
In order to explain how the sun touches the horizon, flat earthers have come up with some peculiar theories related to "perspective".
In particular, this video (https://youtu.be/AfgbqFyiisQ) claims that the "side view diagram" of the angle between the sun and the horizon is not accurate. To arrive at his conclusion, the author hopelessly confuses how angles and perspective relate to each other.
What is an angle?
An angle is simply a difference in direction between two lines. Let's say you have three objects: A, B, and C. Stretch a string between object A and B. Now stretch a string between object A and C. Use a protractor to measure the angle between them. This will give you the actual angle between the objects. Now, depending on where an observer is standing, the angle might appear to be different, but the actual angle measured by the physical protractor gives us the actual, physical angle between the objects.
What is an orthographic projection?
First, let's talk about the "side view diagram". The creator of the video claims that the "side view diagram" doesn't accurately reflect "perspective". This is true. It's not supposed to show exactly what the person sees, because it isn't from the perspective of the person looking at the sun.
The "side view diagram" is an orthographic projection. (https://www.quora.com/Whatisthedifferencebetweenorthographicprojectionandisometricprojection) An orthographic projection has a few useful properties:
1. All lines that are parallel in reality are also parallel in an orthographic projection.
2. For all lines that are parallel to the projection plane, the angles between the lines are accurate. They are exactly the same as the angles measured in reality.
An orthographic projection isn't meant to show perspective. It's specifically meant not to show perspective, so that we can accurately portray angles. The reason we use an othographic projection is because it accurately portrays the angles we are interested in.
How do the angles change when we actually take into account perspective?
(http://imgur.com/Tt6gsDL.png)
The diagram on the left is an orthographic projection. If we had 3 objects located at the ends of the red lines on a grid in reality, the angle portrayed by the orthographic projection will exactly match the angle measured in reality. Now, imagine the grid rotating away from us. The two diagrams on the right represent what you would see, taking into account perspective. Notice the "perspective lines" that have appeared.
1. The grid lines are no longer parallel.
2. The angle between the red lines is not the same.
These perspective diagrams are no longer useful for measuring angles. The angle that the red lines appear to meet at is completely dependent on the angle at which we view the grid. The same goes for the angle of the "perspective lines".
Now imagine that the grid continues to rotate until it is pointing directly away from us. The "perspective lines" are now just a single vertical line. The red lines are now just a single vertical red line. This is representative of what we would see from a first person perspective. There is no point in drawing perspective lines, or measuring any angle with a protractor, because they are both just vertical lines. Keep in mind, throughout this entire process, the physical angle between the red lines has not changed.
At 2:40 (https://youtu.be/AfgbqFyiisQ?t=2m40s), the creator of the video tries to incorporate "perspective lines" into an orthographic view in order to make it more representative of what we would see. However, from a first person perspective, the "perspective lines" are just a single vertical line, and there is no discernible angle to measure. He is trying to combine a first person view with an orthographic side view, and settled somewhere arbitrarily in the middle. The angles he is measuring are completely arbitrary. The "perspective lines" he is using are completely arbitrary.
Orthographic projections are useful for accurately portraying angles.
Perspective projections are useful for portraying what we see.
To sum up:
When we ask "what is the angle between the sun and the horizon?" we are talking about the actual, physical angle. If you point one stick at the sun, and another stick at the horizon, what angle do those sticks meet at? This angle is accurately portrayed by the "side view diagram" (orthographic projection), and has absolutely nothing to do with perspective.

As you turned the scene the height of the far end of the terminal arm dropped in height, due to perspective.
(http://i68.tinypic.com/30u3ci1.png)

As you turned the scene the height of the far end of the terminal arm dropped in height, due to perspective.
Indeed it did! Very perceptive! What's your point?

As you turned the scene the height of the far end of the terminal arm dropped in height, due to perspective.
Indeed it did! Very perceptive! What's your point?
Incidentally, we can calculate the exact drop in height using basic math!
Angular diameter between object and horizon = arctan(height of object / ground distance to object) < Notice, this gets smaller as the distance increases. This is where the "perspective lines" come from.
For example, the sun:
Angular diameter between sun and horizon = arctan(3000 miles / (5600 miles)) = 28 degrees, which would be a rough estimate of the height of the sun at sunset/sunrise on a flat earth.

Draw it. Your math assumes an outofuniverse SIDE VIEW without regards to perspective. We saw in your illustration that when the side view angle was turned it became lower to the horizon.
Considering that the horizon is where everything merges at the vanishing point, I would say that at the horizon the sun is 0 degrees and whatever math you are using is flawed in the face of reality.

As you turned the scene the height of the far end of the terminal arm dropped in height, due to perspective.
Indeed it did! Very perceptive! What's your point?
Just to give you a heads up, Tom does not think math works after a certain distance and Pi=4.
At least that is how I interpreted posts he has made.

Draw it.
Wish granted.
(http://imgur.com/BB4hGoj.png)
Top is the orthographic side view. Bottom is from the perspective of the camera. Notice how the angles in the orthographic side view correspond to the dimensions and placement of the objects in the first person view. That's why the orthographic view is useful. Keep in mind, this isn't 100% accurate, sense any camera will have a bit of distortion due to the shape of the lens or sensor (https://photographylife.com/whatisdistortion). As long as the camera isn't very wide angle, the distortion should be minimal.
Your math assumes an outofuniverse SIDE VIEW without regards to perspective. We saw in your illustration that when the side view angle was turned it became lower to the horizon.
Funnily enough, an orthographic view does assume the viewer is "outofuniverse", in a way. In an orthographic projection, if you trace the rays of light between the object and the projection plane, they are all parallel. This implies that the viewer is at an infinite distance from the object. This is why it is technically impossible to take a perfectly orthographic picture of an object using a traditional camera. That's why we generally have orthographic diagrams, and not orthographic pictures.
This is completely beside the point though. An orthographic diagram is not meant to represent what we see. It is meant to accurately portray angles and distances. This is why it is used by engineers. So they can accurately measure angles and distances from a diagram.
Considering that the horizon is where everything merges at the vanishing point, I would say that at the horizon the sun is 0 degrees and whatever math you are using is flawed in the face of reality.
I agree that the sun is at 0° with the horizon when it sets. The reason the result of my math is wrong (28°) is because it started with a flawed assumption: that the earth is flat. The reason the math doesn't agree with reality is because the earth isn't flat in reality.
This is how models are tested. You make predictions assuming the model is correct. If the predictions disagree with reality, there is probably something wrong with the model. Since my prediction based on the flat earth model is wrong, there is probably something wrong with the flat earth model. Obviously.

Wish granted.
(http://imgur.com/BB4hGoj.png)
As we see, in order to see the angles indicated we must go to a SIDE VIEW scene which takes place outside of the universe.
In your "first person" scene we do not see any angles to measure, being impossible to draw and exist in that orientation. The positions of the objects in the first person scene are impossible to justify and calculate with just that "first person" scene alone  making your approximations entirely arbitrary without your supporting side view image to attach with it.
Top is the orthographic side view. Bottom is from the perspective of the camera. Notice how the angles in the orthographic side view correspond to the dimensions and placement of the objects in the first person view. That's why the orthographic view is useful. Keep in mind, this isn't 100% accurate, sense any camera will have a bit of distortion due to the shape of the lens or sensor (https://photographylife.com/whatisdistortion). As long as the camera isn't very wide angle, the distortion should be minimal.
At which point, according to your math, will the sun touch the vanishing point? If you check your math you will find that it is impossible for the sun to ever touch the vanishing point. It will just keep slowing down and never get to the horizon. In fact, under that math, it's impossible for anything to intersect at a vanishing point.
Since the vanishing point exists, the math is clearly an inaccurate representation of perspective.
I agree that the sun is at 0° with the horizon when it sets. The reason the result of my math is wrong (28°) is because it started with a flawed assumption: that the earth is flat. The reason the math doesn't agree with reality is because the earth isn't flat in reality.
This is how models are tested. You make predictions assuming the model is correct. If the predictions disagree with reality, there is probably something wrong with the model. Since my prediction based on the flat earth model is wrong, there is probably something wrong with the flat earth model. Obviously.
We know that it is possible for perspective lines to intersect. Under your math it is impossible for any perspective lines to intersect. Therefore your math is wrong.
Not only does it lack the intersection of perspective lines, your math has not been demonstrated to be an accurate portrayal of large scale perspective in the real world. You bring us math created by ancient civilizations to us and expect us to just assume that it is correct for the situation. Where is the evidence that it is correct for this purpose?

Draw it. Your math assumes an outofuniverse SIDE VIEW without regards to perspective. We saw in your illustration that when the side view angle was turned it became lower to the horizon.
Considering that the horizon is where everything merges at the vanishing point, I would say that at the horizon the sun is 0 degrees and whatever math you are using is flawed in the face of reality.
We have to completely disagree with your claim that "the horizon is where everything merges at the vanishing point" it clearly is not.
(http://i1075.photobucket.com/albums/w433/RabDownunder/nonLeaning%20Buildings%20%20image_zpshwffllih.png) Those buildings seem past the horizon They have certainly have not merged to a vanishing point.   (http://i1075.photobucket.com/albums/w433/RabDownunder/Diamond%20Princess%206%20about%2016%20miles%20ship%20gone%20haze_zpsrcfrwerl.jpg) The Diamond Princess (16 miles away seems past the horizon It certainly has not merged to a vanishing point..   (http://i1075.photobucket.com/albums/w433/RabDownunder/Sailing%20Boat%20nearer%20and%20Buildings%20behind%20Visible%20Horizon_zpsvtmrawto.png) Those buildings seem past the horizon They have certainly have not merged to a vanishing point.   (https://lh3.googleusercontent.com/VuqBe8otbL2RHP18oWj5poK1MToC0Zq8Xp3AxSpLrBQ=w600h392no) Chicago seems well past the horizon and yet it certainly has not merged to a vanishing point. 
Objects simply do not vanish at the horizon. There is absolutely evidence that they do, nor reason why they should.
Objects merge to a vanishing point when their visual size and contrast makes them merge with the background. Those with a very high contrast to the background can be seen from extreme distances. Stars and planets (whichever cosmology you use) have a size far smaller the quoted 1 minute of arc of the eyes resolution, yet can be easily seen by the unaided eye.
Many figures have been quoted for the visibility distance of a candleflame, the most wellfounded claims 1.6 miles.
Typical flame dimensions are about 1 cm x 3 cm.
This subtends vertical and horizontal angles at the eye of 0.8 sec arc x 2.4 sec arc, far below the 1 minute of arc of the eye's resolution, yet the candle is visible.
But a 32 mile diam extremely bright sun at the horizon distance of around 10,000 miles has a visual angle of 11 minutes of arc.
(Mind you it SHOULD be about 52 minutes f arc, but that's another issue.)
No, there is no possible way for "perspective" to make the sun disappear.
And before you claim that we don't know what happens at these "immense distances", I should remind you that the moon, planets and stars, which are far, far dimmer than the sun, can be seen at the almost the same position soon after,

There are imperfections on the earth's surface behind which things can hide, just as a dime can hide an elephant if you hold it out in front of you and the elephant is far enough away. No matter how small of an imperfection, as the perspective lines merge to 0, the imperfection will become apparent.
See the chapter Perspective on the Sea (http://www.sacredtexts.com/earth/za/za33.htm) in Earth Not a Globe by Samuel Birley Rowbotham for additional information.

At which point, according to your math, will the sun touch the vanishing point? If you check your math you will find that it is impossible for the sun to ever touch the vanishing point. It will just keep slowing down and never get to the horizon. In fact, under that math, it's impossible for anything to intersect at a vanishing point.
The sun never touched the vanishing point because the sun never disappears by shrinking to an apparent size smaller than your visual acuity. In fact the sun never changes size at all. Your question is basically nonsensical.
Since the vanishing point exists, the math is clearly an inaccurate representation of perspective.
And this conclusion is also nonsensical because your premise is nonsensical.
We know that it is possible for perspective lines to intersect. Under your math it is impossible for any perspective lines to intersect. Therefore your math is wrong.
Perspective lines intersect? Can you please prove this? How do you differentiate it from a lack of acuity in the observation?
Not only does it lack the intersection of perspective lines, your math has not been demonstrated to be an accurate portrayal of large scale perspective in the real world. You bring us math created by ancient civilizations to us and expect us to just assume that it is correct for the situation. Where is the evidence that it is correct for this purpose?
Wow, there are a ton of problems here. First, you have not demonstrated clearly that perspective lines intersect. Second, it does not matter if the math is ancient or created 1 second ago. What matters is that it works. No one expects you to assume it works, but it does. If you don't believe it, we can't prove it here, but questioning trigonometry doesn't make you a maverick, this much I know. There are a number of sciences that avail themselves of trigonometry. You should research them.

Wish granted.
(http://imgur.com/BB4hGoj.png)
As we see, in order to see the angles indicated we must go to a SIDE VIEW scene which takes place outside of the universe.
What do you mean it "takes place outside of the universe"?? It's just a diagram. It is used to portray angles and distances. If you go outside and physically measure the angle in reality, that angle should agree exactly with the one portrayed in an orthographic view. I don't understand why you are so vehemently opposed to a simple diagram. (Actually I do know... it's because you don't want to acknowledge that the earth isn't flat.;))
In your "first person" scene we do not see any angles to measure, being impossible to draw and exist in that orientation. The positions of the objects in the first person scene are impossible to justify and calculate with just that "first person" scene alone  making your approximations entirely arbitrary without your supporting side view image to attach with it.
It certainly wasn't arbitrary. I measured and labelled quite carefully, as you can see for yourself. It's all just simple proportions. The camera's vertical field of view is 52°. If an object has an angular diameter of 26° in the orthographic diagram, then the object will take up half the vertical room in the photo. In the above picture, the conversion is 7.7 pixels per degree. The 2.3° wide object takes up 18 pixels. No, it certainly is not arbitrary.
However, the real question is: does it correspond to reality? Yes, it does, assuming negligible distortion in the camera. You can easily go test this out for yourself. Grab a camera and set up several objects. Carefully measure their distances, heights, widths, etc. Calculate their angular diameter, and angular distance from each other relative to the camera. Compare these angles to their size in the picture. Assuming the camera has a relatively narrow field of view, then I can assure you, it does.
Top is the orthographic side view. Bottom is from the perspective of the camera. Notice how the angles in the orthographic side view correspond to the dimensions and placement of the objects in the first person view. That's why the orthographic view is useful. Keep in mind, this isn't 100% accurate, sense any camera will have a bit of distortion due to the shape of the lens or sensor (https://photographylife.com/whatisdistortion). As long as the camera isn't very wide angle, the distortion should be minimal.
At which point, according to your math, will the sun touch the vanishing point?
Assuming it continues to travel in a straight path? Technically, never. However, it can get arbitrarily close to the vanishing point. And our eyes can't distinguish arbitrarily small details. A commonly quoted value for the minimum angular diameter that our eyes can distinguish is 0.02°. If the sun is indeed 3000 miles high, this would correspond to a distance of 8.6 million miles. So, on a flat earth, the sun could possibly appear to touch the horizon when it travels 8.6 million miles away.
If you check your math you will find that it is impossible for the sun to ever touch the vanishing point. It will just keep slowing down and never get to the horizon. In fact, under that math, it's impossible for anything to intersect at a vanishing point.
Yes, technically. See above.
Since the vanishing point exists, the math is clearly an inaccurate representation of perspective.
Good grief, subtle shades of meaning isn't your strong point, is it? The vanishing point exists on the projection, not in 3D reality. The vanishing point is the point on the projection that objects approach, as they travel an infinite distance away from you. You can calculate where the vanishing point will be on the projection using the very same math that you claim the vanishing point disproves.
Please read the following carefully, because this seems to be where you keep getting confused:
No, an object technically can never reach the vanishing point. However, the object can get arbitrarily close to the vanishing point. It can get so close to the vanishing point that we can't tell the difference with our eyes. We can predict exactly how close to the vanishing point it will be using simple trigonometry like I have used. You can test this yourself with some parallel lines, a few objects, a camera, and careful measurements. Stop claiming that the math doesn't work when you can easily verify for yourself that it does work.
I agree that the sun is at 0° with the horizon when it sets. The reason the result of my math is wrong (28°) is because it started with a flawed assumption: that the earth is flat. The reason the math doesn't agree with reality is because the earth isn't flat in reality.
This is how models are tested. You make predictions assuming the model is correct. If the predictions disagree with reality, there is probably something wrong with the model. Since my prediction based on the flat earth model is wrong, there is probably something wrong with the flat earth model. Obviously.
We know that it is possible for perspective lines to intersect. Under your math it is impossible for any perspective lines to intersect. Therefore your math is wrong.
No. The lines can and do intersect at the vanishing point on the projection. However, for an object to reach the vanishing point by following the lines, they would have to travel an infinite distance, in theory. In reality, since our eyes can't tell the difference between two objects 0.00001° apart, it is possible for objects to appear to reach the vanishing point. See above.
Not only does it lack the intersection of perspective lines, your math has not been demonstrated to be an accurate portrayal of large scale perspective in the real world. You bring us math created by ancient civilizations to us and expect us to just assume that it is correct for the situation. Where is the evidence that it is correct for this purpose?
Oh please, not this again.
"All our calculations agree with the round earth model. But maybe... math suddenly stops working for objects the size of the earth, and the earth is actually flat! No, I have no evidence to suggest that the math suddenly stops working except that I really, really, really want the earth to be flat. Please believe me."
Good luck with that argument.
Edit:
"Warning  while you were typing a new reply has been posted. You may wish to review your post."
Rama also makes good points about the sun not changing size. Anything travelling along the "perspective lines" decreases in size as it approaches the vanishing point. Since the sun doesn't change size, as we have already thoroughly established in other threads, it clearly is not getting farther away.
And rabinoz also makes a good point, although I would really rather not get into the "ship/building is behind the horizon" argument in this thread.

Perspective lines intersect? Can you please prove this?
A demonstration that the perspective lines intersect can be found by taking to rulers against a perspective scene. If we hold rulers over the perspective lines they will overlap each other. And if we draw lines along the perspective lines to the horizon and at some point they will intersect. The same will apply to a real world scene, if one were inclined to draw lines on that.
(http://www.technologystudent.com/despro_flsh/singperp1.png)
Wow, there are a ton of problems here. First, you have not demonstrated clearly that perspective lines intersect. Second, it does not matter if the math is ancient or created 1 second ago. What matters is that it works. No one expects you to assume it works, but it does. If you don't believe it, we can't prove it here, but questioning trigonometry doesn't make you a maverick, this much I know. There are a number of sciences that avail themselves of trigonometry. You should research them.
There are many different types of math. It is possible to apply different maths to the same problem and get a different result. Mathematics is also constrained by the model in which computation takes place. It absolutely needs to be demonstrated that the math chosen is fit for purpose.

What do you mean it "takes place outside of the universe"?? It's just a diagram. It is used to portray angles and distances. If you go outside and physically measure the angle in reality, that angle should agree exactly with the one portrayed in an orthographic view. I don't understand why you are so vehemently opposed to a simple diagram. (Actually I do know... it's because you don't want to acknowledge that the earth isn't flat.;))
The first person view scene you presented doesn't make any sense without the accompanying side view scene. It can't. In the first person scene the angles you represented don't even exist from that view. You are in an entirely different dimension when you switch between the two scenes.
It certainly wasn't arbitrary. I measured and labelled quite carefully, as you can see for yourself. It's all just simple proportions. The camera's vertical field of view is 52°. If an object has an angular diameter of 26° in the orthographic diagram, then the object will take up half the vertical room in the photo. In the above picture, the conversion is 7.7 pixels per degree. The 2.3° wide object takes up 18 pixels. No, it certainly is not arbitrary.
It is arbitrary because, without the side view scene, it is impossible to represent why the suns should be where they are on the first person view. Suddenly the side view scene doesn't mean much if it can't be translated to another universe with different dimensions.
No, an object technically can never reach the vanishing point.
Where is the evidence of this? We see that they do. What kind of evidence is there that they do not?
However, the object can get arbitrarily close to the vanishing point. It can get so close to the vanishing point that we can't tell the difference with our eyes. We can predict exactly how close to the vanishing point it will be using simple trigonometry like I have used. You can test this yourself with some parallel lines, a few objects, a camera, and careful measurements. Stop claiming that the math doesn't work when you can easily verify for yourself that it does work.
Surely if this math is so tested and true for this purpose, you can provide evidence justifying it.
This is how models are tested. You make predictions assuming the model is correct. If the predictions disagree with reality, there is probably something wrong with the model. Since my prediction based on the flat earth model is wrong, there is probably something wrong with the flat earth model. Obviously.
Actually, incorrect. The models used for predictions have to be able to agree with reality. Whatever mathematical construct you use needs to reflect the real world. You can start this journey from fiction to truth by showing that it's impossible to reach the vanishing point, for example.
No. The lines can and do intersect at the vanishing point on the projection. However, for an object to reach the vanishing point by following the lines, they would have to travel an infinite distance, in theory. In reality, since our eyes can't tell the difference between two objects 0.00001° apart, it is possible for objects to appear to reach the vanishing point. See above.
Is there any evidence for these infinite distances besides what an ancient man calculated?

Prove to me that a bunch of trivially simple mathematical concepts are true. Yes, I know that I could test these out for myself, but I'm not going to, because doing so would disprove my flat earth model. (loosely paraphrased)
Wow. The mental gymnastics you go through is truly impressive. Seriously. Bravo. I'll respond in more detail tomorrow. Good night.

You are telling us that the Ancient Greeks calculated infinite distances and we should take that as an unquestionable truth. This has not been demonstrated. This type of math is founded on a shaky premise which exists only in imagination.
It is well known that the math and physics of the Ancient Greeks don't really work. For example, they also predict the concept of line and point graphs, which are infinitely indivisible, and that space and time can be represented on them to explain physical actions. We are taught this in school and are encouraged to use their methods. For some simple high level things it may seem to work. But this math it is also makes it impossible to walk through a door, or for a rabbit to overcome a tortoise in a race. See: Zeno's Paradox (http://barang.sg/index.php?view=achilles)
Any continuous mathematical model like this which predicts infinities should be looked at with scrutiny and demands justification.

A demonstration that the perspective lines intersect can be found by taking to rulers against a perspective scene. If we hold rulers over the perspective lines they will overlap each other. And if we draw lines along the perspective lines to the horizon and at some point they will intersect. The same will apply to a real world scene, if one were inclined to draw lines on that.
(http://www.technologystudent.com/despro_flsh/singperp1.png)
The big problem is that you (and Rowbotham) treat "perspective" as though it "is reality".
Perspective is nothing more than a way to represent reality, and it should never be used "as reality".
You still seem to be claiming that the "vanishing point" has some magical connection with the visual horizon. It demonstrably has no such connection.
While that resolution of eye of 1 minute is really a guide as to what angular separation is needed to see if is one or two objects, this value is still a good guide to the distance to the "vanishing point".
Using this for the vanishing points of railway tracks and roads etc, gives quite reasonable answers.
Using this "guide" puts the vanishing point of something the size of the flat earth sun puts it far further away than the visible horizon, and even much further away than the 10,000 miles to the location or the setting sun. According to Rowbotham's 3,000 x object size it would be at 32 x 3,000 = 96,000 miles.
So, I can see no possible justification for sunsets being due to the sun vanishing due to perspective.
This is especially so considering that we know that the sun stays the same angular size all day. And please don't drag up the silly "known magnification due to the intense. . . . , ".
That argument does not wash. The photos I showed were through a filter which removed any glare. And the moon shows the same behaviour, and that could never be ascribed to "known magnification due to the intense glare".
Saying this to you is I know a waste of time, though maybe someone else might see the logic in what I am saying.

