[Bobby Shafto]

So, the flat earth theory is Perspective (explanation of H and converging sight lines) and Surface Irregularities at H (waves/swell).

Now, how do we calculate distance to H?

I've got an idea for that... let's find that with an experiment!
https://forum.tfes.org/index.php?topic=10016.0

I don't think that will find the value of H per Flat Earth (ENaG-version) Theory. Though I have yet to get confirmation, my interpretation of Rowbotham's reasoning for Perspective and the horizon is:



If that's correct, then for a 100' vantage point, H will be around 65 miles away.

[ICanScienceThat]

So, the flat earth theory is Perspective (explanation of H and converging sight lines) and Surface Irregularities at H (waves/swell).

Now, how do we calculate distance to H?

I've got an idea for that... let's find that with an experiment!
https://forum.tfes.org/index.php?topic=10016.0

I don't think that will find the value of H per Flat Earth (ENaG-version) Theory. Though I have yet to get confirmation, my interpretation of Rowbotham's reasoning for Perspective and the horizon is:



If that's correct, then for a 100' vantage point, H will be around 65 miles away.

I don't think I understand this diagram. What are these lines? I thought these lines are the imagined (ie incorrect) projections of perspective lines as seen from a side-view.  No?

[Bobby Shafto]

these lines are the imagined (ie incorrect) projections of perspective lines as seen from a side-view.  No?

Yes, but the key is to see all lines don't converge at a single distance. Depends on origin of perspective line's vertical distance away from the eye, with point of convergence forming a distant angle of 1 arcminute (that's for avg human sight).

But the opaque ground plane creates a limit on convergence distance. Beyond that, lines higher than the eye's level above that ground plane meet at convergent points beyond H, where the ground plane appears to level off.

Vanishing Point is not a single point but variable, with H just being special because the ground is not transparent.

It's mainly described in http://www.sacred-texts.com/earth/za/za32.htm]this chapter of Earth Not a Globe. I just updated the diagram with a formula for calculating distance to H.

[Bobby Shafto]

Surface irregularities at the horizon (H) and the Perspective Theory for why there is H at all are just two components of a Flat Earth explanation for what appears in that first picture. There's another one that is evident in the picture from vantage point #1 but not vantage point #2 so much.

[ICanScienceThat]

these lines are the imagined (ie incorrect) projections of perspective lines as seen from a side-view.  No?

Yes, but the key is to see all lines don't converge at a single distance. Depends on origin of perspective line's vertical distance away from the eye, with point of convergence forming a distant angle of 1 arcminute (that's for avg human sight).

But the opaque ground plane creates a limit on convergence distance. Beyond that, lines higher than the eye's level above that ground plane meet at convergent points beyond H, where the ground plane appears to level off.

Vanishing Point is not a single point but variable, with H just being special because the ground is not transparent.

It's mainly described in http://www.sacred-texts.com/earth/za/za32.htm]this chapter of Earth Not a Globe. I just updated the diagram with a formula for calculating distance to H.

This is all fine, but why are they drawn as lines rather than curves?

[Bobby Shafto]

This is all fine, but why are they drawn as lines rather than curves?

Why should the light lines be curved?

The challenge is to explain a visual phenomenon. Is the light curving?

In the Perspective explanation, light is straight, and it's the angle at convergence and the limits of optics that determine what is perceived. vanishing points, like those lying at a horizon, and those beyond it but aligned with the horizon, are perceptual.

[ICanScienceThat]

This is all fine, but why are they drawn as lines rather than curves?

Why should the light lines be curved?

The challenge is to explain a visual phenomenon. Is the light curving?

In the Perspective explanation, light is straight, and it's the angle at convergence and the limits of optics that determine what is perceived. vanishing points, like those lying at a horizon, and those beyond it but aligned with the horizon, are perceptual.

In the diagram, the point labeled "H" is a point at eye-level where an object of height "C" above eye level appears to be 1 arcminute above eye-level. Correct? The straight-line side-view path between them should show the line between C and H as horizontal because C is still physically above H by the exact same height it started.

