Flat Earth in Bent Space

[grapes]

Orthogonal to what?

ETA: In post #28, you say "orthogonal to the viewpoint" but I'm not sure what you're meaning by that. Is it in that article? I'll check.

[John Jones]

Orthogonal to what?

That's what I was wondering. I was taught that orthogonal meant mutually perpendicular.

[Bob Angstrom]

To me that clearly says it was curving downward to follow the curvature of the earth (as it should from perspective distortion) - perhaps the OP may wish to clarify, but I can't see how that can be interpreted as curving upwards...

The OP statement is lacking in detail and it would be nice for some clarification but, as one who has seen the illusion, the laser beam appears to curve strongly upward. He says the laser beam appeared curved but failed to mention if it curved up or down. A downward curve towards the horizon would hardly be apparent since the beam appeared to be about twenty feet overhead. A laser beam that appeared to curve downward would not give one the impression that they were looking at a flat Earth in curved space. A straight laser beam running above a curved surface that looks flat would appear to be curved upward by comparison. Therefore the conclusion that he was looking at a flat Earth in curved space. To quote from the OP again:

Question: I was in the Black Rock Desert, a huge salt flat in NV, and over head, maybe twenty feet, some guys were shooting a massive green laser out across the desert. Looking out at it, it looked like the laser beam was curved and. The best explanation I could come up with is that the Earth is flat. What do you think? Is it simply an optical illusion. Is the Earth being round an illusion?

I understand the OP as asking if space (the laser beam) is straight while the Earth is curved or if the Earth is flat while space is curved. I can see from your examples how one can interpret the mentioned curvature as being downward but it would take an apparent upward curvature counter to the curvature of the Earth to prompt the question.

[Luckmeister]

To quote from the OP again:I understand the OP as asking if space (the laser beam) is straight while the Earth is curved or if the Earth is flat while space is curved. I can see from your examples how one can interpret the mentioned curvature as being downward but it would take an apparent upward curvature counter to the curvature of the Earth to prompt the question.

The way I read it, the OP was thinking that if the laser beam curved down following the curve of the Earth and the curving of the laser beam is an illusion, then the curving of the Earth might also be an illusion; therefore the Earth might really be flat.

[Bob Angstrom]

The way I read it, the OP was thinking that if the laser beam curved down following the curve of the Earth and the curving of the laser beam is an illusion, then the curving of the Earth might also be an illusion; therefore the Earth might really be flat.

It never occurred to me to interpret the OP as saying the beam curved downward but I see how someone can read it that way. I was trying to explain why the beam appeared to curve up and others were trying to explain why it appeared to curve down and that is the source of confusion. In my experience, the beam curved obviously upward. It would be nice if the OP or someone who has had a similar observation could describe how a powerful laser beam looks when observed from below.

[chrlzs]

An apology - I'm very busy, so a more formal proof of this will have to wait a day or three.

But orthogonal means exactly what it is supposed to mean. An orthogonal line is one that is to you, as the observer, at eye level (or directly overhead), and extends at right angles to a line drawn from your viewpoint to the closest point on that line. That line will be straight... but a line that (as in the OP's example) is at a different level, eg at 45 degrees 'up' from your viewing plane (using the horizon as that plane) and is therefore *further away* from you despite being parallel to the first.. that's the one that gets 'bent' from your perspective.

This stuff is the keystone of perspective, and is a rough analogy of 'great circles'.. and also the behavior of the ecliptic - it's approximately 'straight' from your perspective when overhead or at the horizon, but curved when at any other angle to your viewpoint. I will demonstrate all this by way of diagrams and numbers when I get time. And some of it *is* counter-intuitive! Your brain is *really* effective at correcting this stuff...

[grapes]

But orthogonal means exactly what it is supposed to mean. An orthogonal line is one that is to you, as the observer, at eye level (or directly overhead), and extends at right angles to a line drawn from your viewpoint to the closest point on that line. That line will be straight... but a line that (as in the OP's example) is at a different level, eg at 45 degrees 'up' from your viewing plane (using the horizon as that plane) and is therefore *further away* from you despite being parallel to the first.. that's the one that gets 'bent' from your perspective.