You are telling us that the Ancient Greeks calculated infinite distances and we should take that as an unquestionable truth. This has not been demonstrated. This type of math is founded on a shaky premise which exists only in imagination.
It is well known that the math and physics of the Ancient Greeks don't really work. For example, they also predict the concept of line and point graphs, which are infinitely indivisible, and that space and time can be represented on them to explain physical actions. We are taught this in school and are encouraged to use their methods. For some simple high level things it may seem to work. But this math it is also makes it impossible to walk through a door, or for a rabbit to overcome a tortoise in a race. See: Zeno's Paradox (http://barang.sg/index.php?view=achilles)
Any continuous mathematical model like this which predicts infinities should be looked at with scrutiny and demands justification.
It is for exactly these sorts of issues that calculus was invented 400 years ago. You might be surprised to know that geometry is not the only tool used, although it is very useful in most problems and can get your very close on a bunch of others. Solving Xeno's paradox, or any limit that approaches infinity is the domain of calculus.
Perspective lines intersect? Can you please prove this?
A demonstration that the perspective lines intersect can be found by taking to rulers against a perspective scene. If we hold rulers over the perspective lines they will overlap each other. And if we draw lines along the perspective lines to the horizon and at some point they will intersect. The same will apply to a real world scene, if one were inclined to draw lines on that.
(http://www.technologystudent.com/despro_flsh/singperp1.png)
You do know that as you traverse along the trajectory of the perspective line that two points that appeared to meet are revealed as not actually having met? You can test this by standing on some train tracks and having a friend go to the point you see as having intersected. They will report back that indeed you are mistaken. Don't trust me though, try it yourself!

You do know that as you traverse along the trajectory of the perspective line that two points that appeared to meet are revealed as not actually having met? You can test this by standing on some train tracks and having a friend go to the point you see as having intersected. They will report back that indeed you are mistaken. Don't trust me though, try it yourself!
Please use a closed down railroad when doing so ;D!

You do know that as you traverse along the trajectory of the perspective line that two points that appeared to meet are revealed as not actually having met? You can test this by standing on some train tracks and having a friend go to the point you see as having intersected. They will report back that indeed you are mistaken. Don't trust me though, try it yourself!
Please use a closed down railroad when doing so ;D!
Actually, please don't.

What do you mean it "takes place outside of the universe"?? It's just a diagram. It is used to portray angles and distances. If you go outside and physically measure the angle in reality, that angle should agree exactly with the one portrayed in an orthographic view. I don't understand why you are so vehemently opposed to a simple diagram. (Actually I do know... it's because you don't want to acknowledge that the earth isn't flat.;))
The first person view scene you presented doesn't make any sense without the accompanying side view scene. It can't. In the first person scene the angles you represented don't even exist from that view. You are in an entirely different dimension when you switch between the two scenes.
The angles from the side view are translated to the first person view via the field of view (https://en.wikipedia.org/wiki/Field_of_view) of the camera. Seriously, stop complaining about this, and just go test it. It is easy to test.
No, an object technically can never reach the vanishing point.
Where is the evidence of this? We see that they do. What kind of evidence is there that they do not?
It's common sense. Of course an object can never reach the vanishing point, because the vanishing point is at infinity. An object can never travel an infinite distance away from you. How on earth could it?
However, the object can get arbitrarily close to the vanishing point. It can get so close to the vanishing point that we can't tell the difference with our eyes. We can predict exactly how close to the vanishing point it will be using simple trigonometry like I have used. You can test this yourself with some parallel lines, a few objects, a camera, and careful measurements. Stop claiming that the math doesn't work when you can easily verify for yourself that it does work.
Surely if this math is so tested and true for this purpose, you can provide evidence justifying it.
It's rather difficult to justify anything with you, when you seem to be in denial of basic geometry, number theory, calculus and algebra. I'm not going to personally teach you several years worth of math in order to justify something that you could just walk outside and test with several objects and a camera. Just go test it for yourself. It is trivially easy. Here, I'll give you the exact hypothesis to test:
Assuming our naked eyes can't distinguish anything less than 0.02°, an object will appear to touch another object when the ratio of distance between the objects to distance from our eyes is approximately 1:3000.
So, if they are 1 mm away from each other, they will appear to touch when they are 3 meters away from your eyes. Go test it. Stop begging me to prove it for you.
You are telling us that the Ancient Greeks calculated infinite distances and we should take that as an unquestionable truth. This has not been demonstrated. This type of math is founded on a shaky premise which exists only in imagination.
What on earth do you mean by "calculated infinite distances"? Of course you don't have to accept any of this as unquestionable truth. It helps to have at least a basic understanding of the subject before you declare it to be useless though.
It is well known that the math and physics of the Ancient Greeks don't really work.
By whom? You? You are the only person I have ever heard espouse this opinion. I think you are exaggerating here.
For example, they also predict the concept of line and point graphs, which are infinitely indivisible, and that space and time can be represented on them to explain physical actions. We are taught this in school and are encouraged to use their methods. For some simple high level things it may seem to work. But this math it is also makes it impossible to walk through a door, or for a rabbit to overcome a tortoise in a race. See: Zeno's Paradox (http://barang.sg/index.php?view=achilles)
Lol. No, Zeno's paradox does not prevent us from reaching a destination according to basic geometry. It's mostly just a philosophical thought experiment. Calculus deals with infinities and infinitesimals quite easily.
Any continuous mathematical model like this which predicts infinities should be looked at with scrutiny and demands justification.
Trigonometry does not predict infinity. It assumes continuity. There is a difference. However, you don't have to assume that space is continuous in order for trigonometry to be useful. Just round your answers to the nearest discreet value of space. Problem solved. I assume that's what you are arguing for, right? That space is somehow quantized? If space is in fact quantized, there is no reason to suspect that it would effect the answers significantly for the scale that we are working on (thousands of miles).

Come on, common sense? Is that how this math is justified? "It is this way because of common sense" may have worked in Ancient Greece, but today we require, you know, actual evidence.
This math on perspective should be backed up by mountains of research. As if mathematical infinities are mere common sense and translate perfectly to the real world. That is ridiculous.

Come on, common sense? Is that how this math is justified? "It is this way because of common sense" may have worked in Ancient Greece, but today we require, you know, actual evidence.
Oh irony.
This math on perspective should be backed up by mountains of research. As if mathematical infinities are mere common sense and translate perfectly to the real world. That is ridiculous.
Have you even looked in to this? I thought that was the zetetic way? Anyway, what follows is a link to proofs and texts for projective geometry. The field that was given birth by the exploration of perspective drawing in the renaissance:
http://wwwhistory.mcs.stand.ac.uk/~john/MT4521/Lectures/L20.html
https://wwwm10.ma.tum.de/foswiki/pub/Lehre/WS0809/.../ch1.pdf
www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Dean.pdf
www.math.rug.nl/~piter/KR/Hartshorne.pdf
https://opus4.kobv.de/opus4zib/frontdoor/deliver/index/docId/101/.../SC9305.pdf
www.mit.edu/~alexrem/ProjectiveGeometry.pdf
Indeed as you look over this (you probably won't), you will be astonished to find that projective geometry has advanced substantially since the renaissance, when perspective drawing was first explored formally and gave birth to projective geometry. Now do you have anything other than arguments from personal credulity? Can you actually attack the substance of perspective diagrams or orthographic diagrams? Better yet, can you explain how the FE model utterly fails to explain the sun passing behind the horizon from the bottom up without ever changing apparent size? And even better than that, can you do it in a meaningful way, like real physics and mathematics can?

Come on, common sense? Is that how this math is justified? "It is this way because of common sense" may have worked in Ancient Greece, but today we require, you know, actual evidence.
This math on perspective should be backed up by mountains of research. As if mathematical infinities are mere common sense and translate perfectly to the real world. That is ridiculous.
There are no "mathematical infinities" involved.
The sun is a (supposedly) 3,000 miles high, quite a finite height.
At the time of sunset the distance to the sun at that time depends entirely on the season and the location of the observer, but a 10,000 mile horizontal distance is a resonable value.
Like it or not this puts the sun at about 17° above the horizon and that calculation is what perspective is  you can't then "bring" the sun down further.
You still seem to labour under the misapprehension that the horizon is always the "vanishing point". It is not and there is no justification, theoretical or otherwise for claiming that.
And on top of this there is this critical point that you will never admit. It is quite provable that on a given day the angular size is the same
from everywhere on earth from where the sun can be seen.
Claiming that this can be explained away by "due to a known magnification effect caused by the intense rays of light passing through the strata of the atmolayer" is simply a meaningless statement!
There might be glare around the sun, which in clear conditions can be removed with a filter, but there simply is no "known magnification effect caused by the intense rays of light passing through the strata of the atmolayer". Glare around a bright can hardly be called "magnification"!
There is simply no validity in that. These simply observations show that it cannot explain what we see.
 When suitable filters are used to remove the glare, we certainly do see the sun size remain the same size  always around 0.52° (it varies a little with the season.
 When a filter is used and enough magnification to observe the sunspots the pattern is the same from all over the earth, quite unaffected by location.

Come on, common sense? Is that how this math is justified?
No. Everything in mathematics is justified with tediously rigorous proofs, even stuff that appears to be common sense. I appealed to common sense because you really shouldn't need a rigorous proof to realize this is true. It is not a complicated issue. If you really want the proofs, you can probably find it in the papers Rama provided. Or, as I have said numerous times, you can just go outside with a camera, a measuring tape, a protractor, and several objects and test it yourself.
This math on perspective should be backed up by mountains of research. As if mathematical infinities are mere common sense and translate perfectly to the real world. That is ridiculous.
Mountains of research has been provided. Thanks Rama.
Time to face the music Tom. You are waaaay beyond grasping at straws here.

It appears that FEs Tom, etal and GEs Rabinoz, etal have all exerted efforts and tried to prove their respective claims, but all in vain... sorry, i just could not see that both parties can ever prove anything on what really is the truth about the shape or form of earth. I suggest raise or bring this topic to the next level as I always suggested. Each one of you should recognize and be aware that the law on perspective and vanishing point (LP/VP) is inherent in or governs our unaided eyes in seeing or viewing objects at a distance, whether or not the objects are moving towards or away from our position. You both GEs and FEs here appears to go around a circle like in merry go round.
What we, truth seekers, observe is that FEs keep on proving their claims just like seeing the number "6" from their position, while the GEs also keep on proving their claims just like seeing the same number "6" as number "9" from their position at the opposite side. From your arguments and explanations, it's a fact that all of you have seen distant objects, e.g. sun, bldgs, moon, etc., all by and through the LP/VP, NO ONE with eyes can escape this law. So whether you have a GE mindset or FE mindset, you all have more or less similar observation when it comes to seeing objects at a distance on this enormously HUGE earth as you are only a virus size compared to it.
For GEs even though you have a spherical or global shape in mind, you cannot really prove curvature with unaided eyes because you can only see flatness just like what the flat minded FEs see or wanted to see. Both of you are actually seeing flatness in general with unaided eyes, plus the fact that objects seen are governed by LP/VP, hence, a very limited view. For GEs, you should both consider curvature and LP/VP in explaining why objects disappear from horizon, while FEs are expected to consider only LP/VP. But with the huge earth surface appearing to be flat for both withtheir unaided eyes, they'll just end up at nothing worth their efforts. It's a pity... The truth is out there people, but we have to change strategy, and use tools like high powered telescope or camera to significantly extend this LP/VP much farther for us to find the absolute truth. Better think out of the box in resolving this issue.. Wake up... :)

It appears that FEs Tom, etal and GEs Rabinoz, etal have all exerted efforts and tried to prove their respective claims, but all in vain... sorry, i just could not see that both parties can ever prove anything on what really is the truth about the shape or form of earth. I suggest raise or bring this topic to the next level as I always suggested. Each one of you should recognize and be aware that the law on perspective and vanishing point (LP/VP) is inherent in or governs our unaided eyes in seeing or viewing objects at a distance, whether or not the objects are moving towards or away from our position. You both GEs and FEs here appears to go around a circle like in merry go round.
What we, truth seekers, observe is that FEs keep on proving their claims just like seeing the number "6" from their position, while the GEs also keep on proving their claims just like seeing the same number "6" as number "9" from their position at the opposite side. From your arguments and explanations, it's a fact that all of you have seen distant objects, e.g. sun, bldgs, moon, etc., all by and through the LP/VP, NO ONE with eyes can escape this law. So whether you have a GE mindset or FE mindset, you all have more or less similar observation when it comes to seeing objects at a distance on this enormously HUGE earth as you are only a virus size compared to it.
For GEs even though you have a spherical or global shape in mind, you cannot really prove curvature with unaided eyes because you can only see flatness just like what the flat minded FEs see or wanted to see. Both of you are actually seeing flatness in general with unaided eyes, plus the fact that objects seen are governed by LP/VP, hence, a very limited view. For GEs, you should both consider curvature and LP/VP in explaining why objects disappear from horizon, while FEs are expected to consider only LP/VP. But with the huge earth surface appearing to be flat for both withtheir unaided eyes, they'll just end up at nothing worth their efforts. It's a pity... The truth is out there people, but we have to change strategy, and use tools like high powered telescope or camera to significantly extend this LP/VP much farther for us to find the absolute truth. Better think out of the box in resolving this issue.. Wake up... :)
There's nothing to prove. This whole notion and community builds on the assumption that every single person that actually witnessed earth's curvature are liars, including, but not limited to, the Apollo project participants (Read: thousands of people).
When that's said and done, it relies on Rowbotham, even though his fan boys on this board misquote him constantly. Even with statements that contradicts what Rowbotham stated.
I took a photo of earth's curvature from 24.218 meters of altitude. That's a "lie" as well. So you see, this is only a debate because:
1. Someone's hardcore trolling
2. Lack of intellect
3. People refusing to admit they're wrong for whatever reason. Broken marriages, mental issues, whatever.
There's nothing to prove. Just facts. Reproducible results and facts. Flat earth religion has none of those.

It appears that FEs Tom, etal and GEs Rabinoz, etal have all exerted efforts and tried to prove their respective claims, but all in vain... blah blah blah same post as all your other posts blah blah REers and FEers are 69'ing each other blah
All I have ever seen you do is declare that our arguments are not good enough. Perhaps contribute to the actual discussion? This thread is about how angles relate to perspective when viewing the sun at sunset. Be specific. No more blanket declarations. And no, I'm not going to buy you an expensive telescope.
Each one of you should recognize and be aware that the law on perspective and vanishing point (LP/VP) is inherent in or governs our unaided eyes in seeing or viewing objects at a distance, whether or not the objects are moving towards or away from our position.
Yes, I am aware. If you had read this thread, you would know that I was very specific about how perspective and the vanishing point effects how we see angles. That's the whole point of this thread, in case you didn't notice.

All I have ever seen you do is declare that our arguments are not good enough. Perhaps contribute to the actual discussion? This thread is about how angles relate to perspective when viewing the sun at sunset. Be specific. No more blanket declarations. And no, I'm not going to buy you an expensive telescope.
Each one of you should recognize and be aware that the law on perspective and vanishing point (LP/VP) is inherent in or governs our unaided eyes in seeing or viewing objects at a distance, whether or not the objects are moving towards or away from our position.
Yes, I am aware. If you had read this thread, you would know that I was very specific about how perspective and the vanishing point effects how we see angles. That's the whole point of this thread, in case you didn't notice.
I think you missed my point. All I want to convey to you both GEs and FEs is that you've got good enough arguments and/or proofs supporting your claims, your shortcoming is that you missed to see that you're both seeing or observing the same object or thing with differing mindsets which blinded or shielded you both from the real thing or truth out there (FE or GE?; moving or stationary sun or moon, etc.). What I want people in this thread to understand is that both your arguments or explanations are quite good enough to support each other's claims or proposition, but you missed or don't want to see the "forest". Haven't you wondered why is this so? It's simple. Each of your biased mindsets prevent you from seeing things with open mind. Try seeing what truth seekers see that the number "6" can well be seen also as "9" depending on where the vantage point of the observer or reader is. No one should argue against this, for it is the truth. This is the kind of mindset we want you both to have inorder to arrive at what's really the truth about the earth in relation to the sun, moon or its shape...
Ok, you know what I think where you are right now in your debate re LP/VP (law of perspective/vanishing point), angles etc., you're both arguing for something you observed within only the limits of LP/VP governing human eyes' capacity. You've not even gone beyond such limitation towards what it is really like without the LP/VP limitation. To have this limitation expanded, i think you view things with aided eyes. Also, sun's angles, perspective (setting or rising) can be analyzed well with empirical data taken real time frm all over the world. I think if you have those kind of data, no one dares debunking them as they're what people see in real time. What we can do is to understand and explain the results, whether or not they're inconsistent with our preconceived beliefs, mindsets or biases. Be a truth seeker... :)

All I have ever seen you do is declare that our arguments are not good enough. Perhaps contribute to the actual discussion? This thread is about how angles relate to perspective when viewing the sun at sunset. Be specific. No more blanket declarations. And no, I'm not going to buy you an expensive telescope.
Each one of you should recognize and be aware that the law on perspective and vanishing point (LP/VP) is inherent in or governs our unaided eyes in seeing or viewing objects at a distance, whether or not the objects are moving towards or away from our position.
Yes, I am aware. If you had read this thread, you would know that I was very specific about how perspective and the vanishing point effects how we see angles. That's the whole point of this thread, in case you didn't notice.
I think you missed my point. All I want to convey to you both GEs and FEs is that you've got good enough arguments and/or proofs supporting your claims, your shortcoming is that you missed to see that you're both seeing or observing the same object or thing with differing mindsets which blinded or shielded you both from the real thing or truth out there (FE or GE?; moving or stationary sun or moon, etc.). What I want people in this thread to understand is that both your arguments or explanations are quite good enough to support each other's claims or proposition, but you missed or don't want to see the "forest". Haven't you wondered why is this so? It's simple. Each of your biased mindsets prevent you from seeing things with open mind. Try seeing what truth seekers see that the number "6" can well be seen also as "9" depending on where the vantage point of the observer or reader is. No one should argue against this, for it is the truth. This is the kind of mindset we want you both to have inorder to arrive at what's really the truth about the earth in relation to the sun, moon or its shape...
Ok, you know what I think where you are right now in your debate re LP/VP (law of perspective/vanishing point), angles etc., you're both arguing for something you observed within only the limits of LP/VP governing human eyes' capacity. You've not even gone beyond such limitation towards what it is really like without the LP/VP limitation. To have this limitation expanded, i think you view things with aided eyes. Also, sun's angles, perspective (setting or rising) can be analyzed well with empirical data taken real time frm all over the world. I think if you have those kind of data, no one dares debunking them as they're what people see in real time. What we can do is to understand and explain the results, whether or not they're inconsistent with our preconceived beliefs, mindsets or biases. Be a truth seeker... :)
You still don't get it.
Science is explaining these things with proven math and geometry, test, tests, and more tests. While new findings might be the target of some bias in the scientific community, the shape of the earth isn't. The shape of the earth is as unbiased a fact as they'll ever come.
You seem to miss the point of why some of the more eloquent and obviously educated "GE'ers" participate in debates on this site. To me, as a "GE'er", it's quite clear:
We want to make sure, that the next person who joins the board to ask questions about their newly found belief, aren't misinformed. That the social presence of TFES doesn't lead young people to believe what TFES is actually claiming. To me, that's just as dangerous as religious indoctrination.

Why this discussion about the setting sun? It will be rising at the same time for someone else hence the proof of a round earth.

All I have ever seen you do is declare that our arguments are not good enough. Perhaps contribute to the actual discussion? This thread is about how angles relate to perspective when viewing the sun at sunset. Be specific. No more blanket declarations. And no, I'm not going to buy you an expensive telescope.
Each one of you should recognize and be aware that the law on perspective and vanishing point (LP/VP) is inherent in or governs our unaided eyes in seeing or viewing objects at a distance, whether or not the objects are moving towards or away from our position.
Yes, I am aware. If you had read this thread, you would know that I was very specific about how perspective and the vanishing point effects how we see angles. That's the whole point of this thread, in case you didn't notice.
I think you missed my point. All I want to convey to you both GEs and FEs is that you've got good enough arguments and/or proofs supporting your claims, your shortcoming is that you missed to see that you're both seeing or observing the same object or thing with differing mindsets which blinded or shielded you both from the real thing or truth out there (FE or GE?; moving or stationary sun or moon, etc.). What I want people in this thread to understand is that both your arguments or explanations are quite good enough to support each other's claims or proposition, but you missed or don't want to see the "forest". Haven't you wondered why is this so? It's simple. Each of your biased mindsets prevent you from seeing things with open mind. Try seeing what truth seekers see that the number "6" can well be seen also as "9" depending on where the vantage point of the observer or reader is. No one should argue against this, for it is the truth. This is the kind of mindset we want you both to have inorder to arrive at what's really the truth about the earth in relation to the sun, moon or its shape...
Ok, you know what I think where you are right now in your debate re LP/VP (law of perspective/vanishing point), angles etc., you're both arguing for something you observed within only the limits of LP/VP governing human eyes' capacity. You've not even gone beyond such limitation towards what it is really like without the LP/VP limitation. To have this limitation expanded, i think you view things with aided eyes. Also, sun's angles, perspective (setting or rising) can be analyzed well with empirical data taken real time frm all over the world. I think if you have those kind of data, no one dares debunking them as they're what people see in real time. What we can do is to understand and explain the results, whether or not they're inconsistent with our preconceived beliefs, mindsets or biases. Be a truth seeker... :)
You still don't get it.
Science is explaining these things with proven math and geometry, test, tests, and more tests. While new findings might be the target of some bias in the scientific community, the shape of the earth isn't. The shape of the earth is as unbiased a fact as they'll ever come.
You seem to miss the point of why some of the more eloquent and obviously educated "GE'ers" participate in debates on this site. To me, as a "GE'er", it's quite clear:
We want to make sure, that the next person who joins the board to ask questions about their newly found belief, aren't misinformed. That the social presence of TFES doesn't lead young people to believe what TFES is actually claiming. To me, that's just as dangerous as religious indoctrination.
What you've just said is also exactly what your FE opponents have been saying. You both are using science, scientific explanation, empirical data, tests, etc..... About your intention, it's noble, but let me tell you this, the more you engage into debate and defend your claims like the way you do here or perhaps in other venues as well (in not so clear, flawless and empirically convincing manner), the more people become more curious. This is a fact. The FEs will thank you for that. You can never control people from being curious. All has access to internet, youtube, facebook, this one, etc... It's impossible to control this now.
All you have to do, i think, if you really want to stop these FEs from further exposing their claims/proofs is to really show an irrefutable proof of at least one of your claims such as presenting real time unedited video showing that at, say, 100+ mi away (all other given data should be confirmed true), you cannot anymore see a tower, bldg, etc... by using highpowered telescopic camera. If you can show this with precision, you'll really be the "global man" for GEs and people around, and they, even FEs, will believe. How can this be refuted? Be a truth seeker then... be challenged... :)

This ISS has a live feed. Case closed.