If this diagram is attempting to show how the apparent height of C above eye-level varies with distance, that is a curve and not a straight line. I cannot come up with any reasonable explanation for this diagram. What am I missing? I must be completely misunderstanding the diagram.

[Bobby Shafto]

In the diagram, the point labeled "H" is a point at eye-level where an object of height "C" above eye level appears to be 1 arcminute above eye-level. Correct?

I would articulate it that the object (or portion of an object) that is of height "C" above eye level at the location of the observer appears to be less than 1/60 of a degree (1 arcminute) at some distance, called H. So, I think "yes" to your question, but maybe clarified a bit.

The straight-line side-view path between them should show the line between C and H as horizontal because C is still physically above H by the exact same height it started.

No. That would be a separate line, like one parallel to AB in this diagram, but originating at C. (Rowbotham simply didn't draw that it.)



And I didn't either on my adaptations. But there is acknowledged the actual line (non-changing height) as distinct from perceptual (diminishing heights).

If this diagram is attempting to show how the apparent height of C above eye-level varies with distance, that is a curve and not a straight line. I cannot come up with any reasonable explanation for this diagram. What am I missing? I must be completely misunderstanding the diagram.

The line represents what we see. Not what we graph. Though the line may curve on a graph, it's because the x axis plot points are equidistant (1x, 2x, 3x, etc.) And that's true in actuality, just as the height doesn't really change. But perceptually, not only is the height appearing to diminish, but the distance between points as they recede into the distant doesn't appear equal but compressing. And if height and distance intervals appear to be decreasing in a direct relation, the line stays straight.

Imagine replacing the (1x, 2x, 3x, ....) intervals in your experiment with measured intervals as they appear on a 2D projection or photograph. The line doesn't curve because the intervals appear to be decreasing too.

H is just a perceptual threshold. Any vanishing point is an apparent one. As Tom put it, it's how the world "presents itself to us" (or something like that). The world doesn't really end at a vanishing point, but vanishing points aren't infinitely away. A horizon isn't an infinite distance away. It's finite. I've only added a calculation to it that I don't believe has been expressed before. It's a simple geometric calculation based on optical resolution, and for the human eye, Rowbotham's source cited put that at 1/60th of a degree.

It's just that instead of all points appearing to compressing at the same vanishing point (or multiple points on a vanishing line), lines that originate higher will merge/compress at a distance further than the one that defines H. And because the earth is not transparent, those points will appear to sink behind it.

Look, you do understand that I don't hold to this myself. I'm only trying to honestly present it as best I understand it. I think I apprehend the rationale, but if I'm botching it, a real FE proponent is always free to take over. My main point is that I put a distance to H and not just leave it vague. It may not be a physical point in space, but from the point of the observer, H must occur at a finite distance away from him or her. I'm putting a value to it, based on the ENaG explanation for phenomena at the horizon.

[ICanScienceThat]

In the diagram, the point labeled "H" is a point at eye-level where an object of height "C" above eye level appears to be 1 arcminute above eye-level. Correct?

I would articulate it that the object (or portion of an object) that is of height "C" above eye level at the location of the observer appears to be less than 1/60 of a degree (1 arcminute) at some distance, called H. So, I think "yes" to your question, but maybe clarified a bit.

The straight-line side-view path between them should show the line between C and H as horizontal because C is still physically above H by the exact same height it started.

No. That would be a separate line, like one parallel to AB in this diagram, but originating at C. (Rowbotham simply didn't draw that it.)



And I didn't either on my adaptations. But there is acknowledged the actual line (non-changing height) as distinct from perceptual (diminishing heights).

If this diagram is attempting to show how the apparent height of C above eye-level varies with distance, that is a curve and not a straight line. I cannot come up with any reasonable explanation for this diagram. What am I missing? I must be completely misunderstanding the diagram.

The line represents what we see. Not what we graph. Though the line may curve on a graph, it's because the x axis plot points are equidistant (1x, 2x, 3x, etc.) And that's true in actuality, just as the height doesn't really change. But perceptually, not only is the height appearing to diminish, but the distance between points as they recede into the distant doesn't appear equal but compressing. And if height and distance intervals appear to be decreasing in a direct relation, the line stays straight.