By that defintion, all infinite straight lines are orthogonal to a single Point Of View. Another line drawn from the POV to the closest point on that infinite line will be at right angles to the infinite line. All finite straight lines can be extended to be a part of an infinite line.

All 2D objects viewed from within their plane look like straight lines, including disks (as you've said). A straight line is always viewed within the plane from a POV

That is why great circles look straight from the center of a sphere. It is also why all straight lines look straight.

That said, environmental clues and confusions can contribute to impressions that the straight lines are not straight--an optical illusion.

Your brain is *really* effective at correcting this stuff...

Or, getting it confused.

[chrlzs]

I'll very frankly admit that I am unsure of the best way to get to the nub of this. I'm not a perspective genius nor trained in the science, but I do work with this stuff to get the panoramic images I create, and it is something I have considered at some length..

So, who knows, I may prove myself completely wrong with this approach and you guys may be able to poke huge holes in my logic... but.. we'll see! I may have some comments to make at the end of this about some of the responses above..

So, do follow along and if you have any issues with my approach, be it relevance, broken logic, mathematical or geometric errors, whatever .. I would genuinely appreciate your constructive criticisms.. There is actually a fair bit to cover here and I apologise in advance for the slowness with which I will be proceeding.. if you don't like it, come back to the thread in a week or so and pore over the wreckage... I may also invite some other folks over who are a little more educated/respected than me, if they will agree.. The more the merrier!

Before starting, I would just like to note that I don't see this topic covered very well (or easily!) elsewhere - and I believe that is largely because:
- it is actually a bit more complicated than it looks
- what we are usually trying to do is correct perspective so that 3d spaces can be represented by 2d images/drawings/ projections, not try to define why our 2d view is so flawed
- in the vast majority of cases, we are concerned with a simple front on view that covers less than 90 degrees. For those angles, perspective correction isn't all that difficult. But once you start trying to work with 'fisheye' views, and infinitely long lines like the laser in the OP, or crepuscular/anticrepuscular rays, or the "Moon Tilt Illusion", things start to break down and sometimes become counter-intuitive.

Anyway, for further reading while waiting for me to possibly self-destruct , you may like to also look at this page:
http://termespheres.com/6-point-perspective/
in particular the discussion of 4-, 5- and 6-point perspectives, and critically, why those techniques are required, and what is said/shown about supposedly 'straight' lines.

I'm still a bit time challenged, so this will be in pieces, but I'll start with defining a basic situation - single observation point, two parallel, infinitely long, straight lines.. Let me know if you think I haven't clearly defined anything, if you dispute anything or need clarification. I apologise in advance in that it will take a few posts to get to the crux, but I need to lay some groundwork properly first. So...

You are an observer located at position O. (At least, your EYE is there.)

To simplify matters, let's say you have zero height and your eye is at 'ground' level. That will be our basic reference (but as you will see, we can and will change it later and see what happens..) I'll arbitrarily pick some nice round, handy numbers for the next part - I'll talk in kilometres for no particular reason other than that usually we are talking about distant and long lines. If you would prefer to think in metres, inches or furlongs, or use different numbers, be my guest.

In the distance, there is an infinitely long line. It is 4 km away from your location, at ground level. On that line the closest point to your location I shall nominate as point A. So, obviously the distance O-A is 4km.

Directly above that line (perpendicular to the ground) and perfectly parallel to that first line, there is another infinitely long line. The closest point to your location (directly above A) is point B. Let's say the line is 3km higher than the first line, ie the distance A-B is 3km.

Using a little Pythagoras, the distance from your viewpoint to the nearest point on the upper line O-B is 5 km.

To the observer at O, the altitude angle (call it Ө_B)to point B is ~37°.. (I'm rusty on my trig, but I think that's right..)

Next, let's put some points a reasonable distance along the two lines, say 8km to the right and left... I'll call the first of these points LL, for the one on the lower/closer line at the left and LR to the right, and UL on the upper line to the left, UR on the right.

Now, to the observer at O, the angle (call it Ө_UL) to point UL is ~18.5°.. (As is the angle from O to point UR, Ө_UR.)