What you've just said is also exactly what your FE opponents have been saying. You both are using science, scientific explanation, empirical data, tests, etc.....
I'll come back on this, but please stop saying that FE "opponents" have provided anything using "science", "scientific explanation" and "empirical data". All that have been presented is either really bad science, profound misanderstanding of basic maths or a complete joke. If you were a real "truth seeker", instead of trying to explain that both side have good argument, you should analyze in depth all arguments and check there validity.
Some basic maths have been provided in another thread (http://"http://forum.tfes.org/index.php?topic=5291.0"). This is clearly a proof that the earth is not flat. There's no debate! Anyone can check the maths, it's only basic trigonometry, and anyone can observe that the sun is not accelerating in the sky up to noon and deccelerating afterwards. Therefore the Earth is not flat.
And finally, let me remind you that only one counterexample is necessary to prove that an affirmation is wrong and that tens have been presented to FE believers. If you really think that the arguments advanced by FE are legitimate, it means that you are denying the laws governing our world (beginning with very simple maths as geometry).
So please, instead of trying to explain that everyone have good arguments (according to you) in every single thread. Could you please try to be more constructive and try by yourself some calculation or providing some "scientific explanation" (either to explain how the world is in FE or GE), so that we would have something concrete and scientific to discuss? First excercie could be to reproduce the topic about the elevation and the speed of the sun and confirm ("or not") the results? Just do that and you will be convincs that the earth is a sphere. Don't bother to ask question if you have, I would be glad to help you (if needed of course!)

...Try seeing what truth seekers see that the number "6" can well be seen also as "9" depending on where the vantage point of the observer or reader is. No one should argue against this, for it is the truth. This is the kind of mindset we want you both to have inorder to arrive at what's really the truth about the earth in relation to the sun, moon or its shape...
(https://media.giphy.com/media/N2rLxtwaU9rBC/giphy.gif)
All you have to do, i think, if you really want to stop these FEs from further exposing their claims/proofs is to really show an irrefutable proof of at least one of your claims such as presenting real time unedited video showing that at, say, 100+ mi away...
This ISS has a live feed. Case closed.
Wish granted (https://youtu.be/Gy5PC5Auoak)
you cannot anymore see a tower, bldg, etc... by using highpowered telescopic camera. If you can show this with precision, you'll really be the "global man" for GEs and people around, and they, even FEs, will believe. How can this be refuted? Be a truth seeker then... be challenged... :)
Wish granted (http://forum.tfes.org/index.php?topic=5225.msg101836#msg101836)
You have one wish left. Make it count.

Come on, common sense? Is that how this math is justified? "It is this way because of common sense" may have worked in Ancient Greece, but today we require, you know, actual evidence.
Oh irony.
This math on perspective should be backed up by mountains of research. As if mathematical infinities are mere common sense and translate perfectly to the real world. That is ridiculous.
Have you even looked in to this? I thought that was the zetetic way? Anyway, what follows is a link to proofs and texts for projective geometry. The field that was given birth by the exploration of perspective drawing in the renaissance:
http://wwwhistory.mcs.stand.ac.uk/~john/MT4521/Lectures/L20.html
https://wwwm10.ma.tum.de/foswiki/pub/Lehre/WS0809/.../ch1.pdf
www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Dean.pdf
www.math.rug.nl/~piter/KR/Hartshorne.pdf
https://opus4.kobv.de/opus4zib/frontdoor/deliver/index/docId/101/.../SC9305.pdf
www.mit.edu/~alexrem/ProjectiveGeometry.pdf
Indeed as you look over this (you probably won't), you will be astonished to find that projective geometry has advanced substantially since the renaissance, when perspective drawing was first explored formally and gave birth to projective geometry. Now do you have anything other than arguments from personal credulity? Can you actually attack the substance of perspective diagrams or orthographic diagrams? Better yet, can you explain how the FE model utterly fails to explain the sun passing behind the horizon from the bottom up without ever changing apparent size? And even better than that, can you do it in a meaningful way, like real physics and mathematics can?
Regarding those links, a mathematical proof is not the same thing as a real world evidence of application. A mathematical proof validates an equation or concept against that mathematical framework, not the real world.

Come on, common sense? Is that how this math is justified?
No. Everything in mathematics is justified with tediously rigorous proofs, even stuff that appears to be common sense. I appealed to common sense because you really shouldn't need a rigorous proof to realize this is true. It is not a complicated issue. If you really want the proofs, you can probably find it in the papers Rama provided. Or, as I have said numerous times, you can just go outside with a camera, a measuring tape, a protractor, and several objects and test it yourself.
This math on perspective should be backed up by mountains of research. As if mathematical infinities are mere common sense and translate perfectly to the real world. That is ridiculous.
Mountains of research has been provided. Thanks Rama.
Time to face the music Tom. You are waaaay beyond grasping at straws here.
Again, a mathematical proof doesn't have anything to do with reality.

Come on, common sense? Is that how this math is justified?
Again, a mathematical proof doesn't have anything to do with reality.
So, what do we base things on "common sense", though you seem to have ruled that out,
Or "a mathematical proof" that "doesn't have anything to do with reality"?
You don't seem to have left much.
So, what do you base you ideas of perspective on? Whatever you need to prop up the Flat Earth it seems.

Apparently you and several people are completely uneducated as to what a mathematical proof actually is. I do not wish to continue this discussion.

Apparently you and several people are completely uneducated as to what a mathematical proof actually is. I do not wish to continue this discussion.
I guess you wouldn't.
It is no point with your weird ideas on perspective and inability to accept that we can use simple mathematics to calculate "vanishing points", which you just assume are always on the horizon.

Again, a mathematical proof doesn't have anything to do with reality.
You just asked for a proof. Now that it has been given to you, you are making a blanket declaration that mathematical proofs have nothing to do with reality?? What??? If you don't think it has to do with reality, then go test it in reality, like I have stated numerous times. Your entire argument seems to be based on incredulity that angles can be measured from our point of view, which is ridiculous. Stop whining about the ancient Greek mathematicians and go test it.
I do not wish to continue this discussion.
Cognitive dissonance starting to get uncomfortable? ;)

Again, a mathematical proof doesn't have anything to do with reality.
You just asked for a proof. Now that it has been given to you, you are making a blanket declaration that mathematical proofs have nothing to do with reality?? What??? If you don't think it has to do with reality, then go test it in reality, like I have stated numerous times. Your entire argument seems to be based on incredulity that angles can be measured from our point of view, which is ridiculous.
Come on, didn't you pass middle school algebra? Look up the definition of a mathematical proof:
https://en.wikipedia.org/wiki/Mathematical_proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to selfevident or assumed statements, known as axioms, along with accepted rules of inference.
Stop whining about the ancient Greek mathematicians and go test it.
Test what, that there aren't any hidden infinite distances as predicted by the ancients? That sounds more like a positive claim that you have to prove to support your claim.

Again, a mathematical proof doesn't have anything to do with reality.
You just asked for a proof. Now that it has been given to you, you are making a blanket declaration that mathematical proofs have nothing to do with reality?? What??? If you don't think it has to do with reality, then go test it in reality, like I have stated numerous times. Your entire argument seems to be based on incredulity that angles can be measured from our point of view, which is ridiculous. Stop whining about the ancient Greek mathematicians and go test it.
I do not wish to continue this discussion.
Cognitive dissonance starting to get uncomfortable? ;)
Come on, didn't you pass middle school algebra? Look up the definition of a mathematical proof:
https://en.wikipedia.org/wiki/Mathematical_proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to selfevident or assumed statements, known as axioms, along with accepted rules of inference.
Yes, I know what a mathematical proof is. Yes, I am aware that if you start with bad assumptions then the conclusion may not agree with reality, just like any other kind of proof. Obviously.
Let's be specific. This is what I am arguing is true:
(http://www.mathportal.org/calculators/planegeometrycalculators/triangleRightAngle.gif)
Given objects at locations A, B, and C, with distances between the objects given by a and b. The angle α can be measured by a person at location A. This angle will be equal to arctan(a/b).
There are several ways of measuring this angle, all of which should agree:
1. Using a theodolite.
2. Stretching a string between the objects and using a protractor.
3. Pointing two sticks at the objects and measuring the angle between them with a protractor.
4. Measuring the distance between the objects in a picture, and calculating the angle based on the FOV of the camera. (Approximation based on the distortion of the camera.)
Forget the mathematical proof if you want. This is common sense to most people, but if it isn't common sense to you, then go test it. It is easily testable in reality.

Test what, that there aren't any hidden infinite distances as predicted by the ancients? That sounds more like a positive claim that you have to prove to support your claim.

Test what, that there aren't any hidden infinite distances as predicted by the ancients? That sounds more like a positive claim that you have to prove to support your claim.
Ok, I am just going sidestep the whole "hidden infinite distances" thing, because you are just trying to use Zeno's Paradox to pretend that nothing can be measured.
Here is what you test:
1. Set up two objects so that they make a right triangle with each other and you. (In the above diagram, the objects go at locations B and C. Place yourself at location A.)
2. Measure the distances between the objects. (a and b in the above diagram)
3. Measure the angle between the objects using one or more of the prescribed methods.
4. Calculate the angle between the objects using arctan(a/b). Does the measurement agree with the calculation?
5. Change the position of the objects. Repeat steps 14 until you are convinced that this process works.
Did the process ever not work? Did Zeno's Paradox ever hinder the process? Did "hidden infinite distances" ever hinder the process? If the process always works, then why are you assuming that it won't work for the sun?

You must test long distance perspective, not something else entirely.

You must test long distance perspective, not something else entirely.
Ah, and here we get to the crux of your argument. So your argument is this:
"Yes, this process works for any relatively short, easily testable distance. No, there is no reason to believe it stops working at long distances. However, if the process does work at long distances, then my model would be proved false. Therefore, I am going to assume it stops working at long distances, despite the lack of evidence."
Is this correct? Am I missing something?

You must test long distance perspective, not something else entirely.
Ah, and here we get to the crux of your argument. So your argument is this:
"Yes, this process works for any relatively short, easily testable distance. No, there is no reason to believe it stops working at long distances. However, if the process does work at long distances, then my model would be proved false. Therefore, I am going to assume it stops working at long distances, despite the lack of evidence."
Is this correct? Am I missing something?
Of course things work differently at different scales. We have an entire different branch of physics for the very small and the very large. Any model is nothing more than an approximation for reality within a limited range.

You must test long distance perspective, not something else entirely.
Ah, and here we get to the crux of your argument. So your argument is this:
"Yes, this process works for any relatively short, easily testable distance. No, there is no reason to believe it stops working at long distances. However, if the process does work at long distances, then my model would be proved false. Therefore, I am going to assume it stops working at long distances, despite the lack of evidence."
Is this correct? Am I missing something?
Of course things work differently at different scales. We have an entire different branch of physics for the very small and the very large. Any model is nothing more than an approximation for reality within a limited range.
True enough, but it is important to know exactly how and when a model stops being useful. We know exactly how inaccurate Newtonian physics is at any given scale. We know why it is inaccurate at that scale. On the other hand, you have no notion of why geometry stops working. You have no notion of at what scale geometry stops working. All you know is that it must stop working or else your model is wrong. You are just using this as an excuse to ignore evidence that contradicts your model.

Apparently you and several people are completely uneducated as to what a mathematical proof actually is. I do not wish to continue this discussion.
You think Trig doesn't work in the real world? Wow. Good thing you are leaving the conversation. The fields of surveying and engineering would be fraught with failures at every turn if this were the case. Bye Tom!
It's really sad that you try and make this argument but will hold up Rowbotham's examples from ENaG as valid. Do you not see how you internal contradictions make you untrustworthy?

Apparently you and several people are completely uneducated as to what a mathematical proof actually is. I do not wish to continue this discussion.
You think Trig doesn't work in the real world? Wow. Good thing you are leaving the conversation. The fields of surveying and engineering would be fraught with failures at every turn if this were the case. Bye Tom!
It's really sad that you try and make this argument but will hold up Rowbotham's examples from ENaG as valid. Do you not see how you internal contradictions make you untrustworthy?
It's amazing the kind of arguments that someone can justify in his mind when his belief system is on the line. People sometimes ask why anyone would come to this forum if they don't believe the earth is flat. For me, this right here is it. Some people like watching olympic gymnastics. I like watching mental gymnastics. :)

True enough, but it is important to know exactly how and when a model stops being useful. We know exactly how inaccurate Newtonian physics is at any given scale. We know why it is inaccurate at that scale. On the other hand, you have no notion of why geometry stops working. You have no notion of at what scale geometry stops working. All you know is that it must stop working or else your model is wrong. You are just using this as an excuse to ignore evidence that contradicts your model.
In your example, you did not demonstrate the mathematical infinities of perspective. How is it supposed to prove the subjectmatter true?
Will one trigonometric equation prove the rest of trigonometry true? I think not.

I have seen this conversation with Tom play out before.
It always ended with Tom claiming something along the lines at some unspecified distance math stops working. Except when it supports his belief. I believe he accepts the calculations used to determine the Sun' altitude be various FE's.
I think we can define this distance as how ever far the horizon is from the observer when ships disappear from the bottom up. So about 212 miles in most cases.

I have seen this conversation with Tom play out before.
It always ended with Tom claiming something along the lines at some unspecified distance math stops working.
The math actually claims that something infinite happens at long distances. This has not been demonstrated to be true. You are expecting us to place our faith in the intellectual prowess of a group of people who believed that flies spontaneously generate from rotting meat.

I have seen this conversation with Tom play out before.
It always ended with Tom claiming something along the lines at some unspecified distance math stops working.
The math actually claims that something infinite happens at long distances. This has not been demonstrated to be true. You are expecting us to place our faith in the intellectual prowess of a group of people who believed that flies spontaneously generate from rotting meat.
Then you have to answer what distance math fails. I certainly can accurately predict the angles things will appear to be to an observer at different distances and locations. It is easy to verify this with a simple experiment. So at what distance does math fail and why? 1 mile? 100? 1,000? 10,000?
The math certainly seems to work for celestial navigation. The entire method involves predicting what angle stars will appear to the observer. How come? Math is certainly needed to get a fix and the stars are further away than the Sun in the FE models I am aware of. They are certainly much further away in the RE model.

The math actually claims that something infinite happens at long distances.
What?? You are going to have to be more specific. Stop being vague. Are you still trying to drag Zeno's Paradox into this somehow?
True enough, but it is important to know exactly how and when a model stops being useful. We know exactly how inaccurate Newtonian physics is at any given scale. We know why it is inaccurate at that scale. On the other hand, you have no notion of why geometry stops working. You have no notion of at what scale geometry stops working. All you know is that it must stop working or else your model is wrong. You are just using this as an excuse to ignore evidence that contradicts your model.
In your example, you did not demonstrate the mathematical infinities of perspective. How is it supposed to prove the subjectmatter true?
Will one trigonometric equation prove the rest of trigonometry true? I think not.
Your logical progression is bewildering. I see no logical connection between these statements. Please stop dodging the question. I would like to reemphasize Woody's question:
At what distance does the math fail? Why does it fail? In what way, specifically, does it fail? What is the correct way to calculate the angle?

I have seen this conversation with Tom play out before.
It always ended with Tom claiming something along the lines at some unspecified distance math stops working. Except when it supports his belief.
Oh, I know. But it is fascinating and hilarious every time.

What?? You are going to have to be more specific. Stop being vague. Are you still trying to drag Zeno's Paradox into this somehow?
Please follow the thread. The math claims that it is impossible for an overhead body to reach the place where the perspective lines intersect. It predicts that the horizon would be an infinite distance away on a plane. This has not been demonstrated.
Please stop dodging the question. I would like to reemphasize Woody's question:
At what distance does the math fail? Why does it fail? In what way, specifically, does it fail? What is the correct way to calculate the angle?
The correct way to determine the truth is by taking our que from the real world. If train tracks seem to intersect then they do, according to our present perception. If two train tracks are laid out in front of you are at angles pointing towards each other, then obviously, two lines oriented in that position that will intersect at some point. Only parallel lines can continue into infinity and never intersect.
To calculate where they intersect take the distance between the tracks and the angle of their progression and determine where they would meet in the distance. Calculate based on what we experience, not on some theoretical dimension outside of the universe.

How about answer how celestial navigation can be used to get a fix?
The method used predicts what angle certain stars should appear to an observer at a location at a certain time.
This is real world application that was and is(much rarely now) with accurate and reliable results. This suggest that the math you say fails at those distances does work at them.
If celestial navigation works what changes when predicting what angle the Sun should appear to a person?

What?? You are going to have to be more specific. Stop being vague. Are you still trying to drag Zeno's Paradox into this somehow?
Please follow the thread. The math claims that it is impossible for an overhead body to reach the place where the perspective lines intersect. It predicts that the horizon would be an infinite distance away on a plane. This has not been demonstrated.
No, it does not. It predicts that two objects that are not touching will always have a nonzero angular diameter between them. It does not predict anything to be at infinity.
Please stop dodging the question. I would like to reemphasize Woody's question:
At what distance does the math fail? Why does it fail? In what way, specifically, does it fail? What is the correct way to calculate the angle?
The correct way to determine the truth is by taking our que from the real world. If train tracks seem to intersect then they do.
If the sun seems to touch the horizon then it does.
If an object seems to be hidden behind the horizon then it is.
Am I determining the truth correctly now?
If two train tracks are laid out in front of you at an angle pointing towards each other, then obviously, two lines oriented in that position that will intersect at some point. Only parallel lines can continue into infinity and never intersect.
Yes, we agree on this. Parallel lines never intersect. Brilliant deduction Sherlock.
To calculate where they intersect take the distance between the tracks and the angle of their progression and determine where they would intersect in the distance. Calculate based on what we and experience, not on some theoretical side view dimension outside of the universe.
"take the distance between the tracks and the angle of their progression"  Um... how do I take a distance between an object and an angle?
"determine where they would intersect in the distance"  How? Is there a special Bishop equation that I can use?
"Calculate based on what we and experience"  Ok. Great. How do I perform this calculation?

How about answer how celestial navigation can be used to get a fix?
The method used predicts what angle certain stars should appear to an observer at a location at a certain time.
This is real world application that was and is(much rarely now) with accurate and reliable results. This suggest that the math you say fails at those distances does work at them.
If celestial navigation works what changes when predicting what angle the Sun should appear to a person?
I would prefer to keep these threads on topic.

No, it does not. It predicts that two objects that are not touching will always have a nonzero angular diameter between them. It does not predict anything to be at infinity.
It predicts that the objects will continue to forever approach the horizon, but never touch it.
If the sun seems to touch the horizon then it does.
If an object seems to be hidden behind the horizon then it is.
Am I determining the truth correctly now?
Sure, reality is always a good barometer of truth. The train tracks meet on the horizon due to perspective, so it makes sense that the sun can meet the horizon due to perspective as well. Also, as I mentioned on page 1 (http://forum.tfes.org/index.php?topic=5346.msg103837#msg103837), it is part of Earth Not a Globe that the sinking effect is explained by hiding behind things on the horizon. So far, so good.
If two train tracks are laid out in front of you at an angle pointing towards each other, then obviously, two lines oriented in that position that will intersect at some point. Only parallel lines can continue into infinity and never intersect.
Yes, we agree on this. Parallel lines never intersect. Brilliant deduction Sherlock.
The type of math you are using says that the train tracks should approach each other but NEVER meet.
But from what we see and experience the tracks are angled toward each other in a way that they MUST meet.
So what's right? Are our experiences correct, or is a theoretical calculation which takes place outside of the universe and is missing a dimension correct?

If two train tracks are laid out in front of you at an angle pointing towards each other, then obviously, two lines oriented in that position that will intersect at some point. Only parallel lines can continue into infinity and never intersect.
Yes, we agree on this. Parallel lines never intersect. Brilliant deduction Sherlock.
The type of math you are using says that the train tracks should approach each other but NEVER meet.
But from what we see and experience the tracks are angled toward each other in a way that they MUST meet.
So what's right? Are our experiences correct, or is a theoretical calculation which takes place outside of the universe and is missing a dimension correct?
We have been through this already. Stop arguing in a circle.
Yes, parallel lines are angled towards each other from our perspective.
No, they will never actually meet each other.
Yes, it appears that they meet each other because the angle between them becomes too small for our eyes to distinguish. Using a telescope can extend the range that they appear to not touch at, obviously.
Yes, we can calculate exactly what this angle is using trigonometry, as shown previously on this thread. (http://forum.tfes.org/index.php?topic=5346.msg104144#msg104144)
Yes, the "out of this universe" diagram can correctly portray this angle, as shown previously on this thread.
No, the math doesn't predict an infinity. It predicts that they will never touch, because they can never reach infinity.
Yes, this math can be demonstrated to work at small, testable scales.
No, you have no evidence that it magically stops working at larger scales, other than blind faith in your model.
Ok, now that I have brought us back full circle, can you stop dodging the question, and just answer how you think this stuff can be calculated, if the math is indeed wrong?
"take the distance between the tracks and the angle of their progression"  Um... how do I take a distance between an object and an angle?
"determine where they would intersect in the distance"  How? Is there a special Bishop equation that I can use?
"Calculate based on what we and experience"  Ok. Great. How do I perform this calculation?

If two train tracks are laid out in front of you at an angle pointing towards each other, then obviously, two lines oriented in that position that will intersect at some point. Only parallel lines can continue into infinity and never intersect.
Yes, we agree on this. Parallel lines never intersect. Brilliant deduction Sherlock.
The type of math you are using says that the train tracks should approach each other but NEVER meet.
But from what we see and experience the tracks are angled toward each other in a way that they MUST meet.
So what's right? Are our experiences correct, or is a theoretical calculation which takes place outside of the universe and is missing a dimension correct?
We have been through this already. Stop arguing in a circle.
Yes, parallel lines are angled towards each other from our perspective.
No, they will never actually meet each other.
Yes, it appears that they meet each other because the angle between them becomes too small for our eyes to distinguish. Using a telescope can extend the range that they appear to not touch at, obviously.
Yes, we can calculate exactly what this angle is using trigonometry, as shown previously on this thread. (http://forum.tfes.org/index.php?topic=5346.msg104144#msg104144)
Yes, the "out of this universe" diagram can correctly portray this angle, as shown previously on this thread.
No, the math doesn't predict an infinity. It predicts that they will never touch, because they can never reach infinity.
Yes, this math can be demonstrated to work at small, testable scales.
No, you have no evidence that it magically stops working at larger scales, other than blind faith in your model.
Ok, now that I have brought us back full circle, can you stop dodging the question, and just answer how you think this stuff can be calculated, if the math is indeed wrong?
"take the distance between the tracks and the angle of their progression"  Um... how do I take a distance between an object and an angle?
"determine where they would intersect in the distance"  How? Is there a special Bishop equation that I can use?
"Calculate based on what we and experience"  Ok. Great. How do I perform this calculation?
No, two lines angled towards each other WILL meet. The angle is not "just too small that you can't see it and it actually continues forever". That is not possible. We have two lines angled towards each other. Think about it. What you are describing is impossible. The only way for the lines to continue forever without meeting is if they appeared as PARALLEL lines.

