segment (left-a-bit, right-a-bit) in such way that it appears in your field of view in the place where it does not seem to converge upon the other.
The track will of course be unusable by anything like a normal train and will only look right from where you are sitting.
I think you may be missing that you can have more than one vanishing point. Only mutually parallel objects share the same vanishing point. Take the example of standing at a crossroads in a city: the buildings along each street seem to converge, don’t they? So they have their own vanishing points and do not share the same vanishing point.
I did think about mentioning that, but the OP seemed only concerned about keeping them from converging.
If you widen them to make them appear a constant width, apart from consuming a phenomenal amount of materials, the effect would be that of looking at a short section of track, sticking up vertically out of the ground and ending in line with the horizon.
I think you’d need to make the rails radiate out from you, so that you’re the honest-to-god intersection point, like the spokes on a wheel. This would result in you having to turn, say, 6 degrees to the left to see ANY point on the left rail and some other number of degrees to the right to see any point there. That fact, I believe, will make the separation seem constant, though the environment will distort things. No idea what effect that would have on the illusion, or if it would spoil things.
Yes, that’s exactly what it would look like . . . except getting hazy toward the top. And there could be people standing on the ties, the upper ones looking very tiny. I wonder if anyone’s actually done this. I think it would have to be done in a desert or prairie.
Yes, this is an instance of what I described in post #7. Though, taking “you” to be your eye, this is the specific case where each rail collapses to a single point in your visual field, no?
The illusion provides lots of extra information in the form of all the trompe l’oeil painting - depth perception may try to tell your brain that you’re looking into an odd, irregular-shaped box, but this perception is shouted down by the fact that it looks a lot like a normal room.
Holy crap. Let’s try to answer the actual question.
Say you dig a hole 2 meters in diameter. The hole goes to the center of the earth, something like 12,700 km. For the hole at the other end to appear the same as 2 meters with regard to the perspective of a person 1.75 m tall standing at the edge looking down the center, the other end of the hole would have to be approximately 14,514 km. Unfortunately this exceeds the diameter of the earth, so you couldn’t do it.
You could do it if you got on a ladder to look down the hole, or made a smaller hole. This problem is easily solved using similar triangles. The function giving the diameter of the hole at the other end is
d is the diameter of the hole
p is the depth of the hole
v is the distance of the viewer from the hole
f(d,p,v) = dp/v
The same formula works for railroad tracks, where the distance between the rails at the distance v from the viewer replaces the hole diameter as d. The angle between the tracks can be determined as