Imagine replacing the (1x, 2x, 3x, ....) intervals in your experiment with measured intervals as they appear on a 2D projection or photograph. The line doesn't curve because the intervals appear to be decreasing too.

H is just a perceptual threshold. Any vanishing point is an apparent one. As Tom put it, it's how the world "presents itself to us" (or something like that). The world doesn't really end at a vanishing point, but vanishing points aren't infinitely away. A horizon isn't an infinite distance away. It's finite. I've only added a calculation to it that I don't believe has been expressed before. It's a simple geometric calculation based on optical resolution, and for the human eye, Rowbotham's source cited put that at 1/60th of a degree.

It's just that instead of all points appearing to compressing at the same vanishing point (or multiple points on a vanishing line), lines that originate higher will merge/compress at a distance further than the one that defines H. And because the earth is not transparent, those points will appear to sink behind it.

Look, you do understand that I don't hold to this myself. I'm only trying to honestly present it as best I understand it. I think I apprehend the rationale, but if I'm botching it, a real FE proponent is always free to take over. My main point is that I put a distance to H and not just leave it vague. It may not be a physical point in space, but from the point of the observer, H must occur at a finite distance away from him or her. I'm putting a value to it, based on the ENaG explanation for phenomena at the horizon.

Ok I get you. Perhaps easiest of we don't sweat those lines too much at all and just work the math then. Does that work? It's quite trivial to do this. The angular size of the object is simply arctan(height/distance). Simple as that. So the distance at which an object appears to be 1 arcmin is simply its height/tan(1/60). So that does mean the distance at which an object is 1 arcmin tall varies linearly with the height of the object. (I'm saying "height" only because of the side-view... works for height or width of course.)

Is that what we're talking about? I see you've mentioned that... 100' object vanishes at 65 miles... right! I'm not really certain what the point of the diagram is TBH. It doesn't show anything about objects vanishing bottom-up, and it can't explain the sun going past the horizon as far as I can tell.

[HorstFue]

I neither do understand this diagram.
Rowbotham is jumping back and forth from real world to perceptual view/image plane.

• Real world lines A-B and C-D would have to be extended by far (near infinity)
• Only than both lines C-H and A-W in perceptual view/image plane could be a representation of lines A-B and C-D with possible vanishing points at either H or W. (there's only one vanishing point possible)
• Now back in real world Robotham applies eye resolution from real world for the distance to the vanishing point H from perceptual view
• And now comes the ultimate kick: Rowbotham is turning around perception. The distance, defined by the limit of eye resolution is given as real world distance to the vanishing point in perceptual view and is defined by observers hight and not by the size of the observed object.

Think about this again: "Real world distance to the vanishing point in perceptual view"

[Bobby Shafto]

Ok I get you. Perhaps easiest of we don't sweat those lines too much at all and just work the math then. Does that work? It's quite trivial to do this. The angular size of the object is simply arctan(height/distance). Simple as that. So the distance at which an object appears to be 1 arcmin is simply its height/tan(1/60). So that does mean the distance at which an object is 1 arcmin tall varies linearly with the height of the object. (I'm saying "height" only because of the side-view... works for height or width of course.)

Yes; height above the observer's eye level.

Is that what we're talking about? I see you've mentioned that... 100' object vanishes at 65 miles... right!

I didn't say it that way, but yes, sort of. (I think?) With eye level at 100', the horizon figures to be 65 miles away. So anything that doesn't exceed that 100' height off the ground will have "vanished" on or before the horizon at 65 miles away or less. And it will have "vanished" as you would expect: to a speck and then gone. No "bottom up" vanishing.

Such "vanishing" you can telescope back into view though because your optical resolution improves to better than 1/60th of a degree, and H is extended (within limits imposed by atmosphere).  Those things (or details) aren't beyond the horizon defined by eye level height.

But objects or aspects of objects that are above 100' above a 100' observer don't "vanish" at 65 miles. Vanishing point for them is at a distance from the observer >65 miles, but they do so beyond that apparent horizon, which eclipses the lower portions first before they vanish because of that 65 mile horizon point.