No, two lines angled towards each other WILL meet. The angle is not "just too small that you can't see it and it actually continues forever". That is not possible. We have two lines angled towards each other. Think about it. What you are describing is impossible. The only way for the lines to continue forever without meeting is if they appeared as PARALLEL lines.
Ok, let's say an object is travelling along one of these lines. For simplicity's sake, let's say that these lines make a right angle with each other.
A  B



C
Let's say Bob is travelling from B to A. There are 10 dashes between A and B. Let's say Bob is travelling at a rate of 10/t^{2} dashes per second. t is the time in seconds and starts at 1. How long before Bob intersects the line AC?
Obligatory stop dodging the question:
"take the distance between the tracks and the angle of their progression"  Um... how do I take a distance between an object and an angle?
"determine where they would intersect in the distance"  How? Is there a special Bishop equation that I can use?
"Calculate based on what we and experience"  Ok. Great. How do I perform this calculation?

yo tom: do you think this photo is of a woman who is taller than the leaning tower of pisa? why or why not?
(http://i.imgur.com/V0XJgPQ.png)

How about answer how celestial navigation can be used to get a fix?
The method used predicts what angle certain stars should appear to an observer at a location at a certain time.
This is real world application that was and is(much rarely now) with accurate and reliable results. This suggest that the math you say fails at those distances does work at them.
If celestial navigation works what changes when predicting what angle the Sun should appear to a person?
I would prefer to keep these threads on topic.
How is it off topic?
I am asking about predicting the angle stars appear to the observer. This thread is talking about predicting the angle of the nearest star to us, the Sun.
Your claim the math does not work or is unreliable. Celestial navigation being used to get a fix using math suggest you are wrong about distance causing the math to fail.
Your last post on the subject of celestial navigation was only partially correct. You can get a line of position without using math. To get a fix math is involved. I really suggest you look into how to navigate and finding your position using the stars. It will give you an insight on the distances of where math still works. Since long before GPS, LORAN and computers people figured out using the math you claim is wrong to determine their position anywhere on Earth as long as they could see the stars and horizon. (When someone figured out they could use an artificial horizon they only needed to see the stars)

How is it off topic?
I am asking about predicting the angle stars appear to the observer. This thread is talking about predicting the angle of the nearest star to us, the Sun.
Your claim the math does not work or is unreliable. Celestial navigation being used to get a fix using math suggest you are wrong about distance causing the math to fail.
Your last post on the subject of celestial navigation was only partially correct. You can get a line of position without using math. To get a fix math is involved. I really suggest you look into how to navigate and finding your position using the stars. It will give you an insight on the distances of where math still works. Since long before GPS, LORAN and computers people figured out using the math you claim is wrong to determine their position anywhere on Earth as long as they could see the stars and horizon. (When someone figured out they could use an artificial horizon they only needed to see the stars)
You're going to have to show us an example of what you mean.
yo tom: do you think this photo is of a woman who is taller than the leaning tower of pisa? why or why not?
(http://i.imgur.com/V0XJgPQ.png)
According to our perspective the woman is taller.
No, two lines angled towards each other WILL meet. The angle is not "just too small that you can't see it and it actually continues forever". That is not possible. We have two lines angled towards each other. Think about it. What you are describing is impossible. The only way for the lines to continue forever without meeting is if they appeared as PARALLEL lines.
Ok, let's say an object is travelling along one of these lines. For simplicity's sake, let's say that these lines make a right angle with each other.
A  B



C
Let's say Bob is travelling from B to A. There are 10 dashes between A and B. Let's say Bob is travelling at a rate of 10/t^{2} dashes per second. t is the time in seconds and starts at 1. How long before Bob intersects the line AC?
Obligatory stop dodging the question:
"take the distance between the tracks and the angle of their progression"  Um... how do I take a distance between an object and an angle?
"determine where they would intersect in the distance"  How? Is there a special Bishop equation that I can use?
"Calculate based on what we and experience"  Ok. Great. How do I perform this calculation?
I don't know what you are getting at, and I don't have an equation for you. I was explaining the method of calculation. You would calculate based on what we see and experience  angled lines clearly approaching each other to a point in the distance, not on what is theorized. Two lines angled towards each other will clearly meet at some point. They do not go on and on for infinity.

No, two lines angled towards each other WILL meet. The angle is not "just too small that you can't see it and it actually continues forever". That is not possible. We have two lines angled towards each other. Think about it. What you are describing is impossible. The only way for the lines to continue forever without meeting is if they appeared as PARALLEL lines.
Ok, let's say an object is travelling along one of these lines. For simplicity's sake, let's say that these lines make a right angle with each other.
A  B



C
Let's say Bob is travelling from B to A. There are 10 dashes between A and B. Let's say Bob is travelling at a rate of 10/t^{2} dashes per second. t is the time in seconds and starts at 1. How long before Bob intersects the line AC?
Obligatory stop dodging the question:
"take the distance between the tracks and the angle of their progression"  Um... how do I take a distance between an object and an angle?
"determine where they would intersect in the distance"  How? Is there a special Bishop equation that I can use?
"Calculate based on what we and experience"  Ok. Great. How do I perform this calculation?
I don't know what you are getting at, and I don't have an equation for you. I was explaining the method of calculation. You would calculate based on what we see and experience  angled lines clearly approaching each other to a point in the distance, not on what is theorized. Two lines angled towards each other will clearly meet at some point. They do not go on and on for infinity.
Ugh, should've stuck with middle school math rather than high school math. Whatever. The answer is never. Bob will never reach A at that rate, even though he is following a line going towards A.
The key here is that there is a difference between the lines meeting and objects following the lines meeting. Obviously, the lines themselves meet on the projection. In fact, the lines cross each other and then spread apart. Are you implying that at some point, two objects following parallel paths will not only meet, but cross each other and then appear to separate? Because that is the logical conclusion of your reasoning.

yo tom: do you think this photo is of a woman who is taller than the leaning tower of pisa? why or why not?
(http://i.imgur.com/V0XJgPQ.png)
According to our perspective the woman is taller.
this is oddly evasive. do you agree that the woman in this photo is, physically, probably not taller than the tower of pisa?

yo tom: do you think this photo is of a woman who is taller than the leaning tower of pisa? why or why not?
(http://i.imgur.com/V0XJgPQ.png)
According to our perspective the woman is taller.
this is oddly evasive. do you agree that the woman in this photo is, physically, probably not taller than the tower of pisa?
If you change the perspective you might get a different result. But from this perspective she is taller.

Ugh, should've stuck with middle school math rather than high school math. Whatever. The answer is never. Bob will never reach A at that rate, even though he is following a line going towards A.
The key here is that there is a difference between the lines meeting and objects following the lines meeting. Obviously, the lines themselves meet on the projection. In fact, the lines cross each other and then spread apart. Are you implying that at some point, two objects following parallel paths will not only meet, but cross each other and then appear to separate? Because that is the logical conclusion of your reasoning.
Yes, the lines meet on the projection. Therefore they meet in visual reality. The railroad tracks appear to meet in the distance, and if you were to shine a laser pointer at that point where they meet, a second observer would see that the dot would spread out to cover the entire railroad and a good chunk of the land around it, not merely one track at a time. The point where the tracks merge is very real for all visual purposes.
As per your claim that the perspective lines meet and then separate in opposite directions, I don't believe anything has been observe to that effect and therefore your interpretation must be wrong.

yo tom: do you think this photo is of a woman who is taller than the leaning tower of pisa? why or why not?
(http://i.imgur.com/V0XJgPQ.png)
According to our perspective the woman is taller.
this is oddly evasive. do you agree that the woman in this photo is, physically, probably not taller than the tower of pisa?
If you change the perspective you might get a different result. But from this perspective she is taller.
cool. do you agree that the woman in this photo is, physically, probably not taller than the tower of pisa?

yo tom: do you think this photo is of a woman who is taller than the leaning tower of pisa? why or why not?
(http://i.imgur.com/V0XJgPQ.png)
According to our perspective the woman is taller.
this is oddly evasive. do you agree that the woman in this photo is, physically, probably not taller than the tower of pisa?
If you change the perspective you might get a different result. But from this perspective she is taller.
cool. do you agree that the woman in this photo is, physically, probably not taller than the tower of pisa?
Maybe not. But we would need to change the perspective to see that.

yo tom: do you think this photo is of a woman who is taller than the leaning tower of pisa? why or why not?
(http://i.imgur.com/V0XJgPQ.png)
According to our perspective the woman is taller.
this is oddly evasive. do you agree that the woman in this photo is, physically, probably not taller than the tower of pisa?
If you change the perspective you might get a different result. But from this perspective she is taller.
cool. do you agree that the woman in this photo is, physically, probably not taller than the tower of pisa?
Maybe not. But we would need to change the perspective to see that.
lol maybe not? so you think maybe this woman is taller than the tower of pisa?

<irrelevant crap about lasers removed>
As per your claim that the perspective lines meet and then separate in opposite directions, I don't believe anything has been observe to that effect and therefore your interpretation must be wrong.
Another brilliant deduction! That was my point. I was using your logic to come to that conclusion. Remember saying this?
Yes, the lines meet on the projection. Therefore they meet in visual reality.
Yes, the lines cross on the projection. Therefore they cross in visual reality.
Both use the same logic. Both are just as wrong.
I don't have an equation for you. I was explaining the method of calculation. You would calculate based on what we see and experience  angled lines clearly approaching each other to a point in the distance, not on what is theorized.
You think I should "calculate based on what I see", but not on what is theorized, but you don't have an equation for me? Perhaps you can do an example for me, to show me how it is done? I have a strong suspicion that you have no earthly idea what you are talking about.

yo tom: do you think this photo is of a woman who is taller than the leaning tower of pisa? why or why not?
(http://i.imgur.com/V0XJgPQ.png)
According to our perspective the woman is taller.
this is oddly evasive. do you agree that the woman in this photo is, physically, probably not taller than the tower of pisa?
If you change the perspective you might get a different result. But from this perspective she is taller.
cool. do you agree that the woman in this photo is, physically, probably not taller than the tower of pisa?
Maybe not. But we would need to change the perspective to see that.
lol maybe not? so you think maybe this woman is taller than the tower of pisa?
You seem to have some difficulty here. I obviously agreed with you.

Another brilliant deduction! That was my point. I was using your logic to come to that conclusion. Remember saying this?
My logic is that we must take our queues for how things are from reality. We don't see the effect you claim exists, therefore it does not exist.
Yes, the lines cross on the projection. Therefore they cross in visual reality.
Both use the same logic. Both are just as wrong.
I don't see any perspective lines crossing over each other in reality. I just see that they meet and end where they meet. Not sure where you got your crossover ideas from.
You think I should "calculate based on what I see", but not on what is theorized, but you don't have an equation for me? Perhaps you can do an example for me, to show me how it is done? I have a strong suspicion that you have no earthly idea what you are talking about.
Why should I do any equations? Do you really need an equation to know that two lines which are angled towards each other will eventually meet?

I don't see any perspective lines crossing over each other in reality. I just see that they meet and end where they meet. Not sure where you got your crossover ideas from.
Your argument was that they are angled towards each other, therefore they must touch. By that logic, they are angled towards each other, therefore they must cross. It's bad logic.
Your other argument was that "they appear to touch, therefore they touch, therefore the math is wrong". However, the math predicts that as they move away from you, they will appear to get closer together. Close enough that there is no way to visually distinguish whether they are touching or not. The math predicts that they can become so close that they will appear to touch. So why on earth are you using "they appear to touch" as an argument that the math is wrong? More bad logic.
You think I should "calculate based on what I see", but not on what is theorized, but you don't have an equation for me? Perhaps you can do an example for me, to show me how it is done? I have a strong suspicion that you have no earthly idea what you are talking about.
Why should I do any equations? Do you really need an equation to know that two lines which are angled towards each other will eventually meet?
What? No, that's not what I was asking at all. Good grief, do I have to spell out everything 12 times?
If the math is wrong, as you claim, what is the correct way to determine the angle between the sun and the horizon? Or any 2 objects for that matter? In other words, if I told you the location of several objects relative to you, how would you calculate the apparent angle between them?

Your argument was that they are angled towards each other, therefore they must touch. By that logic, they are angled towards each other, therefore they must cross. It's bad logic.
Two lines angled towards each other must touch, that is logically sound, and the lines are seen to touch, which places the concept in reality as well.
Your crossover idea does not have a component in reality, and is a mere theoretical concept like the bad mathematics which you have presented in the OP.
Your other argument was that "they appear to touch, therefore they touch, therefore the math is wrong". However, the math predicts that as they move away from you, they will appear to get closer together. Close enough that there is no way to visually distinguish whether they are touching or not. The math predicts that they can become so close that they will appear to touch. So why on earth are you using "they appear to touch" as an argument that the math is wrong? More bad logic.
If they appear to touch, they touch, okay? The human eye can see a single photon in a dark room. That is very good resolution. If the tracks are appearing to merge at a point then it means that black photons are arriving side by side without any gap. There is no "almost" touching. The gap is gone.
What? No, that's not what I was asking at all. Good grief, do I have to spell out everything 12 times?
If the math is wrong, as you claim, what is the correct way to determine the angle between the sun and the horizon? Or any 2 objects for that matter? In other words, if I told you the location of several objects relative to you, how would you calculate the apparent angle between them?
The correct way to determine the angle of the sun is to make our determinations based on reality, not theoretical mathematics which lack a dimension. Take a protractor. When the sun is overhead at noontime the sun is at 90 degrees and at sunset the sun is at 0 degrees. There are your angles for the sun. It's quite simple.

If I am following this right Tom is saying the math is wrong because if what ever you are viewing like railroad and do not have enough resolution tracks can appear to touch in the distance.
I do not know what to say if this is his reasoning.
Tom what I was saying about celestial navigation is the entire method is based on angles. The sextant is used to measure the apparent angle of a star above the horizon. Then everything that follows is using math to predict where on Earth at that time that star will be viewed at that angle.
My reasoning is if you are claiming the math fails after a certain distance. Like predicting what angle the Sun should be above the horizon. On a flat earth with a Sun 3k miles high the math says it should never appear to touch the horizon, let alone appear to go below it.

When the sun is overhead at noontime the sun is at 90 degrees and at sunset the sun is at 0 degrees. There are your angles for the sun. It's quite simple.
Yes we agree on that, this is a fact! On flat earth to do so, the sun should seems to accelerate up to noon and decelerate afterwards. Which is not the case as any observer could verify. I prove it on the other thread.
So please prove it otherwise.

You seem to have some difficulty here. I obviously agreed with you.
i seem to be having difficulty? it took you three posts to admit that the woman in the photo "maybe" isn't as tall as the tower of pisa.
so, just to be clear, you agree that this woman only appears as tall as the tower because of the orientation of the camera, yes? why do you suppose it is that she can appear to be larger than the tower and yet physically not be taller than the tower? what is the cause of the disparity?

Tom, you are using two arguments to justify your conclusion. Argument 1 will be labelled in blue, argument 2 will be in red.
(1)Two lines angled towards each other must touch, that is logically sound, and (2)the lines are seen to touch, which places the concept in reality as well.
...
(2)If they appear to touch, they touch, okay? The human eye can see a single photon in a dark room. That is very good resolution. If the tracks are appearing to merge at a point then it means that black photons are arriving side by side without any gap. There is no "almost" touching. The gap is gone.
Argument 1, by itself, was already shown to be bad logic by my "crossover" example. You are using argument 2 to justify argument 1, which is fine, but that means if there is a problem with argument 2, both arguments have to be tossed out.
And there most certainly is a problem argument 2. It assumes we have perfect vision. That is possibly the dumbest assumption I have ever seen anyone make on this website, which is impressive considering the trolls that come by. Do you really think your eyes are good enough to see any arbitrarily small detail? Can you see a hair from 100 meters away? 1000 meters? 10000000 meters? Lol.
The correct way to determine the angle of the sun is to make our determinations based on reality, not theoretical mathematics which lack a dimension. Take a protractor. When the sun is overhead at noontime the sun is at 90 degrees and at sunset the sun is at 0 degrees. There are your angles for the sun. It's quite simple.
Good grief, are you purposely misunderstanding the point of the question?
Yes, the sun is at 0 degrees at sunset and 90 degrees when directly overhead. Thanks for stating the obvious. The point is to be able to figure out our distance from the sun based on the sun's height and angle. We can do this easily using trigonometry, or an orthographic diagram. And the angles/distances that we calculate agree with reality for any testable distance. Your only argument is that the math doesn't work (which it does, for testable distances) or that it doesn't work for long distances that are conveniently too long to test.
So, if the math doesn't work, what is the correct way to determine the distance of the sun based on its angle and height? Or, alternatively, what is the correct way to determine its angle based on its distance and height? The person in the video clearly attempts to do this with his orthographic diagram overlayed with perspective lines. However, his lines seemed arbitrary and his reasoning was vague. So, what is the correct way to do it? Show us. Use these numbers:
Object A is 500 meters away from you on the ground. Object B is 50 meters directly over object A. What is the angle between object A and B from your perspective?

You should probably address Tom's assertion, in reply 79, that since we do not see perspective lines crossover in reality, it is illogical to assume they should. Otherwise he will just behave as if he has the higher ground.

You should probably address Tom's assertion, in reply 79, that since we do not see perspective lines crossover in reality, it is illogical to assume they should. Otherwise he will just behave as if he has the higher ground.
Since there is a decent chance he will miss the point, I will clarify:
"Two lines angled towards each other must touch" is no more logical than "two lines angled towards each other must cross". I think both are illogical. Therefore, he is only left with the argument "the lines are seen to touch, therefore they touch", which assumes his eyes are perfect. Which is just silly.

You should probably address Tom's assertion, in reply 79, that since we do not see perspective lines crossover in reality, it is illogical to assume they should. Otherwise he will just behave as if he has the higher ground.
Since there is a decent chance he will miss the point, I will clarify:
"Two lines angled towards each other must touch" is no more logical than "two lines angled towards each other must cross". I think both are illogical. Therefore, he is only left with the argument "the lines are seen to touch, therefore they touch", which assumes his eyes are perfect. Which is just silly.
I think Tom is not aware of asymptote!

You should probably address Tom's assertion, in reply 79, that since we do not see perspective lines crossover in reality, it is illogical to assume they should. Otherwise he will just behave as if he has the higher ground.
Since there is a decent chance he will miss the point, I will clarify:
"Two lines angled towards each other must touch" is no more logical than "two lines angled towards each other must cross". I think both are illogical. Therefore, he is only left with the argument "the lines are seen to touch, therefore they touch", which assumes his eyes are perfect. Which is just silly.
Assuming the existence of a hidden pocket of infinite distance in the railroad tracks, undetectable to man and machine, is even sillier.
We have more evidence that they touch than they do not. Empirically, they touch. It is only by an interpretation of ancient mathematics, that they do not.

Empirically they touch? Wow. That's like saying, "empirically viruses don't exist because that's what I observe with the naked eye. M
Tom, simple experiment to falsify your claim: find a point where two rails appear to converge. Have a friend stay there. You walk alongside the rails with radio contact with your friend until he sees you converge with the rails as well. Check to see if the rails have converged in reality.
Report back with your findings please.

Empirically they touch? Wow. That's like saying, "empirically viruses don't exist because that's what I observe with the naked eye.
That was true, until someone found empirical evidence for viruses.
Tom, simple experiment to falsify your claim: find a point where two rails appear to converge. Have a friend stay there. You walk alongside the rails with radio contact with your friend until he sees you converge with the rails as well. Check to see if the rails have converged in reality.
The friend has a different perspective than I do. Things will be differently for him. His experience of being 5 feet away from the tracks has nothing to do with the touching perspective lines I see.

Empirically they touch? Wow. That's like saying, "empirically viruses don't exist because that's what I observe with the naked eye.
That was true, until someone found empirical evidence for viruses.
Exactly.
Tom, simple experiment to falsify your claim: find a point where two rails appear to converge. Have a friend stay there. You walk alongside the rails with radio contact with your friend until he sees you converge with the rails as well. Check to see if the rails have converged in reality.
The friend has a different perspective than I do. Things will be differently for him. His experience of being 5 feet away from the tracks has nothing to do with the touching perspective lines I see.
You could simply apply this idea to the case of viruses. Someone looking under magnification has a different perspective and has nothing to do with invisible viruses I don't see.

Enough with the stupid railway lines and 20 feet high walls that converge due to perspective over long distances. Yes, that is a fact due to the limitations of the human eyes! We are talking about the FE sun 3000 miles above a flat earth. If there were lines parallel to the earth at 1000, 2000 and 3000 miles high, they would converge very slowly, far beyond any vanishing point we could see and far beyond 12000 miles. The sun would always be above the horizon on a flat earth. Don't use small scale perspective models to try and explain large scale real life logic and mathematics.
... and Tom, there is a big difference between queue and cue, which you don't seem to understand (que and queues used instead of cue and cues in this thread)

Enough with the stupid railway lines and 20 feet high walls that converge due to perspective over long distances. Yes, that is a fact due to the limitations of the human eyes! We are talking about the FE sun 3000 miles above a flat earth. If there were lines parallel to the earth at 1000, 2000 and 3000 miles high, they would converge very slowly, far beyond any vanishing point we could see and far beyond 12000 miles. The sun would always be above the horizon on a flat earth.
This is all very true, but...
Don't use small scale perspective models to try and explain large scale real life logic and mathematics.
I disagree with this. The math scales up quite nicely. There is no reason to think it doesn't. His entire argument rests on "maybe the math suddenly stops working at long distances for no apparent reason".
The simple fact is that congruent triangles have the same angles. A 3000x6000 mile right triangle has the same angle as a 30x60 meter right triangle and a 3x6 mm right triangle. This is the fact that Tom is desperately trying not to admit. (Edit: The ratio of the sides is what matters, not the absolute length of the sides.)

I agree with the above completely. Small or large makes no difference to the calculations. It is just that using small scale limits lots of people's conception on a grand scale.

"Two lines angled towards each other must touch" is no more logical than "two lines angled towards each other must cross". I think both are illogical. Therefore, he is only left with the argument "the lines are seen to touch, therefore they touch", which assumes his eyes are perfect. Which is just silly.
Assuming the existence of a hidden pocket of infinite distance in the railroad tracks, undetectable to man and machine, is even sillier.
There is no hidden pocket of infinite distance. I have made no such assumptions. Your understanding of infinity and basic geometry is childish at best, intentionally delusional at worst. I won't even bother countering this argument. Read below.
Empirically they touch? Wow. That's like saying, "empirically viruses don't exist because that's what I observe with the naked eye.
That was true, until someone found empirical evidence for viruses.
Oh good, I am glad that is cleared up.
Now let's take a step back for a moment, since you are clearly desperate to misinterpret any argument having to do with infinity. What is the purpose of your "hidden infinity" argument? The purpose is to cast doubt on the basic mathematics of perspective. However, we have empirical evidence that the math does work, at least at any testable distance. Therefore, your "hidden infinity" argument is wrong. Sorry. We can quibble about the subtleties of infinity all we want, but at the end of the day, you are the one who insists empirical evidence trumps all else.
We have empirical evidence for viruses, and we have empirical evidence that the math works. "But I can't see the virus! But I can't see the separation of the train tracks!" is no longer a valid argument.
Therefore, you are left only with your argument that "maybe the math suddenly stops working at large distances". We have already establish that you have no evidence for this either.

I agree with the above completely. Small or large makes no difference to the calculations. It is just that using small scale limits lots of people's conception on a grand scale.
It's not the scale that matters. It's the fact that he is trying to compare a triangle with side ratio of 3000:1 to a triangle with a side ratio of 2:1.
Tom Bishop's logic: "Two objects separated by a 0.01 degree angle appear to be touching, therefore the sun should appear to touch the horizon, even though they are separated by a 20 degree angle!" Derp.
Edit: Ok, I get what you are saying. Using a large scale makes it difficult to imagine/conceptualize. Yep.

You could simply apply this idea to the case of viruses. Someone looking under magnification has a different perspective and has nothing to do with invisible viruses I don't see.
The person looking at the virus under magnification has a different perspective, and will see different things. It's not that the virus doesn't exist at other scales, it's that we are not in a perspective to see it.
I disagree with this. The math scales up quite nicely. There is no reason to think it doesn't. His entire argument rests on "maybe the math suddenly stops working at long distances for no apparent reason".
The simple fact is that congruent triangles have the same angles. A 3000x6000 mile right triangle has the same angle as a 30x60 meter right triangle and a 3x6 mm right triangle. This is the fact that Tom is desperately trying not to admit. (Edit: The ratio of the sides is what matters, not the absolute length of the sides.)
No, that is not true that a 3000x6000 mile right triangle always has the same angle as a 30x60 meter right triangle. It also depends how you are looking at it.
The angles change depending on your perspective and how you look at them. You showed this change of angle yourself with the illustration you provided in the OP where the angle changed:
(http://imgur.com/Tt6gsDL.png)
A 40 degree angle turned into a completely different angle.