Here's a picture (neither to scale or perspective-corrected, obviously!) - if it isn't clear, or I've typo'd anything, let me know...
perspective_1.gif

Before I move on (which may be a day or so..), is there anything that is unclear or disputed with the image or my statements above?

Please note, I will be covering what happens when you change the 'ground' level later - I'd appreciate it if we don't start changing that aspect around just yet, for reasons that will become clear (I hope!) later - but don't worry, I am NOT avoiding it! By that time, the problem/anomaly (and it's quite a nasty one!) should be making itself clear (if it isn't already) - and if not, I'll have to concede defeat..

[Hornblower]

All we have from Biogroovy in the OP is that he saw a laser beam that appeared curved to him, and that it appeared to match the curvature of the Earth’s surface. Since he presumably expected the laser beam to be straight, he wondered if the Earth might actually be flat.

I don’t know what he meant by “what appeared to be the curvature of the Earth”, and since he has not been back it is hard to get a clarification. In general a large salt flat will appear flat, especially if surrounded by higher ground that prevents seeing an effect like that of a ship disappearing over the horizon at sea. At least that was my perception in the Bonneville Salt Flats in Utah.

I just now checked out the perspective effects of looking at parallel lines extending far in both directions from a nearby viewpoint. I stood close to a long straight wall in my living room and initially just stared at the center directly in front of me. The top and bottom edges looked reasonably straight and parallel, though having them in my upper and lower peripheral vision made them rather indistinct because of the inherent lack of focus. When I looked directly at the midpoint of each one, that line definitely looked straight.

Next I turned toward the corner and stared at it. Now the top and bottom edges looked straight and diverging toward me, just as the railroad track did in an earlier post. Looking at the other corner gave the same view except for being reversed.

So far these are static views. When I started sweeping my vision from one corner to the other the visual perception underwent a major addition. Now I had a progressive sensation built up by scanning and memory. As I swept along the wall, the top and bottom edges started out as diverging, leveled off as my vision swept over the midpoint, and finished by converging. The overall perception now was vividly that of a pair of curved lines, very similar to what the fisheye image showed with the crepuscular rays. The crucial difference in viewing the latter is that it is a static view of a distorted image of a 180 degree scene, reduced to a much smaller field of view, while the former is not a single static view but rather a sequence built up by the brain into an integrated perception. When I stare at any part of the scene, I don’t see much of anything in the peripheral area unless it is moving or otherwise changing, or has very high contrast over relatively large areas. I need to be scanning to build a detailed visual perception of the whole scene.

[chrlzs]

When I looked directly at the midpoint of each one, that line definitely looked straight.
...
As I swept along the wall, the top and bottom edges started out as diverging, leveled off as my vision swept over the midpoint, and finished by converging.
..
The overall perception now was vividly that of a pair of curved lines, very similar to what the fisheye image showed with the crepuscular rays. The crucial difference in viewing the latter is that it is a static view of a distorted image of a 180 degree scene, reduced to a much smaller field of view, while the former is not a single static view but rather a sequence built up by the brain into an integrated perception. When I stare at any part of the scene, I don’t see much of anything in the peripheral area unless it is moving or otherwise changing, or has very high contrast over relatively large areas. I need to be scanning to build a detailed visual perception of the whole scene.

That's a really useful and perceptive (this time no pun intended!) set of observations - thanks, Hornblower! I've bolded a couple of bits that I will come back to later - they allude to what I am trying to demonstrate... You have also hinted at the superb ability of your mind to sort/straighten all this stuff out so that you aren't aware of what it is up to... Which makes my job of proving that it *isn't* illusory, a fair bit harder..

But I'll get there!

[grapes]

Thanks Hornblower, nice observations!

Before I move on (which may be a day or so..), is there anything that is unclear or disputed with the image or my statements above?

All dimensions look OK, to nearest two decimals.

I would add another orthogonal line at C, 1.25 "behind" (from the observer, line C is completely hidden by line A) A, but you don't need to add it to the illustration. Since B is also orthogonal, the angle OBC will be 90 degrees, so the diagram with only B and C (with O) will be identical to your A and B (with O) diagram, with all dimensions multiplied by 1.25. We can go one step further and use units of 1.25 kilometer for the OBC diagram, and the OBC diagram will be exactly the same as your OAB diagram, just turned over.