I'm not really certain what the point of the diagram is TBH. It doesn't show anything about objects vanishing bottom-up, and it can't explain the sun going past the horizon as far as I can tell.

I not doing it justice, and that's probably because even just typing it out I see flaws in the explanation. If my 218mm focal length is improving optical acuity to better than 1/60th of a degree, then that should extend H out to some greater distance and 65 miles is no longer the point at which "hull first vanishing" is happening.

I'm also not keen on the reasoning for why anything that is merging with the horizon but beyond the point of H would disappear "bottom first," though I'm trying to include waves/swell or whatever irregularities and undulations are at the flat plane that could impact appearance at H (including one big factor I've been holding for last). But this particular ship was only 25 miles away at the time, so whether with the naked eye (H~65 miles) or camera zoom (H>65 miles), I would not expect it to appear the way Rowbotham/ENaG explain. If the ocean surface is flat, then something else has to be going on, either to supplement or replace what ENaG covers in chapter XIV.

[Bobby Shafto]

I neither do understand this diagram.
Rowbotham is jumping back and forth from real world to perceptual view/image plane.

• Real world lines A-B and C-D would have to be extended by far (near infinity)
• Only than both lines C-H and A-W in perceptual view/image plane could be a representation of lines A-B and C-D with possible vanishing points at either H or W. (there's only one vanishing point possible)

I'm tracking with you, but that last parenthetical isn't true. Right? There are infinite vanishing points. Even if just focusing on the horizontal plane, the Rowbotham explanation of the natural law of perspective is that a single VP at which all things appear to converge is wrong. The horizon is a vanishing point, but vanishing points for other things lie beyond it.

He uses the example of a disc with a smaller circle feature on it. As the disc gets more distant, the circle on the disc will reach 1 arcminute before the disc itself. The VP for the circle is sooner (closer) than the disc as a whole. You will see the disc but not be able to make out its smaller feature. It's all a function of that ability to resolve angular distance of a certain measure.

But the "bottom first" element has to do with proximity to an opaque plane. If you look upward, perspective works the same way, but there's no eclipsing by a non-transparent planar surface. That's not true when looking along a ground plane you can't see through. Then, those elements with angular dimensions smaller than what you can optically resolve "blend" or "merge" (whatever the preferred term is) with the opaque, obscuring plane, and will "vanish" before the elements that are still angularly resolvable -- I'm "word salading" now trying to convey this in stream of thought -- from the opaque plane surface.

I got off track. Point is there isn't one VP.  It's this point that Rowbotham makes to explain why not everything vanishes to a point and how an object can become obscured at a horizon line bottom-up.

• Now back in real world Robotham applies eye resolution from real world for the distance to the vanishing point H from perceptual view
• And now comes the ultimate kick: Rowbotham is turning around perception. The distance, defined by the limit of eye resolution is given as real world distance to the vanishing point in perceptual view and is defined by observers hight and not by the size of the observed object.

Think about this again: "Real world distance to the vanishing point in perceptual view"

I have no good response to this. I will say that as a devils' advocate presenting the Rowbotham reasoning, I've applied a distance calculation to H as this eye resolution reasoning goes, but I've never seen actual advocates do that. It's supposed to be a finite point, but one I never see given a finite value. How "the world presents itself" in a perceptual sense vs. a "real world distance" is off-putting to me too. I have to leave that to Tom or someone vested in the idea to defend. I can't.

[Bobby Shafto]

Surface irregularities at the horizon (H) and the Perspective Theory for why there is H at all are just two components of a Flat Earth explanation for what appears in that first picture. There's another one that is evident in the picture from vantage point #1 but not vantage point #2 so much.

No one has mentioned "Convergence Zone" yet. That's the 3rd element that I believe works in conjunction with surface undulations + perspective theory to create the appearance that objects sink beyond the horizon.

Look at the picture and notice the distortion in the band right at the apparent horizon. 

[SiDawg]

Also "swell" would not magically form a reasonably static line in front of a boat... like the ocean just swells up at one point and stays there? Or like it's continually "swelling" up and down and yet somehow the line in front of the boat doesn't rise up and down i.e. there's a huge swell but also no motion to the swell at all? hmmmmm. Obviously static photos don't show this lack of movement (ironically) but plenty of videos that do.