You could simply apply this idea to the case of viruses. Someone looking under magnification has a different perspective and has nothing to do with invisible viruses I don't see.
The person looking at the virus under magnification has a different perspective, and will see different things. It's not that the virus doesn't exist at other scales, it's that we are not in a perspective to see it.
Similarly, it is not that the perspective lines are touching, it is that you are in the wrong perspective to see it.
I disagree with this. The math scales up quite nicely. There is no reason to think it doesn't. His entire argument rests on "maybe the math suddenly stops working at long distances for no apparent reason".
The simple fact is that congruent triangles have the same angles. A 3000x6000 mile right triangle has the same angle as a 30x60 meter right triangle and a 3x6 mm right triangle. This is the fact that Tom is desperately trying not to admit. (Edit: The ratio of the sides is what matters, not the absolute length of the sides.)
No, that is not true that a 3000x6000 mile right triangle always has the same angle as a 30x60 meter right triangle. It also depends how you are looking at it.
The angles change depending on your perspective and how you look at them. You showed this change of angle yourself with the illustration you provided in the OP where the angle changed:
(http://imgur.com/Tt6gsDL.png)
A 40 degree angle turned into a completely different angle.
The angles change but congruent triangles being viewed in the same projection will have identical angles.

You could simply apply this idea to the case of viruses. Someone looking under magnification has a different perspective and has nothing to do with invisible viruses I don't see.
The person looking at the virus under magnification has a different perspective, and will see different things. It's not that the virus doesn't exist at other scales, it's that we are not in a perspective to see it.
I disagree with this. The math scales up quite nicely. There is no reason to think it doesn't. His entire argument rests on "maybe the math suddenly stops working at long distances for no apparent reason".
The simple fact is that congruent triangles have the same angles. A 3000x6000 mile right triangle has the same angle as a 30x60 meter right triangle and a 3x6 mm right triangle. This is the fact that Tom is desperately trying not to admit. (Edit: The ratio of the sides is what matters, not the absolute length of the sides.)
No, that is not true that a 3000x6000 mile right triangle always has the same angle as a 30x60 meter right triangle. It also depends how you are looking at it.
The angles change depending on your perspective and how you look at them. You showed this change of angle yourself with the illustration you provided in the OP where the angle changed:
(http://imgur.com/Tt6gsDL.png)
A 40 degree angle turned into a completely different angle.
Oh goodie, back to angles 101. That picture represents an angle between 3 objects seen from an outside perspective. The physical angle doesn't change. The angle as we see it can change depending on where we are relative to the triangle.
When we are talking about the angle between the sun and the horizon, what we actually mean is the angle between the sun, our eyes, and the horizon. From a first person perspective, we can't actually see an angle like we do in the above picture, since we are physically located at the corner of the triangle. HOWEVER, we can measure the angle based on their apparent distance from each other in our vision. For example, if we have 180 degree vision, and the distance between two objects takes up half our vision, then there is a 90 degree angle between the two objects. Someone looking at us and the objects from the side would be able to measure that angle as 90 degrees with a protractor. This is easier to calculate through a camera, since a camera has a much more well defined field of view.
Once we know the angle based on the above method, we can calculate distances.
All of this is easily testable with a camera, several objects, and a tape measure.

You could simply apply this idea to the case of viruses. Someone looking under magnification has a different perspective and has nothing to do with invisible viruses I don't see.
The person looking at the virus under magnification has a different perspective, and will see different things. It's not that the virus doesn't exist at other scales, it's that we are not in a perspective to see it.
Similarly, it is not that the perspective lines are touching, it is that you are in the wrong perspective to see it.
I think you have gone off track a bit with this analogy.
The virus does not have zero width.
Similarly, the math predicts a nonzero width between the train tracks.
However, the width of a virus is too small to see.
Similarly, the distance between the train tracks is too small to see. Because our eyes aren't perfect. Obviously.

No, that is not true that a 3000x6000 mile right triangle always has the same angle as a 30x60 meter right triangle. It also depends how you are looking at it.
The angles change depending on your perspective and how you look at them. You showed this change of angle yourself with the illustration you provided in the OP where the angle changed:
(http://imgur.com/Tt6gsDL.png)
A 40 degree angle turned into a completely different angle.
Oh goodie, back to angles 101. That picture represents an angle between 3 objects seen from an outside perspective. The physical angle doesn't change. The angle as we see it can change depending on where we are relative to the triangle.
We don't live in a two dimensional world. We might be looking at the angle on the far right, but you want us to measure it like we're looking at it like the angle on the far left. That makes no sense at all. You need to do the math from its appearance in reality, not from a hypothetical side view universe. The hypothetical side view is missing a dimension and certain aspects of perspective.
Your position that all angles are absolute and do not ever change is ridiculous. In fact, the idea that all angles should be measured from a specific side in a universe with higher dimensions is entirely arbitrary and bias. The illustration shows it very clearly. The angle goes from 40 degrees, to 51 degrees, to 73 degrees, as well as changes its height, depending on where we look!

No, that is not true that a 3000x6000 mile right triangle always has the same angle as a 30x60 meter right triangle. It also depends how you are looking at it.
The angles change depending on your perspective and how you look at them. You showed this change of angle yourself with the illustration you provided in the OP where the angle changed:
(http://imgur.com/Tt6gsDL.png)
A 40 degree angle turned into a completely different angle.
Oh goodie, back to angles 101. That picture represents an angle between 3 objects seen from an outside perspective. The physical angle doesn't change. The angle as we see it can change depending on where we are relative to the triangle.
We don't live in a two dimensional world. We might be looking at the angle on the far right, but you want us to measure it like we're looking at it like the angle on the far left. That makes no sense at all.
No, I did not say I want you to measure it like on the far left. Nor are we looking at it from the perspective of the far right. We are looking at it from a first person perspective, so I said to measure it from first person perspective, which is not portrayed in the above image. I gave a method for measuring it based on the FOV of the camera in a previous post. It just so happens that the angle measured from a first person perspective agrees with the angle on the far left.
(Edit: clarification)
You need to do the math from its appearance in reality, not from a hypothetical side view universe. The hypothetical side view is missing a dimension and certain aspects of perspective.
Your position that all angles are absolute and do not ever change is ridiculous. In fact, the idea that all angles should be measured from a specific side in a universe with higher dimensions is entirely arbitrary and bias. The illustration shows it very clearly. The angle goes from 40 degrees, to 51 degrees, to 73 degrees, as well as changes its height, depending on where we look!
Stop putting words in my mouth. I did not say all angles are absolute and don't ever change. In fact, I was the one who originally made the claim that an angle can appear different depending on what angle we are viewing the angle from, even though the physical angle, as measured by a protractor resting up against the triangle, doesn't change.
First it was a side view in a "universe missing a dimension", now it's a "universe with higher dimensions"? And you call my argument arbitrary. Lol.
Look, you can make all the nonsensical, hypothetical arguments that you want. However, as you said, empirical evidence is king. I gave you a method for measuring the angle from a first person perspective. The simple fact is that this angle is equal to the angle measured in the side view diagram, which is equal to the angle measured by a protractor in reality, which is equal to the angle calculated by simple trigonometry. The math works, regardless of your misgivings about orthographic projections.
angle measured from a picture using FOV = side view angle = real angle measured by protractor = angle calculated by math.

We don't live in a two dimensional world. We might be looking at the angle on the far right, but you want us to measure it like we're looking at it like the angle on the far left. That makes no sense at all. You need to do the math from its appearance in reality, not from a hypothetical side view universe. The hypothetical side view is missing a dimension and certain aspects of perspective.
Hey, don't take this up with us: Tell it to Rowbotham. Or do you not know the contents and illustrations of your own Bible (http://www.sacredtexts.com/earth/za/za23.htm)? You send us there often enough that the rest of us all know what's in it, maybe you should too.
(http://www.sacredtexts.com/earth/za/img/fig58.jpg)
While you're at it, you better tell Voliva about it too. (http://i45.tinypic.com/334n0yc.png)
Heck, tell it to YOUR OWN WIKI (http://wiki.tfes.org/Distance_to_the_Sun), for crying out loud! This image is proudly displayed, with no scornful denunciation of "hypothetical side views" anywhere to be seen!
(http://wiki.tfes.org/images/4/40/Flatrth.png)

We don't live in a two dimensional world. We might be looking at the angle on the far right, but you want us to measure it like we're looking at it like the angle on the far left. That makes no sense at all. You need to do the math from its appearance in reality, not from a hypothetical side view universe. The hypothetical side view is missing a dimension and certain aspects of perspective.
Hey, don't take this up with us: Tell it to Rowbotham. Or do you not know the contents and illustrations of your own Bible (http://www.sacredtexts.com/earth/za/za23.htm)? You send us there often enough that the rest of us all know what's in it, maybe you should too.
<lots of side view diagrams from flat earthers>
Whoa whoa whoa, I call foul. Hypothetical side views are only disallowed if they support a round earth. If they support a flat earth, they are fine. Please learn the rules of this forum.

Whoa whoa whoa, I call foul. Hypothetical side views are only disallowed if they support a round earth. If they support a flat earth, they are fine. Please learn the rules of this forum.
That's true, and no better example than Tom Bishop himself, who provided the following "hypothetical side view" in HIS OWN POST (http://forum.tfes.org/index.php?topic=5337.msg103452#msg103452) in support of a troposcatter theory to explain how "satellite" TV works in a world without space flight.
(http://www.tpub.com/neets/book10/NTX226.GIF)
Seriously, Tom, you need to get your act together.
If hypothetical side views are of no value, you're not allowed to use them yourself!

Hey, don't take this up with us: Tell it to Rowbotham. Or do you not know the contents and illustrations of your own Bible (http://www.sacredtexts.com/earth/za/za23.htm)? You send us there often enough that the rest of us all know what's in it, maybe you should too.
Except, of course, if you look at the material you would find that Rowbotham is measuring and comparing angles in the sky as they actually exist from the first person, not predicting where the bodies are or where they might go according to a particular type of math in the third person.
The angles Rowbotham measures are very empirical, taken from direct observation of the sun. The angles are measured directly, and therefore they are true. He does not begin his inquiry by assuming what the angles would be in an outside universe. He does exactly as I suggested; to begin inquiry from reality and to base any and all conclusions on what is experienced.
He uses a side view, but it is not "hypothetical" as I framed the mathematical construct in the OP as a "hypothetical side view universe". Rowbotham's approach to the subject is to begin by observing the angles of the sun directly and seek to understand how they interrelate with each other; not to base his approach on a fantasy. The measurements directly translate into reality, and therefore is a comparatively much more empirical approach to truth, regardless of debatable accuracy.

Hey, don't take this up with us: Tell it to Rowbotham. Or do you not know the contents and illustrations of your own Bible (http://www.sacredtexts.com/earth/za/za23.htm)? You send us there often enough that the rest of us all know what's in it, maybe you should too.
Except, of course, if you look at the material you would find that Rowbotham is measuring and comparing angles in the sky as they actually exist from the first person, not predicting where the bodies are or where they might go according to a particular type of math in the third person.
The angles Rowbotham measures are very empirical, taken from direct observation of the sun. The angles are measured directly, and therefore they are true. He does not begin his inquiry by assuming what the angles would be in an outside universe. He does exactly as I suggested; to begin inquiry from reality and to base any and all conclusions on what is experienced.
He uses a side view, but it is not "hypothetical" as I framed the mathematical construct in the OP as a "hypothetical side view universe". The measurements directly translate into reality, and therefore is a comparatively much more empirical approach to truth, regardless of debatable accuracy.
Rowbotham's approach to the subject is to begin by observing the angles of the sun directly and seek to understand how they interrelate with each other; not to base his approach on a fantasy. I note that in the chapter Rowbotham declines to give exact values for the sun's height, primarily concluding that it must be close to the earth's surface on a Flat Earth.
The measurements that astronomers and surveyors make are also "very empirical", and taken from a first person perspective. They also work from observation. It is almost like your protests in this thread are nonsense.

The measurements that astronomers and surveyors make are also "very empirical", and taken from a first person perspective. They also work from observation. It is almost like your protests in this thread are nonsense.
The empirical measurements of astronomers isn't really being doubted in a lot of the Flat Earth material. It is being argued that the astronomers compute their distance of the sun based on a Round Earth. If you assume a Flat Earth, with a flat baseline, the sun becomes a lot closer to the earth's surface. It is an easy mistake to make when you have been brainwashed from birth into believing that you live on a ball.

The measurements that astronomers and surveyors make are also "very empirical", and taken from a first person perspective. They also work from observation. It is almost like your protests in this thread are nonsense.
The empirical measurements of astronomers isn't really being doubted in a lot of the Flat Earth material. It is being argued that the astronomers compute their distance of the sun based on a Round Earth. If you assume a Flat Earth, with a flat baseline, the sun becomes a lot closer to the earth's surface. It is an easy mistake to make when you have been brainwashed from birth into believing that you live on a ball.
But then starts all the other issues that flat earth can't explain, if the sun is as close to the surface as math would tell, assuming the earth is flat. We've been telling you that a gazillion times.

The empirical measurements of astronomers isn't really being doubted in a lot of the Flat Earth material.
FALSE. Astronomers measure a zero degree angle to the sun at sunset, to which FE materials reply that the sun is at some nonzero elevation and propose an atmospheric / perspective effect to "explain" that the sun isn't actually located where it appears to be located. If that doesn't count as "empirical measurement isn't really being doubted" then I don't know what would count.

Hey, don't take this up with us: Tell it to Rowbotham. Or do you not know the contents and illustrations of your own Bible (http://www.sacredtexts.com/earth/za/za23.htm)? You send us there often enough that the rest of us all know what's in it, maybe you should too.
Except, of course, if you look at the material you would find that Rowbotham is measuring and comparing angles in the sky as they actually exist from the first person, not predicting where the bodies are or where they might go according to a particular type of math in the third person.
Quoting directly from ENAG: Chapter V: The True Distance of the Sun (http://www.sacredtexts.com/earth/za/za23.htm).
The distance from London Bridge to the seacoast at Brighton, in a straight line, is 50 statute miles. On a given day, at 12 o'clock, the altitude of the sun, from near the water at London Bridge, was found to be 61 degrees of an arc; and at the same moment of time the altitude from the seacoast at Brighton was observed to be 64 degrees of an arc, as shown in fig. 58. The baseline from L to B, 50 measured statute miles; the angle at L, 61 degrees; and the angle at B, 64 degrees. In addition to the method by calculation, the distance of the under edge of the sun may be ascertained from these elements by the method called "construction." The diagram, fig. 58, is the above case "constructed;" that is, the baseline from L to B represents 50 statute miles; and the line L, S, is drawn at an angle of 61 degrees, and the line B, S, at an angle of 64 degrees. Both lines are produced until they bisect or cross each other at the point S. Then, with a pair of compasses, measure the length of the baseline B, L, and see how many times the same length may be found in the line L, S, or B, S. It will be found to be
(http://www.sacredtexts.com/earth/za/img/fig58.jpg)
sixteen times, or sixteen times 50 miles, equal to 800 statute miles. Then measure in the same way the vertical line D, S, and it will be found to be 700 miles. Hence it is demonstrable that the distance of the sun over that part of the earth to which it is vertical is only 700 statute miles. By the same mode it may be ascertained that the distance from London of that part of the earth where the sun was vertical at the time (July 13th, 1870) the above observations were taken, was only 400 statute miles, as shown by dividing the baseline L, D, by the distance B, L. If any allowance is to be made for refractionwhich, no doubt, exists where the sun's rays have to pass through a medium, the atmosphere, which gradually increases in density as it approaches the earth's surfaceit will considerably diminish the abovenamed distance of the sun; so that it is perfectly safe to affirm that the under edge of the sun is considerably less than 700 statute miles above the earth.
Uhoh SpaghettiO.
He is doing the exact same thing that we have been doing here. The exact same thing you are so desperate to denounce.
1. Measure the altitude (angle) of the sun from a first person perspective.
2. Draw a side view diagram of the measured angles.
3. Measure the distances in the side view diagram. (Or just calculate them with math. Both will give the same answer.)
Sometimes we reverse the process, and go from distance to angle, instead of angle to distance. We can do it either way though. Both ways prove the absurdity of a 3000 mile high sun. (Or less than 700 miles, according to Rowbotham.)

But then starts all the other issues that flat earth can't explain, if the earth is as close to the surface as math would tell, assuming the earth is flat. We've been telling you that a gazillion times.
Please elaborate. I don't understand this reference.
The empirical measurements of astronomers isn't really being doubted in a lot of the Flat Earth material.
FALSE. Astronomers measure a zero degree angle to the sun at sunset, to which FE materials reply that the sun is at some nonzero elevation and propose an atmospheric / perspective effect to "explain" that the sun isn't actually located where it appears to be located. If that doesn't count as "empirical measurement isn't really being doubted" then I don't know what would count.
I don't believe the FE materials assert that the sun is at a nonzero elevation when the sun is appeared to be at a zero elevation. I think you are wrong on that count. Rowbotham describes the sun going to the eye level horizon.
Uhoh SpaghettiO.
He is doing the exact same thing that we have been doing here. The exact same thing you are so desperate to denounce.
1. Measure the altitude (angle) of the sun from a first person perspective.
2. Draw a side view diagram of the measured angles.
3. Measure the distances in the side view diagram. (Or just calculate them with math. Both will give the same answer.)
Sometimes we reverse the process, and go from distance to angle, instead of angle to distance. We can do it either way though. Both ways prove the absurdity of a 3000 mile high sun. (Or less than 700 miles, according to Rowbotham.)
We've already established that the angles in a side view are different than the angles that are observed. Recall this image where the angles changed when they were turned:
(http://imgur.com/Tt6gsDL.png)
Remember that? The angles changed from 41 degrees, to 51 degrees, to 71 degrees, as well as changed in height. Then in the second image you provided the angles stayed the same between orientations. How is it that the angles change in the first image, but not in the second? See below:
(http://imgur.com/BB4hGoj.png)
How is this possible? In the side view orientation the angles are 11 degrees, 14 degrees, and 20 degrees, and in the first person view the angles are 11 degrees, 14 degrees, and 20 degrees.
You obviously got it completely wrong, and are missing something, such as certain elements of perspective, when you translated the scene.
Rowbotham is taking the first person angles (which are not the same as the side view angles, remember) and illustrating those collected angles on paper, working solely with those values in a nonhypothetical side view. This is not the same as illustrating hypothetical side view angles from the start and forcing it into a halfthoughtout first person view which fails on properly translating perspective.

Uhoh SpaghettiO.
He is doing the exact same thing that we have been doing here. The exact same thing you are so desperate to denounce.
1. Measure the altitude (angle) of the sun from a first person perspective.
2. Draw a side view diagram of the measured angles.
3. Measure the distances in the side view diagram. (Or just calculate them with math. Both will give the same answer.)
Sometimes we reverse the process, and go from distance to angle, instead of angle to distance. We can do it either way though. Both ways prove the absurdity of a 3000 mile high sun. (Or less than 700 miles, according to Rowbotham.)
We've already established that the angles in a side view are different than the angles that are observed. Recall this image where the angles changed when they were turned:
(http://imgur.com/Tt6gsDL.png)
Remember that? The angles changed from 41 degrees, to 51 degrees, to 71 degrees, as well as changed in height. Then in the second image you provided the angles stayed the same between orientations. How is it that the angles change in the first image, but not in the second? See below:
(http://imgur.com/BB4hGoj.png)
How is this possible? In the side view orientation the angles are 11 degrees, 14 degrees, and 20 degrees, and in the first person view the angles are 11 degrees, 14 degrees, and 20 degrees.
You obviously got it completely wrong, and are missing something, such as certain elements of perspective, when you translated the scene.
Goodness, you are super confused about this subject. The first person perspective of the angle is not even illustrated in the first image. In Rowbotham's example, the angle does not change between the first person perspective and the side view either.
Rowbotham is taking the first person angles (which are not the same as the side view angles, remember)
Except they ARE the same angle in Rowbotham's diagram. Read it and weep.
and illustrating those collected angles on paper, working solely with those values in a nonhypothetical side view.
BY DRAWING A SIDE VIEW DIAGRAM, WHICH HAS THE EXACT SAME ANGLES AS MEASURED FROM THE FIRST PERSON PERSPECTIVE. You literally JUST argued that the angles should be different in this same post. Did you not notice that the angles Rowbotham draws in his side view are the same angles he measured from the first person perspective?
This is not the same as illustrating hypothetical side view angles from the start and forcing it into a halfthoughtout first person view. The angles Rowbotham collected and represented are, in fact, much more empirical, as they are more directly correlated to reality.
Normally, we have been starting with the hypothetical distances (according to the flat earth model), calculating the expected angle, and then comparing that with the measured angle. This is the reverse of what Rowbotham is doing, but we can just as easily do it in the same order as Rowbotham, if you think it will make a difference. Here you go:
Yesterday, the sun made an angle of 1 degree with the horizon just before sunset. This is the same type of measurable, first person perspective angle that Rowbotham starts out with. (It actually got even smaller than that, obviously, but we will start with 1 degree.) We will also assume that the sun is 3000 miles high, like in the flat earth model. From there, we can calculate the distance to the sun. We can draw a side view diagram like Rowbotham, or just use trigonometry. Both give the same answer.
3000 miles / tan(1 degree) = 172,000 miles.
So apparently, if the sun is indeed 3000 miles high, it should also be 172,000 miles away when it is only 1 degree from the horizon. This is WAAAAY bigger than the earth. Therefore, the sun probably isn't 3000 miles high. This is the EXACT same process that Rowbotham uses, in the EXACT same order.
By all means, continue arguing. It is quite entertaining to watch you desperately defend Rowbotham and denounce us at the same time, when we are using the EXACT same process. I can only imagine how uncomfortable the cognitive dissonance must be getting.

It's not the same process. You start off with a hypothesis, Rowbotham does not. Rowbotham starts off with reality. There is quite a big difference.
You have not shown why your two examples contradict each other. Are the angles the same when you turn the scene or not? Your examples are entirely contradictory.
Yesterday, the sun made an angle of 1 degree with the horizon just before sunset. This is the same type of measurable, first person perspective angle that Rowbotham starts out with. (It actually got even smaller than that, obviously, but we will start with 1 degree.) We will also assume that the sun is 3000 miles high, like in the flat earth model. From there, we can calculate the distance to the sun. We can draw a side view diagram like Rowbotham, or just use trigonometry. Both give the same answer.
3000 miles / tan(1 degree) = 172,000 miles.
So apparently, if the sun is indeed 3000 miles high, it should also be 172,000 miles away when it is only 1 degree from the horizon. This is WAAAAY bigger than the earth. Therefore, the sun probably isn't 3000 miles high. This is the EXACT same process that Rowbotham uses, in the EXACT same order.
That math is assuming that numbers near 0 approach infinity. But we know that things can touch the horizon. That makes your hypothesis suspect.
The ancient greeks have never observed an example of things traveling infinite distances. That assumption of the math is entirely without merit.