I do that because anything you say about the straightness of A can be applied to B, by the same reasoning. Likewise, anything you say about the curvature of B can be applied to A.

Please note, I will be covering what happens when you change the 'ground' level later - I'd appreciate it if we don't start changing that aspect around just yet, for reasons that will become clear (I hope!) later

I'm not sure if what I've done is what you mean by changing the level, but regardless I think it goes to the key point, quickly.

[Swift]

As Biogroovy has abandoned the discussion, and the thread has evolved into a discussion of mainstream science, the thread is moved from ATM to S&T. Carry on...

[grapes]

I think I see where you are going here. In order to represent both lines in a 2D illustration, you assume one is going to be the straight one ("level"), and the other has to curve. My point is that you can choose either line to be straight, because there is no geometrical bend to either, when you look at each one.
ETA:

Anyway, for further reading while waiting for me to possibly self-destruct , you may like to also look at this page:
http://termespheres.com/6-point-perspective/
in particular the discussion of 4-, 5- and 6-point perspectives, and critically, why those techniques are required, and what is said/shown about supposedly 'straight' lines.

Nice!

I guess you are talking about the discussion in the four-point perspective, where parallel lines are said to be like footballs?

The Termesphere ("6 point perspective"? no that's different) looks like it might be like the gnomic projection in reverse. All straight lines would get mapped to great circles.

[chrlzs]

I think I see where you are going here. In order to represent both lines in a 2D illustration, you assume one is going to be the straight one ("level"), and the other has to curve.

Exactly, but I wish to take this a bit further, in particular to prove my original assertion that it is not an illusion, it is the inevitable distortion that is inherent in not just our perception, but also *any* attempt to represent the mapping of the line onto a projection, be it a camera, line drawing, whatever.

My point is that you can choose either line to be straight

Yes...

because there is no geometrical bend to either, when you look at each one.

I disagree with this because of your proviso! At any given moment you have to look at one, and when you do then the other *is* geometrically bent! That's where I'm heading with the illustration I shall continue with later. There *is* a geometrical bend in *any* line/object that is not on your plane of view. Under normal circumstances it isn't all that significant, and if you constrain your field of view to ~90° or less, it can be corrected for (and is, in high quality lenses, for example) and your brain 'fixes' it. But what such a lens (and your brain) is doing is distorting the actual view! The lines *should* be bent, and if you are making images that span more than 90 degrees like I do, eg a 240° panorama, you quickly learn that lines not in the plane of view *must* bend. If you tilt the camera up/down in order to straighten a line, then all other lines above and below that one are bent (except those now directly above and below, of course).

So that's the anomaly. It's why the Moon can look as if it is being illuminated from above horizontal when the Sun has set. It's why crepuscular/anti-crepuscular rays *do* bend upwards and then back down to the opposing vanishing points. It's why rail lines bend outwards at your feet and then re-converge at each horizon point. Yes, you can line yourself up with one line, but then the others get bent...

I maintain it is not a 'confused brain' issue, except that it is counter-intuitive and extremely weird. But real.

BTW, the Sun and Moon are quite nicely placed at the moment to demonstrate the problem - take a look at the Moon at about an hour after sunrise.. The illumination is clearly coming from *above* horizontal, yet the Sun is lower to the horizon...? If you tilt your head up to 'level' the two of them, it's a straight line alright, but now the horizon is horribly bent even though your brain does its best to hide that..

I've taken some photos this am to illustrate the issue and I'll post them later when I have time to make them look presentable.. Gotta work now..

[grapes]

Yes, you can line yourself up with one line, but then the others get bent...

As I showed earlier, the viewpoint O is lined up with both line A and line B, there's no movement necessary geometrically.

I maintain it is not a 'confused brain' issue, except that it is counter-intuitive and extremely weird. But real.

If someone cannot look at a straight line and see it as a straight line, it is an optical illusion. There's no getting around that. The geometry is not going to differentiate between the two.

[grapes]

Universe Today article today, crepuscular rays and anti-crepuscular rays, at the same time:
http://www.universetoday.com/95762/a...puscular-rays/