[Bobby Shafto]

Also "swell" would not magically form a reasonably static line in front of a boat... like the ocean just swells up at one point and stays there? Or like it's continually "swelling" up and down and yet somehow the line in front of the boat doesn't rise up and down i.e. there's a huge swell but also no motion to the swell at all? hmmmmm. Obviously static photos don't show this lack of movement (ironically) but plenty of videos that do.

I have 3-4 photos of this view. I'd have to check the timestamps, but at no time did the boat or the horizon change vertically.

Ocean turbulence could only possibly account for 2-3'.

But on to the point of "Convergence Zones," there's a surface inversion layer (common as the morning heats up) that's creating an inferior mirage, evident from 100' but not from 400'.

[HorstFue]

I neither do understand this diagram.
Rowbotham is jumping back and forth from real world to perceptual view/image plane.

• Real world lines A-B and C-D would have to be extended by far (near infinity)
• Only than both lines C-H and A-W in perceptual view/image plane could be a representation of lines A-B and C-D with possible vanishing points at either H or W. (there's only one vanishing point possible)

I'm tracking with you, but that last parenthetical isn't true. Right? There are infinite vanishing points. Even if just focusing on the horizontal plane, the Rowbotham explanation of the natural law of perspective is that a single VP at which all things appear to converge is wrong. The horizon is a vanishing point, but vanishing points for other things lie beyond it.

Rowbotham is wrong for parallel lines. All parallel lines from real world, in perceptual view merge at the one and only vanishing point. The vanishing point in perceptual view represents infinity in real world, no other "point" can be beyond infinity, and no matter how far apart these real world lines are they appear to merge at the vanishing point in perceptual view.
Horizon: What horizon?
The horizon defined by perspective, where viewing plane (eye level) and ground plane of the observer merge? Ok, yes this horizon includes the vanishing point for any bundle of parallel lines, which are parallel to observers viewing plane.
The horizon defined by globe earth curvature. No, this horizon clearly is closer than infinity. That's another type of 'horizon'.
The horizon defined by observers eye resolution. Sorry, that's no horizon, that's something blurring your view. Perspective is pure geometry and mathematics, there's nothing in it for limited eye resolution.
You could instead calculate that point, where in real world eye resolution would find it's limit and than apply perspective.

He uses the example of a disc with a smaller circle feature on it. As the disc gets more distant, the circle on the disc will reach 1 arcminute before the disc itself. The VP for the circle is sooner (closer) than the disc as a whole. You will see the disc but not be able to make out its smaller feature. It's all a function of that ability to resolve angular distance of a certain measure.

The vanishing point represents infinity in real world. The ability to resolve angular distance does not define a 'horizon'. Playing with the zoom of your camera does not change the vanishing point, or? It's just more - or less blur. Or is the vanishing point also 'moved' by atmospheric haze?

But the "bottom first" element has to do with proximity to an opaque plane. If you look upward, perspective works the same way, but there's no eclipsing by a non-transparent planar surface. That's not true when looking along a ground plane you can't see through. Then, those elements with angular dimensions smaller than what you can optically resolve "blend" or "merge" (whatever the preferred term is) with the opaque, obscuring plane, and will "vanish" before the elements that are still angularly resolvable -- I'm "word salading" now trying to convey this in stream of thought -- from the opaque plane surface.

Again this is just a blur. Wouldn't the opaque plane near that - I define a new term for it - 'blurring point' also be blurred in a similar way? That all is merged into one bigger blurred zone, where no one can decide, where the plane ends and the 'vanished' object begins.

I got off track. Point is there isn't one VP.  It's this point that Rowbotham makes to explain why not everything vanishes to a point and how an object can become obscured at a horizon line bottom-up.

"Bottom-up": In all examples Rowbotham gives, I see a whole object 'vanish'. It's not the lower part of the ship, it's the hull, clearly distinguishable from the rest of the ship. It's the black boots of the soldiers, in clear contrast to their uniforms. It's the white part of these circles mentioned above, in clear contrast to the bigger black ones.