It's not the same process. You start off with a hypothesis, Rowbotham does not. Rowbotham starts off with reality. There is quite a big difference.
Rowbotham is starting off with a hypothesis as well: that the earth is flat. He is trying to calculate the height and distance of the sun assuming that the earth is flat.
I am starting off with almost the same hypothesis: that the earth is flat, and that the height of the sun is 3000 miles high. The calculations are the same though. The math is the same. The diagram is the same. The process is the same.
You have not shown why your two examples contradict each other. Are the angles the same when you turn the scene or not? Your examples are entirely contradictory.
*Sigh*
The point of the first diagram is to point out how angles change when measured by a protractor that is not parallel to the plane of the angle. In each view, imagine a protractor painted on your screen. In the "side view", the protractor is parallel to the plane of the angle, therefore the angle it measures is correct (40 degrees). In the middle and right views, the protractor on the screen is no longer parallel to the plane of the angle. Therefore, the angles measured by the protractor will be incorrect (51 degrees, 73 degrees). In a first person perspective, the protractor would be completely perpendicular to the plane of the angle. There is no way to measure the angle with a protractor that is perpendicular to the plane of the angle. Therefore, we need a DIFFERENT way to measure this angle from a first person perspective. I gave one possible way to measure that angle from a photo (bottom of the second image). Other possible ways include using a theodolite, sextant, stick and shadow, etc... All of these methods will yield an angle that is the same as the side view angle.
Once again, please notice that the angles measured by Rowbotham from a first person perspective are the same angles as those in his side view diagram.
Yesterday, the sun made an angle of 1 degree with the horizon just before sunset. This is the same type of measurable, first person perspective angle that Rowbotham starts out with. (It actually got even smaller than that, obviously, but we will start with 1 degree.) We will also assume that the sun is 3000 miles high, like in the flat earth model. From there, we can calculate the distance to the sun. We can draw a side view diagram like Rowbotham, or just use trigonometry. Both give the same answer.
3000 miles / tan(1 degree) = 172,000 miles.
So apparently, if the sun is indeed 3000 miles high, it should also be 172,000 miles away when it is only 1 degree from the horizon. This is WAAAAY bigger than the earth. Therefore, the sun probably isn't 3000 miles high. This is the EXACT same process that Rowbotham uses, in the EXACT same order.
That math is assuming that numbers near 0 approach infinity. But we know that things can touch the horizon. That makes your hypothesis suspect.
The ancient greeks have never observed an example of things traveling infinite distances. That assumption of the math is entirely without merit.
Did you already forget that this is the exact same math that Rowbotham uses? That was quick. For the millionth time, the math works for all testable distances, despite your abysmal and naive understanding of infinity and infinitesimals.

The measurements that astronomers and surveyors make are also "very empirical", and taken from a first person perspective. They also work from observation. It is almost like your protests in this thread are nonsense.
The empirical measurements of astronomers isn't really being doubted in a lot of the Flat Earth material. It is being argued that the astronomers compute their distance of the sun based on a Round Earth. If you assume a Flat Earth, with a flat baseline, the sun becomes a lot closer to the earth's surface. It is an easy mistake to make when you have been brainwashed from birth into believing that you live on a ball.
But then starts all the other issues that flat earth can't explain, if the earth is as close to the surface as math would tell, assuming the earth is flat. We've been telling you that a gazillion times.
This, Tom.
And the issues that arises are the question about the size of the sun and so on and so forth.
This is an endless circle of denial and repetition. You don't get it, because you don't want to. You have private life issues that makes you refute the obvious.

The measurements that astronomers and surveyors make are also "very empirical", and taken from a first person perspective. They also work from observation. It is almost like your protests in this thread are nonsense.
The empirical measurements of astronomers isn't really being doubted in a lot of the Flat Earth material. It is being argued that the astronomers compute their distance of the sun based on a Round Earth. If you assume a Flat Earth, with a flat baseline, the sun becomes a lot closer to the earth's surface. It is an easy mistake to make when you have been brainwashed from birth into believing that you live on a ball.
But then starts all the other issues that flat earth can't explain, if the earth is as close to the surface as math would tell, assuming the earth is flat. We've been telling you that a gazillion times.
Did you mean if the sun is as close to the surface as math would tell? I honestly don't understand what you mean either.
Edit:
This is an endless circle of denial and repetition. You don't get it, because you don't want to. You have private life issues that makes you refute the obvious.
But yeah, the endless repetition of arguments that have already been refuted, the contradictions, the denial of the obvious. All because he really, really, reeaaaally wants to believe the earth is flat. It's like the quintessential case study of denial.

The empirical measurements of astronomers isn't really being doubted in a lot of the Flat Earth material. It is being argued that the astronomers compute their distance of the sun based on a Round Earth. If you assume a Flat Earth, with a flat baseline, the sun becomes a lot closer to the earth's surface.
It is an easy mistake to make when you have been brainwashed from birth into believing that you live on a ball.
The last portion, in particular, is what I commented on.
If Flat Earth material does not doubt the empirical measurements, but call it an "easy mistake to make when you have been brainwashed", you just introduce the problems flat earth material can't solve, if we've been brought up "knowing" the earth is flat. For instance, the Sun staying the same size throughout the day, hence my comment about the endless circle of repitition and denial.
Quite frankly, I get offended when a flat earth proponent of Tom's caliber use the words "easy mistake" and "brainwashed" in one sentence that is ment to imply that socalled "round earthers" are wrong. On some level, I have to accept the fact that these authorities will have an influence in future knowledge sharing and debates that my kids take part in. And yes, adult and seemingly intelligent people are authorities in the eyes of a kid.
It's scary.

If Flat Earth material does not doubt the empirical measurements, but call it an "easy mistake to make when you have been brainwashed", you just introduce the problems flat earth material can't solve, if we've been brought up "knowing" the earth is flat. For instance, the Sun staying the same size throughout the day, hence my comment about the endless circle of repitition and denial.
Yes, he has no explanation for the sun staying the same size (as has been thoroughly demonstrated in other threads). Yes, it is related to the topic of perspective. But let's try to keep this thread as narrowly focused as possible. We could bring the size of the sun into it, but we don't really need to, and it will probably just confuse him more if we do.
And I still don't understand what you mean by "if the earth is as close to the surface as math would tell".

And I still don't understand what you mean by "if the earth is as close to the surface as math would tell".
I was refering to the following from Tom's comment:
If you assume a Flat Earth, with a flat baseline, the sun becomes a lot closer to the earth's surface.
If that statement is utilizing math, and we assume the math is correct, nothing in the flat earth material explains the aforementioned problems, like the size of the Sun. Alas, the flat earth math is wrong. Not understood, at best.

And I still don't understand what you mean by "if the earth is as close to the surface as math would tell".
I was refering to the following from Tom's comment:
If you assume a Flat Earth, with a flat baseline, the sun becomes a lot closer to the earth's surface.
I still don't understand that statement. How can the earth be close to its own surface?

And I still don't understand what you mean by "if the earth is as close to the surface as math would tell".
I was refering to the following from Tom's comment:
If you assume a Flat Earth, with a flat baseline, the sun becomes a lot closer to the earth's surface.
I still don't understand that statement. How can the earth be close to its own surface?
Hah! I was so focused on the general meaning of it all, that I actually didn't see my own typo. Fixed below :)
But then starts all the other issues that flat earth can't explain, if the earth Sun is as close to the surface as math would tell, assuming the earth is flat. We've been telling you that a gazillion times.

Okay I'm showing up way late to this topic, I only read most of the first page, but I already [think I] understand the misconception here. Note: I've never heard of an orthographic projection, had to google it; according to google:
a method of projection in which an object is depicted or a surface mapped using parallel lines to project its shape onto a plane.
a drawing or map made using orthographic projection.
(http://www.engineeringdrawing.org/wpcontent/uploads/2012/07/Drawing9.4a.bmp)
I played Dwarf Fortress for a while, so I'm familiar with that type of design, though only from a very shrewd and limited engineer standpoint (no offense, Great Toady One).
The basic misconception I'm seeing here is.... the orthographic projection seems to be a tool for engineers to use, which doesn't typically deal with say, milelong distances in everyday application. This is Point 1.
Now, PBrane (the youtuber who made the video in question), is arguing that orthographic projection's are not applicable to reality on these scales, because all parallel lines converge at a specific distance (https://youtu.be/AfgbqFyiisQ?t=4m34s), due to perspective; a detail omitted by the orthographic projection pbrane claims a 'glober' presented him with  sparking the creation of said video. This is Point 2; if all = lines converge, perspective must be taken into consideration  and by the nature of this debate, as Tom Bishop stated, it is impossible to get to a point where a orthographic projection would be possible on this scale, so we must rely on perspective, knowing that all parallel lines converge due to perspective.
Now, I'm only tentatively holding onto understanding of this at this point, so don't ask me to draw this; but seeing Point 2 (all = lines converge), that means that due to perspective, something moving parallel to a flat plane far (say a few thousand miles) overhead, would eventually reach that point of convergence, and 'disappear' beyond the horizon, before fading  though it may 'fade' somewhere thereafter  which would explain 'dusk' and 'night time' quite conveniently for a Flat Earth; the light source literally extinguished by mechanics of orthographic projection, beyond the horizon.
Now, yes, he does use 'arbitrary definitions' for lines @ 2 minutes 40 seconds (https://youtu.be/AfgbqFyiisQ?t=2m40s); but that isn't an illustration of the argument, just an illustration of Point 2, to avoid having to make such a forum post instead.
Orthographic projections are useful for accurately portraying angles.
Perspective projections are useful for portraying what we see.
To understand the motion of the sun, both of these are needed. That is just basic geometry. The sun sometimes appears to move in straight lines (https://youtu.be/GDaiwG1VGE?t=2m37s), but that is just perspective. Globe science acknowledges this, that's pretty simple; the sun appears to move in straight lines, though we can be fairly sure (do to above mentioned points) that this isn't literally the case. The only way to know 'how it is really moving' is to realize that perspective isn't necessarily always 'what is really happening'.
Disclaimer: Ugh... also I am not a fan of youtube drama so I don't want to get dragged into that... lol... if anyone sees this and claims credit for it (ie pbrane), fine take it I'm not claiming I put the ideas out there, I'm just clarifying a potential oversight I saw here. Feel free to use this as a resource if need be.

First of all, thanks for the nice writeup. It was very readable and your points were quite clear. I apologize in advance for the following wall of text. I tried to explain everything as clearly and precisely as possible, which means it got a little wordy.
The basic misconception I'm seeing here is.... the orthographic projection seems to be a tool for engineers to use, which doesn't typically deal with say, milelong distances in everyday application. This is Point 1.
Perspective occurs at all scales (not just large scales) and orthographic projections are valid and useful at all scales (not just small scales). Orthographic projections very intentionally don't portray perspective in order to allow certain things to be accurately measured.
1. Straight lines in reality are always straight in the diagram.
2. Parallel lines in reality are always parallel in the diagram.
3. Angles parallel to the plane of projection are accurate.
4. Distances parallel to the plane of projection are proportional. Distances in the same direction are proportional.
None of these properties change with scale. If we suddenly decide to add in perspective to the diagram, at any scale, these things will stop being true, and the diagram will no longer be useful for taking measurements.
This is Point 2; if all = lines converge, perspective must be taken into consideration  and by the nature of this debate, as Tom Bishop stated, it is impossible to get to a point where a orthographic projection would be possible on this scale, so we must rely on perspective, knowing that all parallel lines converge due to perspective.
I did take perspective into consideration. Unlike Tom Bishop and the aptly named PBrane, I have more than just a vague understanding of perspective. I know exactly how to calculate it. More on this later.
Now, I'm only tentatively holding onto understanding of this at this point, so don't ask me to draw this; but seeing Point 2 (all = lines converge), that means that due to perspective, something moving parallel to a flat plane far (say a few thousand miles) overhead, would eventually reach that point of convergence, and 'disappear' beyond the horizon, before fading  though it may 'fade' somewhere thereafter  which would explain 'dusk' and 'night time' quite conveniently for a Flat Earth; the light source literally extinguished by mechanics of orthographic projection, beyond the horizon.
This kindasorta makes sense if you only have a vague understanding of how perspective works. Unfortunately for Mr. Bishop and Mr. Brane, it is relatively easy to calculate exactly how far away an object is based on its height and angle with the horizon. This is where the orthographic diagram becomes useful. See this image:
(http://imgur.com/BB4hGoj.png)
On top is a side view diagram of a camera and 3 round objects. On bottom is the same 3 objects as seen from the perspective of the camera. The top DOES NOT take into account perspective. The bottom DOES take into account perspective. So, how are they related?
First: how does perspective arise?
The size of an object in a photo is proportional to the angular diameter (https://en.wikipedia.org/wiki/Angular_diameter#/media/File:Angular_diameter.jpg) of that object. As an object moves farther away, this angular diameter decreases, which causes the object to appear smaller. The same is true for distances between objects. This is why objects appear smaller as they get farther away. It is also why parallel lines appear to converge as they get farther away.
Look at the orthographic diagram. Notice that the 3 objects are the same size. However, the angular diameter of each object is different (4.2°, 3.0°, 2.3°). The farthest object has the smallest angular diameter. Notice that these angles are easily measurable using the orthographic diagram.
Now look at the bottom picture. Notice that the size of each object in the bottom picture corresponds to the angular diameter of the object from the orthographic diagram. We can convert these angles into sizes using the field of view (https://en.wikipedia.org/wiki/Field_of_view) of the camera. For example, if the field of view of the camera is 60°, and an object has an angular diameter of 30°, then the object will take up half the picture. (This assumes that the camera doesn't have a huge field of view or lots of optical distortion.)
We can do the exact same process for the height of each object: (skip the next two paragraphs if you already get the point)
Look at the orthographic diagram. Notice that the 3 objects are the same height. However, the angle between the object and the ground decreases as it gets farther away (20°, 14°, 11°). The farthest object has the smallest angular height. Notice that these angular heights are easily measurable using the orthographic diagram.
Now look at the bottom picture. Notice the height of each object in the bottom picture corresponds to the angular height of the object from the orthographic diagram. Once again, we can convert the angles to heights using the field of view of the camera.
Now, please notice 3 things about how we constructed the bottom image:
1. We determined the height and size of the objects from the angles in the orthographic diagram.
2. The size of the objects naturally decreases as they get farther away.
3. The objects naturally approach the horizon as they get farther away.
4. If we draw a line connecting the right edges of each object, and another line connecting the left edges of each object, those two lines will converge below the objects at the horizon. Just like perspective lines! Coincidence? I think not!
Points 2 and 3 are the effects of what we call perspective. Perspective naturally arises from the angles measured from the orthographic diagram! Neat!
Notice, that there is absolutely no need to overlay any arbitrarily drawn "perspective lines" onto the orthographic diagram in order to calculate the position of each object due to perspective. The perspective lines arise naturally in the bottom picture due to measurements of angles that we took from the orthographic side view diagram.
Tom Bishop's rebuttal:
So far, Tom Bishop's main argument has been to simply deny the mathematical relationship between the two pictures above. To justify this denial, he has made several... um... entertaining... arguments involving "hidden infinities" and generally denounced the usefulness of orthographic diagrams. (Even though his hero, Rowbotham, uses them quite frequently.)
However, you don't have to take my word for it. Go outside with a camera and a few objects and test the math for yourself. You will find that the mathematical relationship between the angles from the orthographic side view and the size and placement of the objects in the picture is quite accurate for any distance that you manage to test. (Use a relatively narrow field of view for more accuracy. Wide angle lens have too much optical distortion.)
Tom Bishop's other argument is that "maybe the math math suddenly stops working at super long distances that are too long to test". However, he has presented no evidence or logical reasoning that would suggest that the math suddenly stops working at a particular distance. I think we can all agree that this is a horrible argument and merely a desperate attempt to ignore evidence that contradicts his flat earth model.

TL;DR:
It is easy to calculate the distance of an object based on its height and angle with the horizon, using an orthographic diagram or this equation:
distance = height / tan(angle)
You can deny that the math is wrong, or that orthographic diagrams are invalid all you want. However, the math works for any testable distance. Test it.

TL;DR:
It is easy to calculate the distance of an object based on its height and angle with the horizon, using an orthographic diagram or this equation:
distance = height / tan(angle)
You can deny that the math is wrong, or that orthographic diagrams are invalid all you want. However, the math works for any testable distance. Test it.
What continually perplexes me is that an explanation and one that can be tested relatively easily like yours will not be accepted or tested by any true FE believer.
Similar how Tom will not do any experiments that will challenge his belief. He will stick to the safe claims of seeing stuff he should not see or shooting a laser at a target without providing any data or other information.

And (just to remind) I am still waiting for an explanation about the constancy of the angular speed of the sun, which is totally inconsistent with FE model!

Tom Bishop's other argument is that "maybe the math math suddenly stops working at super long distances that are too long to test". However, he has presented no evidence or logical reasoning that would suggest that the math suddenly stops working at a particular distance. I think we can all agree that this is a horrible argument and merely a desperate attempt to ignore evidence that contradicts his flat earth model.
No, no, I'm sorry, but no.
The math you are using assumes that values near zero approach infinity. This assumption needs merit in the material world. There must be an example of this somewhere for us to accept it as reality. It is quite decieving to say that "Tom Bishop assumes the math stops working" when this part of the math has not been demonstrated anywhere.
You must show where these pockets of infinities have been seen. It is not my responsibility to "prove it wrong" any more than it is my responsibility to prove that ghosts do not exist in an argument on the existence of ghosts. This is a positive claim of something which has never been observed. It is your responsibility to show that this math is somehow right in contradiction to physical experience.

If we take a line and define a segment then we are no longer assuming infinity.
When we select how accurate a calculation needs to be using something like Pi or 1/3 we are no longer using an infinite number.
The math has been constantly proven to work in real world applications.

Tom Bishop's other argument is that "maybe the math math suddenly stops working at super long distances that are too long to test". However, he has presented no evidence or logical reasoning that would suggest that the math suddenly stops working at a particular distance. I think we can all agree that this is a horrible argument and merely a desperate attempt to ignore evidence that contradicts his flat earth model.
No, no, I'm sorry, but no.
The math you are using assumes that values near zero approach infinity. This assumption needs merit in the material world. There must be an example of this somewhere for us to accept it as reality. It is quite decieving to say that "Tom Bishop assumes the math stops working" when this part of the math has not been demonstrated anywhere.
You must show where these pockets of infinities have been seen. It is not my responsibility to "prove it wrong" any more than it is my responsibility to prove that ghosts do not exist in an argument on the existence of ghosts. This is a positive claim of something which has never been observed. It is your responsibility to show that this math is somehow right in contradiction to physical experience.
But that's what they have been doing all along. The only reason to refute it, is if you assume that Pi = 4 and that proven math has been wrong from day 1. With this, I'm referring specifically to your claims about math and the ancient Greeks for instance.
In that regard, yes, it's still up to you to prove where math is wrong, using real world examples, just as the RE'ers in this and many other threads has shown you how math and real world examples are 1:1.

Tom Bishop's other argument is that "maybe the math math suddenly stops working at super long distances that are too long to test". However, he has presented no evidence or logical reasoning that would suggest that the math suddenly stops working at a particular distance. I think we can all agree that this is a horrible argument and merely a desperate attempt to ignore evidence that contradicts his flat earth model.
No, no, I'm sorry, but no.
The math you are using assumes that values near zero approach infinity.
For the... *counts on fingers... takes off shoes... counts on toes* ...12th time?
1. Your logic is vague and lazy. What "value near zero approaches infinity"? Distance to the object? Size of the object on the projection? Distance from the vanishing point on the projection? Be specific.
2. Your statement is absurd. Near zero. Approaches infinity. Do you not see the contradiction here? The "math" certainly makes no such assumptions.
The math assumes that there is a 1 to 1 relationship between the ratio of sides of a right triangle and the angle of a right triangle. It assumes euclidean geometry. For the math to be applicable and accurate, we assume that light travels in a straight line.
Given 2 objects moving in straight lines parallel to each other away from the observer, the math predicts that the objects will appear closer to the vanishing point. The math predicts that for any arbitrary finite (NOT infinite) distance away, the objects will appear a nonzero distance away from the vanishing point on the projection. This nonzero distance can also become arbitrarily small, thus indistinguishable from zero since our eyes and cameras don't have perfect resolution.
Please notice that nowhere did I say the math predicts or assumers (there is a difference) that any object actually reaches an infinite distance away from the observer. Nor does any other value actually reach infinity.
How many more times do I have to refute this ridiculous argument before you stop bringing it up? 20 times? 100 times? Would a language other than English make it more clear? I am quite proficient in Pig Latin.
It is quite decieving to say that "Tom Bishop assumes the math stops working" when this part of the math has not been demonstrated anywhere.
First of all, these are two unrelated arguments. Second of all, it is an argument (the red part) you have made implicitly in this thread, and quite explicitly in other threads. If you want to disavow this argument, that is perfectly fine by me. However, I'll bet anyone five Itoldyouso's that you will go back to this argument eventually.
You must show where these pockets of infinities have been seen.
No, I don't. I'm not sure how you expect anyone to show that something is infinite. However, I will give you a more apt analogy, for funsies:
Spread your arms in front of you, making a 90 degree angle between your arms. What is the maximum width of an object that will fit inside the visual field between your arms? Go.

Tom, correct me if I'm wrong, but are you essentially saying that the mathematical foundations of visual perspective that are used in highly detailed and accurate systems, like flight simulators, are unproven?

Tom, correct me if I'm wrong, but are you essentially saying that the mathematical foundations of visual perspective that are used in highly detailed and accurate systems, like flight simulators, are unproven?
His argument is at some certain unspecified distance and reason the math fails to work. From what I can gather from his post the math fails at around 12 miles where most sailors on lookout duty would see the horizon and again around 3,000 miles the altitude of the Sun.

You sure seem defensive over math that is so proven. Just show us where these approaches to infinity or ultra long decents have been found in realty, and we can just see that the math is undeniable, OKAY?

You sure seem defensive over math that is so proven. Just show us where these approaches to infinity or ultra long decents have been found in realty, and we can just see that the math is undeniable, OKAY?
Tom, are you suggesting that the sun is infinitely far away?

You sure seem defensive over math that is so proven. Just show us where these approaches to infinity or ultra long decents have been found in realty, and we can just see that the math is undeniable, OKAY?
Who are you talking to? No one claimed anything approaches infinity. And what the heck are "ultra long decents"?

The math you are using says that is is impossible for anything to reach the horizon. A sun would descend forever on a Flat Earth without ever reaching the horizon. The math says that touching the horizon is impossible, as 0 degrees is defined as infinity.
Where has any endless approach to a point like this been reported in any scientific, everyday, or historical observation, or anywhere in the world? Please present some form of evidence that any of this is possible. Provide something, anything from reality, showing that ultra long descents would occur or that the horizon would be an infinite distance away.

The math you are using says that is is impossible for anything to reach the horizon.
Assuming it is traveling in a straight line parallel to a flat earth, sure.
A sun would descend forever on a Flat Earth without ever reaching the horizon.
No it wouldn't. After noon, it would start sinking and turning northwards (because it is circling the north pole). By 6 pm it would be about 28 degrees above the horizon. It would reach a minimum of 20 degrees above the horizon around midnight and would be due north. After that, it would start rising again. (Edit: your bipolar model would be slightly different of course, since it sometimes circles the south pole instead.)
The math says that touching the horizon is impossible, as 0 degrees is defined as infinity.
Strictly speaking, correct. Since an object can never be an infinite distance away, it will never be 0 degrees with the horizon. Please notice: it CAN get arbitrarily close to the horizon, if it is a very long distance away relative to its height. It can be so close to the horizon that we cannot tell the difference with our eyes, because our eyes aren't perfect.
We have covered this already. Stop bringing up the same refuted arguments over and over.
Where has any endless approach to a point like this been reported in any scientific, everyday, or historical observation, or anywhere in the world? Please present some form of evidence that any of this is possible. Provide something, anything from reality, showing that ultra long descents would occur or that the horizon would be an infinite distance away. < I did not claim the horizon is an infinite distance away. Your strawman arguments are getting tiring. Almost every single post contains one.
Chemistry: Try diluting a solute by adding water to it. As long as you keep adding water, the concentration will approach zero, but will never actually reach it.
This is actually quite a relevant comparison. The angle with the horizon is related to the height:distance ratio of the object. For the angle to be zero, the ratio of height to distance has to be zero.
In both examples, you are trying to make a ratio approach zero by adding to one side. In the chemistry example, you are trying to make the ratio of solute to solvent go to zero by adding solvent. The ratio will continually get smaller, but will never actually reach zero since you aren't actually removing any solute. In the perspective example, you are trying to make the ratio of height to distance go to zero by adding distance. It will continually get smaller, but will never actually reach zero since the height isn't actually decreasing.
Counter challenge: Draw a right triangle that has a nonzero height and width and one of the angles is zero. This is what you are implying is happening during sunset on a flat earth. Good luck!