Another try to explain: Think about a bigger object appearing close above a clearly visible horizontal line. With enough resolution you will see this as described. Now decrease resolution: The horizontal line will blur and appear to get wider: There's no significant change on the bottom side, because the plane below is affected by a similar blur. But on the top side of the blur seems to extend into the object above the horizontal line.

[Bobby Shafto]

This is all fine, but why are they drawn as lines rather than curves?

My answer to this above was that though the elevation angle is inversely proportional to distance, so is the perceived interval distance, which offset the "curve" and restores linearity.

But I've been trying to work out a way for the ENaG orthogonal depiction of merging lines to explain a sunset (or appearance of things at a horizon), and something occurred to me that I failed to realize. 

I'm going to post it on the Sunset from a Drone discussion topic, but I thought it was apropos to this discussion as well since I've tried to delve into the Perspective Theory here too, though not about the sun specifically. I'll append a cross link after I post it; I just wanted to make note of it here since it's a counter to my reply to you and references the conclusion of your vanishing point experiment.

-------

Since that other topic was exiled to AR, I'll just amend this post with what I posted there.

-------
Not to imply that the curved earth explanation is the gold standard, but I've not encountered anyone who objects to rationale of where the horizon is and why objects on the surface of a convex surface beyond it can be hidden.

Whether you believe the surface of earth is convex or not, this makes sense:



Increasing "eye level" height above the surface extends the horizon, diminishes geometric drop and changes the amount hidden (or eliminates it entirely).  The question is whether or not this is true for the earth, and that depends on whether or not the earth's surface is convex. But I've never heard anyone argue with the geometry or the logic of why it would work this way IF the earth was spherical.

Instead, an alternative explanation is offered for the horizon and objects or features appearing beyond it. Trying to compare and contrast this Perspective alternative, using Earth Not a Globe's diagram as a starting point, we might get something like this:



Increasing eye level height above surface extends the horizon and changes what's hidden too, but how? The 1/60th degree of resolution is an explanation for the apparent merging of perspective planes/lines that don't actually merge but just appear to. And the opaque flat surface merging to a vanishing point with the transparent plane that is 2x the height of eye level somehow is responsible for the horizon line AT eye level. There is no "dip." There is no calculation for how much is hidden for distances and heights beyond the horizon

It makes sense that increasing eye level height will extend the distance to the horizon, but there is still no alternative explanation to the curved-earth approach to explain why there is a hidden zone beyond the horizon or how to calculate such a thing.

The curved, geometric model has a Drop value from level sight. The flat, perspective model doesn't.
The curved, geometric model can calculate the amount hidden beyond the horizon. The flat, perspective model can't.
The curved, geometric model acknowledges that the horizon doesn't remain level with the eye but dips. The flat, perspective model relies on the horizon always being level with the eye, with no "dip."

I think if you resolve those issues, the flat earth explanation of Perspective gains merit as a competitor to curvature being the reason for horizons and hidden features beyond horizons.

------

This is a rendition of how a parallel overhead plane sun can appear to set, according to the Rowbotham "Natural Law of Perspective."



I've seen this same explanation from the more popular YouTube Flat Earth advocates too using Perspective to explain how the sun appears to descend and merge with a vanishing point horizon.

One of the first critiques of this is that the sun should, according to Perspective, also decrease in size to a point by the time it reaches the vanishing line of the horizon.



Some say it, in fact, does.

This site says it doesn't, but that's because of a magnification effect of the atmosphere as the sun has to pass through greater and greater density of air/moisture, and that effect offsets the expected reduction in sun size.

But also observed with Perspective is that it's not just the apparent size of things that gets smaller with distance away from the viewer. It's also the apparent longitudinal distance compresses. For instance, evenly space things like railroad ties or lamp poles will appear to get closer and they get smaller. To apply this to a setting sun as related to Perspective:



If atmosphere water vapor offsets the diminishing in perceived size, what can account for the offset of diminishing interval distance?

Distance of sun movement along the downward angled line of Perspective is always depicted as being equidistant to its actual straight parallel path position. But if Perspective is at work, the distance should lag too. If not, then why not?