Strictly speaking, correct. Since an object can never be an infinite distance away, it will never be 0 degrees with the horizon.
And where is the evidence of this phenomenon? Why should we assume that there are these pockets of infinity which prevents perspective lines from meeting?
You have repeatedly declined to show any of the real world evidence you have been asked for. Why isn't there anything to show that distances might grow exponentially near zero? If this math is so proven then an example, observation, or experiment should be readily quotable and hiding behind repeated claims of incredulity would be unnecessary.

Chemistry: Try diluting a solute by adding water to it. As long as you keep adding water, the concentration will approach zero, but will never actually reach it.
This is actually quite a relevant comparison. The angle with the horizon is related to the height:distance ratio of the object. For the angle to be zero, the ratio of height to distance has to be zero.
In both examples, you are trying to make a ratio approach zero by adding to one side. In the chemistry example, you are trying to make the ratio of solute to solvent go to zero by adding solvent. The ratio will continually get smaller, but will never actually reach zero since you aren't actually removing any solute. In the perspective example, you are trying to make the ratio of height to distance go to zero by adding distance. It will continually get smaller, but will never actually reach zero since the height isn't actually decreasing.
It could equally be argued that when you replace 100% of a chemical solution with water, the resulting concoction will be 100% water. Your example tells us nothing about what is actually occurring with perspective.
Counter challenge: Draw a right triangle that has a nonzero height and width and one of the angles is zero. This is what you are implying is happening during sunset on a flat earth. Good luck!
The premise here is that the ancient math of the greeks is fallible, so of course their math fails on that point.

Chemistry: Try diluting a solute by adding water to it. As long as you keep adding water, the concentration will approach zero, but will never actually reach it.
This is actually quite a relevant comparison. The angle with the horizon is related to the height:distance ratio of the object. For the angle to be zero, the ratio of height to distance has to be zero.
In both examples, you are trying to make a ratio approach zero by adding to one side. In the chemistry example, you are trying to make the ratio of solute to solvent go to zero by adding solvent. The ratio will continually get smaller, but will never actually reach zero since you aren't actually removing any solute. In the perspective example, you are trying to make the ratio of height to distance go to zero by adding distance. It will continually get smaller, but will never actually reach zero since the height isn't actually decreasing.
It could equally be argued that when you replace 100% of a chemical solution with water, the resulting concoction will be 100% water. Your example tells us nothing about what is actually occurring with perspective.
Counter challenge: Draw a right triangle that has a nonzero height and width and one of the angles is zero. This is what you are implying is happening during sunset on a flat earth. Good luck!
The premise here is that the ancient math of the greeks is fallable, so of course their math fails on that point.
OK, so please Tom, be constructive and with your great wisdom, explain us how it is possible that something, with a finite diameter, at a finite height of a finite plan can reach a zero elevation without modifying its height.

Strictly speaking, correct. Since an object can never be an infinite distance away, it will never be 0 degrees with the horizon.
And where is the evidence of this phenomenon? Why should we assume that there are these pockets of infinity which prevents perspective lines from meeting?
You have repeatedly declined to show any of the real world evidence you have been asked for. Why isn't there anything to show that distances might grow exponentially near zero? If this math is so proven then an example, observation, or experiment should be readily quotable and hiding behind repeated claims of incredulity would be unnecessary.
No one claimed there is a pocket of infinity anywhere. No one claimed distances grow exponentially near zero. You are the only one making arguments based on incredulity here. I have bent over backwards to give you examples, make you drawings, explain the logic in extreme detail. I have told you exactly how you can find the evidence yourself. The only thing I haven't done is physically collected the evidence for you. Good grief.
Fine. What evidence would convince you? If I am able to correctly predict the size of an object on a picture based on its size and distance using the "ancient greek math", would that convince you? I'm not going to waste my time if you are just going to cry "fake".
Chemistry: Try diluting a solute by adding water to it. As long as you keep adding water, the concentration will approach zero, but will never actually reach it.
This is actually quite a relevant comparison. The angle with the horizon is related to the height:distance ratio of the object. For the angle to be zero, the ratio of height to distance has to be zero.
In both examples, you are trying to make a ratio approach zero by adding to one side. In the chemistry example, you are trying to make the ratio of solute to solvent go to zero by adding solvent. The ratio will continually get smaller, but will never actually reach zero since you aren't actually removing any solute. In the perspective example, you are trying to make the ratio of height to distance go to zero by adding distance. It will continually get smaller, but will never actually reach zero since the height isn't actually decreasing.
It could equally be argued that when you replace 100% of a chemical solution with water, the resulting concoction will be 100% water.
Hahahaa... what?!? Yes, and if you forcibly shoved the sun directly into the ground, decreasing its height to zero, then it would most certainly appear to touch the horizon. What's your point? You asked for an example, and I gave it. Don't throw a tantrum and toss out my solution!
Your example tells us nothing about what is actually occurring with perspective.
It tells us exactly what is occurring with perspective. You are just too thickheaded and/or stubborn to admit it.
Counter challenge: Draw a right triangle that has a nonzero height and width and one of the angles is zero. This is what you are implying is happening during sunset on a flat earth. Good luck!
The premise here is that the ancient math of the greeks is fallible, so of course their math fails on that point.
Ah yes. Your model requires a triangle with nonzero sides and a zero angle. Since no one has heard of such an absurd triangle, it must be those darn mathematicians' fault for not inventing such a thing! It certainly is not a problem with your model! No bias here, no sirree! Lol.

Tom, as far as I am concerned, this thread is a perfect example of something that goes on forever but never actually reaches anywhere. At least, YOU aren't going anywhere. You just keep repeating the same arguments over and over that have already been refuted numerous times. If you feel the need to make another vague argument involving infinity, don't bother. There is a very good chance I have already refuted it in this thread. Just reread the thread.
Just answer this final question:
Fine. What evidence would convince you? If I am able to correctly predict the size of an object on a picture based on its size and distance using the "ancient greek math", would that convince you? I'm not going to waste my time if you are just going to cry "fake".

The math you are using says that is is impossible for anything to reach the horizon. A sun would descend forever on a Flat Earth without ever reaching the horizon. The math says that touching the horizon is impossible, as 0 degrees is defined as infinity.
Where has any endless approach to a point like this been reported in any scientific, everyday, or historical observation, or anywhere in the world? Please present some form of evidence that any of this is possible. Provide something, anything from reality, showing that ultra long descents would occur or that the horizon would be an infinite distance away.
The answer on a flat earth is yes, there is no way for the sun to ever approach anywhere near the horizon, assuming that the furthest point below the sun from an Antarctic base would be about 23000 kms away when the sun is over the Tropic of Capricorn on the other side of the north pole. The angle of the sun above the horizon would be 7.4 degrees. Simple trigonometry.
Why do you need to bring infinite distances into a discussion when the distances travelled by the FE sun 'orbit' is finite. Just another way to deflect the discussion as adopted in most unanswerable FE debunks.

Key: (to avoid repeating myself and condense; nothing new is being stated in the KEY)
OP = Orthographic Projection
Point 1/2 = Parrallel lines converge due to perspective.
FOV: Field of View
This is going to be long, so I'm working on explaining it in a youtube video. Funny (for me at least) I am typing this up on September 30, 2016; the day before Obama hands over the internet ICANN to international corporations and countries... so RIP internet, it was fun without censorship. Heregoes...
Totes, Great well written response, thanks; I think I get it better now, though I still take issue. The 2 images posted take into consideration perspective, with a caveat; in order to achieve the perspective of the OP 'side view' (the top image), you must be an impossible/unachievable distance away  Tom's  and partially, my  argument. Thanks for that response, your 3 recent topics here were 3 of my FE favorites prior to your posting them, as I put rather pathetically previously (https://forum.tfes.org/index.php?topic=5237.msg102453#msg102453) when I anticipated the SCP one the day before you posted it; these 3 recent topics (SCP (https://forum.tfes.org/index.php?topic=5269.0), Perspective/Sun (https://forum.tfes.org/index.php?topic=5346.0), Southern Hemisphere December (https://forum.tfes.org/index.php?topic=5237.0)) are really one and the same issue, just seen 'from different angles' :) xD lol
That said, I really don't know much about angles. My experience comes largely from simulated planes (video games), as mentioned above, I admit. From what I can tell, no new info is presented here, other than the mechanics of angles and perspective being more elaborated on. Which you do an honorable effort of, and somewhat improves my opinion of academic institutions everywhere.
SO lets get into the Meat (hopefully no hot potatoes slip in) of this post:
1. Parallel lines are parallel.
2. Parallel lines converge due to perspective; even those in a OP diagram, if drawn to 'infinity'.
3. We only see objects, at best, a few hundred miles in any direction at best from anywhere on Earth (FoV). Caveat: the sun and stars (more on this later).
4. The FE sun is ~3,000 miles above the Earth.
5. It is impossible to demonstrate perspective of 14 within the FoV of 3.
6. My 'few hundred' miles is quite generous. I have seen images where islands or mountains are visible over 125 miles away, but many sources (https://www.quora.com/Howfarcanthehumaneyeseeonperfectlyflatland) cite 50100 miles as the 'limit' of or FoV (2 (https://www.technologyreview.com/s/539826/howfarcanthehumaneyeseeacandleflame/)), (3 (http://www.livescience.com/33895humaneye.html)). With Cameras/telescopes it is notably possible to see much further, ofc.
7. The image presented here has many flaws:
7a) Distance. 3,000 miles to the sun; this image covers many times that; over 12,000 miles; over half the actual equatorial distance.
7b) The perspective, viewing the sun traveling 'parallel' with the Earth, is impossible to achieve.
7c) The sun doesn't move in a straight line as depicted here on the FE model, but rotates East to West (clockwise as viewed from North Pole).
7d) The Orthographic Projection here does not accurately demonstrate the convergence of parallel lines; the top image is an impossible 'god's eye' perspective, for one; and for two, if you trace converging lines, they converge a great distance before 12,000 miles away; what the image here is depicting.
8 ) You can't understand the rest of this post without watching this (brief) youtube video.
https://www.youtube.com/watch?v=ejwrxGs_Y_I
You get the point from the video now? Okay...
At least, that is what I think Tom means about 'infinities' is that the image presented here is an impossible 'infinity' away from what we observe, looking at our observation on the FE model. Alternatively, there is also an 'infinity' in an OP as the sun gets 'further and further' away (a great point against the flat Earth, and what this topic was started to point out, and Tom and myself realize that, no one is arguing). But as stated above, we can't see anything beyond ~50140 miles from the surface Earth (more on that later). But the mechanics here don't work, as the sun has to start coming back 'the other way' somewhere in the neighborhood of 12k18k out on FE model (this is where I don't know the FE model well enough, but we can at least agree the sun doesn't literally travel in a straight line forever  well, you could, technically, but you'd have to create a whole other model for that).
So, lets look at the image. The common Flat Earth model shows the sun as being ~3,000 miles above the surface. This image illustrates an impossible phenomenon then (point 1 above); as Tom Bishop has already stated; no man has ever seen more than a few hundred miles in any direction; and the sun 'sets' either long before reaching, or as it reaches, that point. What Tom is saying here, is it is impossible to attain the perspective in that image, observing an observer observing the sun moving parallel to the ground, in a clockwise circle/spiral, not a straight line as depicted here. Illustration: (yes I borrowed it again sorry)
(http://i.imgur.com/QZODvNS.png)
So, that image is not applicable to our observations; look at the 'ground' beneath the Red, Orange, and Yellow 'dot' or 'ball'; it is much farther than the distance from the ground to them; thus more than 3 thousand miles (even the globe states the circumference at the equator to be no more than 25,000 miles). I'll call this Point 3 for now. Keep in mind that half the circumference is roughly 12,300 miles; the limit of 'globular perspective', before the 'curve' is on 'the other side of the globe'. For FE, this represents approximately onehalf the equatorial circle. The commonly agreedupon 3,000 miles to the sun on Flat Earth is outmatched by the ground covered no matter how it is calculated; and Point 2, parallel lines converge due to perspective (assuming the sun moves parallel to the Earth in FE model). This image is WAAAAY out of proportion. The equator is only 12k miles across; this image goes way further! Up to 18k! The sun is going the other way at that point.
With that ridiculous 20k mile perspective taken into consideration, the line would be more like this:
(http://i.imgur.com/ICRA1XD.png)
Again that is generous at best.... as there is no way for us to know what a 20k mile line would look like, let alone 2 parallel ones, from being sandwiched between them.
Now, just as pbrane did, obviously your image is not meant to be to scale. But again, as Tom said, this 'several thousand mile perspective' is simply impossible. So, lets look at some common blueprints instead, which use principles of orthographic projection, and compare the actual buildings to the blueprints.
(http://www.departments.bucknell.edu/history/carnegie/building/blueprint.jpg)
In this image the lines are parallel. But stand at the foot of the building to the left or right, and the roof and floor 'lines' will be 'converging', like this (https://encryptedtbn2.gstatic.com/images?q=tbn:ANd9GcQQbwZvf6PV58_D0O7NLAcnavflrOuiJCT_PA1xYFsaZNfFnWWG).
Alternatively....
(https://www.marypomerantzadvertising.com/wpcontent/uploads/Blueprint1602548501024x768.jpg)
Uh ooh... the parallel lines of the blueprints... converge... in reality... while the lines are drawn perfectly parallel in a schematic, you can't stop them from converging due to perspective, try as you may; esp. if you continue the line indefinitely; yes, it stays 'parallel' in reality, but it is impossible to see from any point (when continued 'indefinitely') without a 'god's eye view'; even navigating the parallel lines, from every point within it, they appear to converge. No amount of equations can defy that, though they may serve as a red herring to those unable to understand.
Additionally, maybe you can call the 'few hundred mile' FoV the 'Earth's curvature', but the higher you go, the wider the FoV gets, so there's that as well...
Anyway I'm no shill or whatever so I have no angle other than perhaps too much time on my hands. This is a very good topic, I hope I don't insult with my lack of understanding; this is just what I've posted based on how I think I understand it... I could be wrong, but it seems right to me.

On top is a side view diagram of a camera and 3 round objects. On bottom is the same 3 objects as seen from the perspective of the camera. The top DOES NOT take into account perspective. The bottom DOES take into account perspective. So, how are they related?
Just realized I missed this. I'll already reconsidering my previous post. I'm still unsure how relation can be inferred from an impossible perspective, though I understand the hypothetical correlation and how it invalidates FE model. Consider me in 'Winnie the Pooh' mode right now...
As my whole argument in that wall of text relies entirely on the correlation of the two images, my argument may be invalid, I realize now. My apologies if this be the case, I'll probably have to sleep on it. Also apologies for however wrongly I may have referred to users here or my misinterpretation of framing their arguments.

Just realized I missed this. I'll already reconsidering my previous post. I'm still unsure how relation can be inferred from an impossible perspective, though I understand the hypothetical correlation and how it invalidates FE model. Consider me in 'Winnie the Pooh' mode right now...
I saw your this post when I tried to submit this one, which means that everything below might already be out of date. I'll submit it anyway, since it might be relevant to your current thoughts on the matter. My responses get a bit repetitive, since as you guessed above, most of the issues in your post seem to stem from the same misunderstanding.
Edit: Good gravy, my post was longer than I thought. I highlighted the important points in red.
The 2 images posted take into consideration perspective, with a caveat; in order to achieve the perspective of the OP 'side view' (the top image), you must be an impossible/unachievable distance away  Tom's  and partially, my  argument.
Then both you and Tom have missed my point entirely. Human's don't have orthographic vision. Of course we can't see this view in reality. Once again, the point of an orthographic view is NOT to portray what we see. It is to accurately portray angles and distances. We can then use the measured angles to predict how it will look according to perspective.
That said, I really don't know much about angles. My experience comes largely from simulated planes (video games), as mentioned above, I admit. From what I can tell, no new info is presented here, other than the mechanics of angles and perspective being more elaborated on. Which you do an honorable effort of, and somewhat improves my opinion of academic institutions everywhere.
Fun fact: video games use the same math that I am using to generate their perspective projections. Because the math works. And I agree, I am not presenting any new or revolutionary information. Everything I am saying is extremely well known and understood by thousands of engineers, physicists, artists, computer graphics programmers... and pretty much anyone who didn't sleep through high school math, and has bothered to think about the subject for 30 minutes. It really is not a complicated subject.
2. Parallel lines converge due to perspective; even those in a OP diagram, if drawn to 'infinity'.
No no no no no! A thousand times no! Parallel lines are always parallel in an orthographic projection, by definition. If the parallel lines converge, then it isn't an orthographic projection.
3. We only see objects, at best, a few hundred miles in any direction at best from anywhere on Earth (FoV). Caveat: the sun and stars (more on this later).
4. The FE sun is ~3,000 miles above the Earth.
5. It is impossible to demonstrate perspective of 14 within the FoV of 3.
This goes back to Tom's "maybe the math suddenly stops working at some large untestable distance" argument. Sure. Maybe it does. But the math DOES work at testable distances. And there is no evidence or logic to suggest that it suddenly might stop working at a particular distance. You might as well just appeal to magic at that point.
7. The image presented here has many flaws:
7a) Distance. 3,000 miles to the sun; this image covers many times that; over 12,000 miles; over half the actual equatorial distance.
And yet even at that absurd distance, the object still wouldn't come close to touching the horizon according to the math. The math that is testable. The math that works.
7b) The perspective, viewing the sun traveling 'parallel' with the Earth, is impossible to achieve.
That's not the point. Again, the point of an orthographic diagram is NOT to portray what we see. It is to allow us to accurately measure distances and angles. We can then use the measured angles to predict how it will look according to perspective.
7c) The sun doesn't move in a straight line as depicted here on the FE model, but rotates East to West (clockwise as viewed from North Pole).
True. My diagram is somewhat simplified. If I wanted to take into account the clockwise rotation of the sun, I would draw a topview orthographic diagram. I would then go through the same process as before, except the angles would translate to horizontal dimensions in the perspective picture, instead of vertical dimensions. The object would appear to curve off to the right as it sunk. This brings up another problem with the standard flat earth model: the sun doesn't always appear to be curving northward as it sets, like the flat earth model predicts.
7d) The Orthographic Projection here does not accurately demonstrate the convergence of parallel lines;
Good. It isn't supposed to. If it did, it wouldn't be an orthographic projection. It allows us to predict the convergence of parallel lines in a perspective view based on the angles we measure in the orthographic diagram.
if you trace converging lines, they converge a great distance before 12,000 miles away; what the image here is depicting.
Oh really? And how did you calculate this? I assume you are referring to the vanishing point. If so, I can assure you that you are misunderstanding what the vanishing point is.
At least, that is what I think Tom means about 'infinities' is that the image presented here is an impossible 'infinity' away from what we observe, looking at our observation on the FE model. Alternatively, there is also an 'infinity' in an OP as the sun gets 'further and further' away (a great point against the flat Earth, and what this topic was started to point out, and Tom and myself realize that, no one is arguing). But as stated above, we can't see anything beyond ~50140 miles from the surface Earth (more on that later). But the mechanics here don't work, as the sun has to start coming back 'the other way' somewhere in the neighborhood of 12k18k out on FE model (this is where I don't know the FE model well enough, but we can at least agree the sun doesn't literally travel in a straight line forever  well, you could, technically, but you'd have to create a whole other model for that).
If you are going to make an argument about infinity, you will need to be much more specific than this. "the image presented here is an impossible 'infinity' away" does not really make sense. I understand what you are trying to say, but your argument just is not relevant. Once again, orthographic projections are not SUPPOSED to show what we are able to see. They are supposed to allow us to accurately measure distances and angles. You know... that thing that you did when you took the height of the object as 3000 miles, turned it sideways, and then counted out "3, 6, 9, 12, 18" to determine the horizontal width of the diagram? That's exactly what an orthographic diagram is useful for. Good job!
no man has ever seen more than a few hundred miles in any direction; and the sun 'sets' either long before reaching, or as it reaches, that point. What Tom is saying here, is it is impossible to attain the perspective in that image, observing an observer observing the sun moving parallel to the ground, in a clockwise circle/spiral, not a straight line as depicted here. Illustration: (yes I borrowed it again sorry)
(Obligatory: orthographic diagrams aren't supposed to portray what we observe.... blah blah blah, etc etc)
So, that image is not applicable to our observations; look at the 'ground' beneath the Red, Orange, and Yellow 'dot' or 'ball'; it is much farther than the distance from the ground to them; thus more than 3 thousand miles (even the globe states the circumference at the equator to be no more than 25,000 miles). I'll call this Point 3 for now.
You are correct. According to the standard flat earth model, the sun would get no farther away from us than the yellow ball. Therefore, the MINIMUM angle it would make with the horizon is 20 degrees. Since we can observe the sun going much lower than that every single day, this proves that the earth isn't flat. How is this not blatantly obvious?
This image is WAAAAY out of proportion. The equator is only 12k miles across; this image goes way further! Up to 18k! The sun is going the other way at that point.
Fine. Remove the orange and red balls. Congratulations, now it is in proportion!
With that ridiculous 20k mile perspective taken into consideration, the line would be more like this:
(http://i.imgur.com/ICRA1XD.png)
No, it would not. That is no longer an orthographic projection. Also, how on earth did you calculate the slant of that line? Everything having to do with perspective in the bottom image was very carefully calculated. I did not place anything arbitrarily.
Now, just as pbrane did, obviously your image is not meant to be to scale.
My image was not intended to exactly represent the dimensions of a flat earth. I never said it was. If you remove the red and orange objects, then the distances are a good approximation of a flat earth.
The difference between my image and pbrane's, is that the bottom image that I drew was carefully calculated according to the math I have been describing. Nothing about it was arbitrary. The perspective effect portrayed in the bottom image arose naturally from the math. On the otherhand, the perspective lines that pbrane (and you) drew are completely arbitrary. They are nothing more than a vague guess based on a vague/poor understanding of how perspective works.
But again, as Tom said, this 'several thousand mile perspective' is simply impossible. So, lets look at some common blueprints instead, which use principles of orthographic projection, and compare the actual buildings to the blueprints.
(http://www.departments.bucknell.edu/history/carnegie/building/blueprint.jpg)
In this image the lines are parallel. But stand at the foot of the building to the left or right, and the roof and floor 'lines' will be 'converging', like this (https://encryptedtbn2.gstatic.com/images?q=tbn:ANd9GcQQbwZvf6PV58_D0O7NLAcnavflrOuiJCT_PA1xYFsaZNfFnWWG).
Yes. The first image is an orthographic projection. The second image is a perspective projection (a picture from a camera). One shows perspective. One doesn't. What's your point? If you give me the orthographic projection, I can predict how it will look in the perspective projection, using the math that I have described. That's how we know the math works.
Alternatively....
(https://www.marypomerantzadvertising.com/wpcontent/uploads/Blueprint1602548501024x768.jpg)
Uh ooh... the parallel lines of the blueprints... converge... in reality...
Of course they do. I have never claimed otherwise. That's not an orthographic projection. It's a perspective projection.
Here is the irony: that image is not a picture from a camera (obviously). It was generated based on a 3D mapping of points by software that uses... wait for it... THE SAME MATH THAT I HAVE BEEN USING. The fact that you agree that it is a good representation of what we would see in reality is evidence that THE MATH WORKS.
while the lines are drawn perfectly parallel in a schematic, you can't stop them from converging due to perspective, try as you may
No kidding. I have never claimed otherwise.
Yes, it stays 'parallel' in reality, but it is impossible to see from any point (when continued 'indefinitely') without a 'god's eye view'; even navigating the parallel lines, from every point within it, they appear to converge. No amount of equations can defy that...
The "equations" don't defy that parallel lines converge. They PREDICT it. I am not trying to defy that parallel lines converge. I am saying that I can PREDICT exactly how, when, and where those parallel lines converge using math. Perspective is not some mysterious, poorly understood phenomenon. It is easy to predict, easy to calculate, and the math involved is easy to test.
Using the same math, I can also predict the angular diameters of objects that we see based on their size and distance from us. I can also predict the angular distance between two objects based on their distance from us and distance between each other. I can also predict the angle between the sun and the horizon on a flat earth based on the sun's distance and height. The fact that all of these predictions turn out to be correct except the one based on a flat earth is evidence that the earth isn't flat.
Additionally, maybe you can call the 'few hundred mile' FoV the 'Earth's curvature', but the higher you go, the wider the FoV gets, so there's that as well...
FoV doesn't get wider. FoV generally refers to the angle of your vision, not the distance. But yeah, the distance you can see DOES increase as you get higher... because the earth is round... I don't think you thought that argument through very well.
Anyway I'm no shill or whatever so I have no angle other than perhaps too much time on my hands. This is a very good topic, I hope I don't insult with my lack of understanding; this is just what I've posted based on how I think I understand it... I could be wrong, but it seems right to me.
Cheers.

Lol great post, I caught my error before you posted though. I thought you were trying to pull a fast one, but then I realized I was wrong. That is my fault for jumping the gun. I got some really good chuckles reading your post, though ;D spot on. I deserved it.
Most of the 'arguments' I elaborated on, weren't presentations so much as me representing what's already been posted here my own way. I didn't mean to come off as arguing the infinities! Or the difference between parallel and perspective (though I thought it was being passed off that there wasn't a difference, is all; I realize my mistake).
Everything I am saying is extremely well known and understood by [...] pretty much anyone who didn't sleep through high school math, and has bothered to think about the subject for 30 minutes. It really is not a complicated subject.
Yes, and you provided an excellent crash course herein.
2. Parallel lines converge due to perspective; even those in a OP diagram, if drawn to 'infinity'.
No no no no no! A thousand times no! Parallel lines are always parallel in an orthographic projection, by definition. If the parallel lines converge, then it isn't an orthographic projection.
On a lifesized, really big, OP projection, they would converge ;D that's why I stated 'due to perspective' and 'if drawn to infinity'. IE, the highway is a physical representation of it's OP. A orthographic projection of that scale would appear to converge, is all I meant; or, alternatively, if you traced the lines of an orthographic projection into the horizon, they would converge (ie god's eye view). That's all I meant here; I can see how that offends good taste  I did it deliberately to offend, thinking the "God's Eye View" was to assume that parallel lines don't converge (20,000 mile lines). I knew 20,000 mile lines would converge. I thought you were trying to say that they wouldn't.
And yet even at that absurd distance (3,000+ miles), the object still wouldn't come close to touching the horizon according to the math. The math that is testable. The math that works.
Yes, here is where the FE model begins to mathematically come undone, as best I have ever been able to tell. I only brought it up in relation, and to provide context for, the idea of Field of View; my only argument here is not about 'spooky mathematics', but rather that, how the sun disappears at the 50mile mark, when it is supposed to be 12,000+ miles away. Not a typical Flat Earther question, I think.
This brings up another problem with the standard flat earth model: the sun doesn't always appear to be curving northward as it sets, like the flat earth model predicts.
Yup.
if you trace converging lines, they converge a great distance before 12,000 miles away; what the image here is depicting.
Oh really? And how did you calculate this? I assume you are referring to the vanishing point. If so, I can assure you that you are misunderstanding what the vanishing point is.
Most demonstrably shaky part of my argument (other than using 3D generated blueprints). Touche, I regretted typing that part and even omitted it several times in my rough drafts. I don't even know how to calculate the 'convergence point', even utilizing what you provided.
If you are going to make an argument about infinity, you will need to be much more specific than this.
I wasn't making this argument, but offering my speculation about the "infinities' I kept seeing repeated again and again in this thread, since they don't make sense to me. Also why I posted the Toy Story clip (https://www.youtube.com/watch?v=ejwrxGs_Y_I). Was meant to be considered tongue in cheek, though I do like the idea of a completed Flat Earth model, these topics make it seem rather bleak (to me, at least). Perhaps it's due to my (lack of) education ::) ;D I have been skimming Gleason and Parallax's books, but I'm no closer to comprehension than I was when I first decided to pick this topic up (FE).
According to the standard flat earth model, the sun would get no farther away from us than the yellow ball. Therefore, the MINIMUM angle it would make with the horizon is 20 degrees. Since we can observe the sun going much lower than that every single day, this proves that the earth isn't flat. How is this not blatantly obvious?
Honestly, I thought I was most clever in the construction of this paragraph, glad someone pointed it out already. I knew I slid a silent 'checkmate' in there but wasn't sure if it would be noticed. Point 3 has been bothering me for a while now; it seems the sun shouldn't set on a flat earth with these dimensions. Although I admit pbrane did pull me in a bit with his 'perspective' as a solution to why it 'sets' on FE. This is one of my biggest problems with the [dimensions of the] FE model.
Also, how on earth did you calculate the slant of that line? Everything having to do with perspective in the bottom image was very carefully calculated. I did not place anything arbitrarily.
Short answer: MS Paint Magic. Long answer: No calculations were done. Shame on me on this one. My second major blunder in that post. I knew pbrane had his arbitrary position, and the top image was arbitrary as well, so I put an arbitrary line to demonstrate the 'convergence' of the two 20,000 mile lines on the top image. I felt this wasn't too much of a step, but yes I should have calculated I suppose.
My image was not intended to exactly represent the dimensions of a flat earth. I never said it was. If you remove the red and orange objects, then the distances are a good approximation of a flat earth.
Yes, you did say that in your first response to me. I missed it. That's my fault. I wouldn't have posted my previous post if I realized this. I thought about deleting it, but realized I was probably being responded to already (as I correctly guessed). So I'll let my stupidity stand, I figured, and I doubleposted instead to point this out. I thought there was shady logic being passed off as math, then realized I was wrong. I owe an apology there, so sorry. My bad.
The difference between my image and pbrane's, is that the bottom image that I drew was carefully calculated according to the math I have been describing. Nothing about it was arbitrary
Again. Sincerest apologies. See above. You did a great job. I pooped on it. I'm ashamed.
The perspective effect portrayed in the bottom image arose naturally from the math. On the otherhand, the perspective lines that pbrane (and you) drew are completely arbitrary. They are nothing more than a vague guess based on a vague/poor understanding of how perspective works.
Again, above. I thought the top image was trying to convey something else, so I made something up to counter it. That's my own oversight at work there. I see now.

Okay Everything bellow here is just a repeat of the points already corrected, I think. Like I said I'm not going to retract it, though I now see how it was wrong. I underestimated your presentation, didn't read it enough times, thought I saw a representation of what I thought was a deliberate lie, and went to work. I realize that now. Also, my math is terribly vague estimates. I thought about it, and the sun should be 'turning around' before the 12k miles point, I'm fairly certain. Though the fact you didn't bother addressing that speaks louder than I can say anyway. Thanks for the effort, and sorry for the potential aneurysm. I honestly got a great chuckle out of this, realizing how simple my misunderstand was. Still, I can't get it to add up without 'spooky mathematics', as you say, for FE. I feel really bad, but at the same time, this was very cathartic. I laughed pretty good, hope you will too.

Okay, now that I got the apologies out of the way, I think pbrane makes some good points. I totally misconstrued your argument, thus mine was approaching the wrong way... Here is what I meant to say, had I not thought someone was saying parallel lines never converge due to perspective:
Now, again, I don't understand the math of the angles represented in the diagrams he is 'debunking'. But his point at 4:50 (https://youtu.be/W0Gx1vD1CRE?t=4m50s) seems solid, right?
I mean, airplanes do seem to travel slower near the horizon, but they do emerge  and pass over  as it were. The higher the object, the more downward elevating is the angle. That's simple high school geometry. The sun should travel slower, 'but muh vanishing point'. Obviously the sun makes it there (it is night time in contiguous America right now), and pbrane seems to demonstrate how it can get there on a Flat Earth; perspective.
Which is what that video is about; showing how theoretical math predictions about where parallel line locations will be has nothing to do with where they actually are.
I don't know the angular math but I can see what he is saying. The vanishing point is the same for all parallel lines, no matter how high up you draw them; all parallel lines converge on the exact same location. Like so:
(https://upload.wikimedia.org/wikipedia/commons/d/d4/Vanishing_Point.jpg)
Again I find myself thinking that mathematical geometry is just a red herring for uninquisitive and undereducated minds. Which I guess I am. Claiming the sun is exempt from this rule because maths still sounds funny to me. I think he made good points, and I don't see how explaining how angles and perspective works changes the fact that all parallel lines converge in the same spot. Though I imagine I'll shortly be pulling my foot out of my mouth again; in which case I would apologize ahead of time, but I know you wouldn't have made this topic if you didn't already consider the nature of the vanishing point, and are already anticipating this type of response. ;D I'm sure I'm just missing something. I swear it's like I can feel it coming already.

Lol great post, I caught my error before you posted though. I thought you were trying to pull a fast one, but then I realized I was wrong. That is my fault for jumping the gun. I got some really good chuckles reading your post, though ;D spot on. I deserved it.
No worries. I apologize if the tone of my post was a bit snippy. In my defense, I have been arguing in circles with Tom for awhile now.
2. Parallel lines converge due to perspective; even those in a OP diagram, if drawn to 'infinity'.
No no no no no! A thousand times no! Parallel lines are always parallel in an orthographic projection, by definition. If the parallel lines converge, then it isn't an orthographic projection.
On a lifesized, really big, OP projection, they would converge ;D
Oh... I see what you are saying. Lol, yes, since you view everything through the perspective projection created by your own eyes, then even the parallel lines on an orthographic projection will appear to converge. This doesn't change the actual angles measured in the orthographic diagram though. This has more to do with the size or closeness of the physical piece of paper that you draw the diagram on, and nothing to do with the scale of the object being portrayed by the diagram. If you are having problems with perspective while viewing your orthographic diagram, I suggest that you not use a sheet of paper the size of a football field. _
If you are going to make an argument about infinity, you will need to be much more specific than this.
I wasn't making this argument, but offering my speculation about the "infinities' I kept seeing repeated again and again in this thread, since they don't make sense to me.
Here's the thing. All these arguments about infinity are vague, and basically amount to "the idea of infinity is somehow related to this concept, therefore everything involved with this concept is impossible and wrong".
The problem is, we can relate just about ANY concept to infinity if you are vague enough. How many times can an inch be subdivided? An infinite amount of times? Does this mean it's impossible for anything to be an inch long? No, of course not.
Mathematicians deal with infinity all the time, despite the apparent impossibility of any value actually reaching infinity. They do this by being precise with their assumptions and meanings. For example, we can say "as x approaches infinity, 1/x approaches zero". Note that we are not saying that x REACHES infinity. Nor are we saying that 1/x REACHES zero. This is a common topic in introductions to calculus.
If you want some food for thought, I recommend this Vsauce video on the size of numbers (https://www.youtube.com/watch?v=SrU9YDoXE88). Or this one on Zeno's Paradox (https://www.youtube.com/watch?v=ffUnNaQTfZE). Of course, these videos, as awesome as they are, will not give you much more than just a vague understanding of the topics. If you want a working knowledge, you will have to study hard! I recommend starting with courses titled "number theory" or "set theory". Preferably with the word "introduction" in their title. (This post took a long time to make because I got distracted by the Vsauce channel. RIP my Saturday.)
But his point at 4:50 (https://youtu.be/W0Gx1vD1CRE?t=4m50s) seems solid, right?
Sort of. The fact that "they all visually go to the same place" is exactly what is predicted by the math. Notice that in the bottom perspective drawing that I made, the 3 objects are approaching a point on the horizon. They are following perspective lines.
1. 5:17 "These are true perspective lines, these are parallel lines, but the observer, which is this little guy down here, this what these parallel lines would look like."
Absolutely not. The "little guy down here" is not the observer. The camera is the observer. That little guy down there was arbitrarily placed in that scene. He is trying to combine the orthographic diagram (which shows distances between objects, including the observer) and the perspective projection (which shows how things appear, from the observers point of view). Everything he tries to calculate from this combined diagram will be wrong.
I mean, airplanes do seem to travel slower near the horizon, but they do emerge  and pass over  as it were. The higher the object, the more downward elevating is the angle. That's simple high school geometry. The sun should travel slower, 'but muh vanishing point'. Obviously the sun makes it there (it is night time in contiguous America right now), and pbrane seems to demonstrate how it can get there on a Flat Earth; perspective.
Which is what that video is about; showing how theoretical math predictions about where parallel line locations will be has nothing to do with where they actually are.
I don't know the angular math but I can see what he is saying. The vanishing point is the same for all parallel lines, no matter how high up you draw them; all parallel lines converge on the exact same location. Like so:
(https://upload.wikimedia.org/wikipedia/commons/d/d4/Vanishing_Point.jpg)
Ok, instead of telling you how each statement is wrong or right, let me give you this challenge:
Using that diagram, how long would it take for an object to REACH the vanishing point by following the two bottom parallel lines?
Again I find myself thinking that mathematical geometry is just a red herring for uninquisitive and undereducated minds. Which I guess I am.
Well, you certainly aren't uninquisitive. But I disagree. Geometry is a fantastic subject for everyone to understand, not just mathematicians. In fact, I am of the opinion that the method of creating geometric proofs that most people learn in highschool is the single most important subject for promoting good problem solving skills and logical reasoning.
I know you wouldn't have made this topic if you didn't already consider the nature of the vanishing point, and are already anticipating this type of response. ;D I'm sure I'm just missing something. I swear it's like I can feel it coming already.
Good guess... :)
(Edit: Removed Jack Nicholson picture. His smile creeps me out.)
Just do the challenge from above, then we can get back to this.

The math you are using says that is is impossible for anything to reach the horizon. A sun would descend forever on a Flat Earth without ever reaching the horizon. The math says that touching the horizon is impossible, as 0 degrees is defined as infinity.
Where has any endless approach to a point like this been reported in any scientific, everyday, or historical observation, or anywhere in the world? Please present some form of evidence that any of this is possible. Provide something, anything from reality, showing that ultra long descents would occur or that the horizon would be an infinite distance away.
Tom, if the math doesn't fit your model, then maybe it's time to start considering the possibility that your model just might be wrong.

The math you are using says that is is impossible for anything to reach the horizon. A sun would descend forever on a Flat Earth without ever reaching the horizon. The math says that touching the horizon is impossible, as 0 degrees is defined as infinity.
Where has any endless approach to a point like this been reported in any scientific, everyday, or historical observation, or anywhere in the world? Please present some form of evidence that any of this is possible. Provide something, anything from reality, showing that ultra long descents would occur or that the horizon would be an infinite distance away.
Tom, if the math doesn't fit your model, then maybe it's time to start considering the possibility that your model just might be wrong.
Many people have seen a flat earth. No one has seen that aspect of the math that causes perspective lines to perpetually approach each other for infinity. The ball is in your court for this highly theoretical concept, I am afraid.

The math you are using says that is is impossible for anything to reach the horizon. A sun would descend forever on a Flat Earth without ever reaching the horizon. The math says that touching the horizon is impossible, as 0 degrees is defined as infinity.
Where has any endless approach to a point like this been reported in any scientific, everyday, or historical observation, or anywhere in the world? Please present some form of evidence that any of this is possible. Provide something, anything from reality, showing that ultra long descents would occur or that the horizon would be an infinite distance away.
Tom, if the math doesn't fit your model, then maybe it's time to start considering the possibility that your model just might be wrong.
Many people have seen a flat earth. No one has seen that aspect of the math that causes perspective lines to perpetually approach each other for infinity. The ball is in your court for this highly theoretical concept, I am afraid.
You will have to explain how these many people decided they have seen a flat earth, beyond 'it looks flat'. Ask how they explain the path of the sun as seen from many locations.
You need to understand the definition of perspective.
How's the dish measurements going?

Many people have seen a flat earth. No one has seen that aspect of the math that causes perspective lines to perpetually approach each other for infinity.
Still have your fingers firmly implanted in your ears, repeating the same refuted argument I see... Whatever. Just answer the question:
Fine. What evidence would convince you? If I am able to correctly predict the size of an object on a picture based on its size and distance using the "ancient greek math", would that convince you? I'm not going to waste my time if you are just going to cry "fake".
The ball is in your court
Still? We must be playing makeittakeit.
for this highly theoretical concept, I am afraid.
Ah yes. Highly theoretical highschool geometry.

The math you are using says that is is impossible for anything to reach the horizon. A sun would descend forever on a Flat Earth without ever reaching the horizon. The math says that touching the horizon is impossible, as 0 degrees is defined as infinity.
Where has any endless approach to a point like this been reported in any scientific, everyday, or historical observation, or anywhere in the world? Please present some form of evidence that any of this is possible. Provide something, anything from reality, showing that ultra long descents would occur or that the horizon would be an infinite distance away.
Tom, if the math doesn't fit your model, then maybe it's time to start considering the possibility that your model just might be wrong.
Many people have seen a flat earth. No one has seen that aspect of the math that causes perspective lines to perpetually approach each other for infinity. The ball is in your court for this highly theoretical concept, I am afraid.
Tom, you are the one who is claiming that sunsets on a flat earth are the results of perspective. Doesn't it stand to reason that you should be the one show the math that supports your claim? You do realize that perspective does indeed have a mathematical foundation to it, don't you?
https://www.math.utah.edu/~treiberg/Perspect/Perspect.htm

If you are having problems with perspective while viewing your orthographic diagram, I suggest that you not use a sheet of paper the size of a football field. _
Can't help it, I'm a final solution kind of individual. I got tired of running out of toilet paper. ::)
I get it with the infinities, I think now, and why they are running infinitely (http://imgur.com/6ADLV4r.png) through this topic...
The fact that "they all visually go to the same place" is exactly what is predicted by the math. Notice that in the bottom perspective drawing that I made, the 3 objects are approaching a point on the horizon. They are following perspective lines.
1. 5:17 "These are true perspective lines, these are parallel lines, but the observer, which is this little guy down here, this what these parallel lines would look like."
Absolutely not. The "little guy down here" is not the observer. The camera is the observer.
I get this. It's kind of like back to the first image in this thread, we are looking at someone else's perspective, the 'tilted' middle piece (http://i68.tinypic.com/30u3ci1.png).
However, when we are 'that little guy down there', with our face pressed against that wall, the lines still converge... but oh shi I think I see where this is going.
Ok, instead of telling you how each statement is wrong or right, let me give you this challenge:
Using that diagram, how long would it take for an object to REACH the vanishing point by following the two bottom parallel lines?
Ugh. From wikipedia (https://en.wikipedia.org/wiki/Vanishing_point): When the image plane is parallel to two worldcoordinate axes, lines parallel to the axis which is cut by this image plane will meet at infinity i.e. at the vanishing point.
Unrelated but I had nightmares about this image last night, thanks for making me read about vanishing points...
(https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Axonometric_projection.svg/150pxAxonometric_projection.svg.png)
That image makes my brain hurt.

@totesnotreptilianCorrect me if I am wrong but isn't it incorrect to say that "the math says perspective lines meet at inifinity"? Instead, isn't it more accurate to say that the mathematically, the only place parallel lines could meet under Eulcid's axioms is at inifinity? So it is more about excluding the lines intersecting in real space, rather than saying they actually do meet at inifinity?
Hope this makes sense.

Ok, instead of telling you how each statement is wrong or right, let me give you this challenge:
Using that diagram, how long would it take for an object to REACH the vanishing point by following the two bottom parallel lines?
Ugh. From wikipedia (https://en.wikipedia.org/wiki/Vanishing_point): When the image plane is parallel to two worldcoordinate axes, lines parallel to the axis which is cut by this image plane will meet at infinity i.e. at the vanishing point.
Well, yes. My point is that you don't have to take wikipedia's or my word for it. You can come to the same conclusion from that diagram.
@totesnotreptilianCorrect me if I am wrong but isn't it incorrect to say that "the math says perspective lines meet at inifinity"? Instead, isn't it more accurate to say that the mathematically, the only place parallel lines could meet under Eulcid's axioms is at inifinity? So it is more about excluding the lines intersecting in real space, rather than saying they actually do meet at inifinity?
Hope this makes sense.
I wouldn't say the first wording is completely incorrect. But yeah, your wording is much more clear and precise.

You will have to explain how these many people decided they have seen a flat earth, beyond 'it looks flat'. Ask how they explain the path of the sun as seen from many locations.
What observations would those be? Surely you can reference a paper on the subject rather than make up observations.
How's the dish measurements going?
What are you talking about? That is your idea for an experiment and your responsibility.

Still have your fingers firmly implanted in your ears, repeating the same refuted argument I see... Whatever. Just answer the question:
Fine. What evidence would convince you? If I am able to correctly predict the size of an object on a picture based on its size and distance using the "ancient greek math", would that convince you? I'm not going to waste my time if you are just going to cry "fake".
I don't know. Would that prove that the perspective lines perpetually approach each other for infinity?
Ah yes. Highly theoretical highschool geometry.
The same geometry fantasy which says that perfect circles exist when they, in fact, do not.

Still have your fingers firmly implanted in your ears, repeating the same refuted argument I see... Whatever. Just answer the question:
Fine. What evidence would convince you? If I am able to correctly predict the size of an object on a picture based on its size and distance using the "ancient greek math", would that convince you? I'm not going to waste my time if you are just going to cry "fake".
I don't know. Would that prove that the perspective lines perpetually approach each other for infinity?
No. My point is that we can predict the angular distance between two objects based on distance. I can demonstrate that the math works for this.
I ask again. If I am able to correctly predict the size of an object on a picture based on its size and distance using the "ancient greek math", would that convince you that the math works? Please note that this question has nothing to do with anything an infinite distance away.
Ah yes. Highly theoretical highschool geometry.
The same geometry fantasy which says that perfect circles exist when they, in fact, do not.
*sigh*, fine, I'll bite. Which part of geometry states that perfect circles exist? Please be specific.

I would think that since Pi is an irrational that you would never be able to get an exact answer.

Ah yes. Highly theoretical highschool geometry.
The same geometry fantasy which says that perfect circles exist when they, in fact, do not.
As opposed to the geometry fantasy which says that the sun can recede into the far distance without decreasing in angular size?

The actual fact of the matter is that not all parallel lines seem to converge AT the vanishing point of a person, they would only converge along the midpoint of the projected line in the middle of these two parallel lines! If someone was standing in the middle of a flat piece of desert, and in the middle of parallel lines were stretched out in front of him at widths apart of one inch, one foot, one yard, 100 yards, 1000 yards, 5000 yards, 3 miles, and some further outside his vision, the first 3 would appear to touch well within his vanishing point, the 4th maybe near the VP, while the others would only appear to be converging inwards to meet way beyond the VP or will not even be visible yet. Add two parallel lines 32 miles high, but 6000 miles apart, visible just above the horizon (ie the size the sun appears to be) and these lines will not even seem to converge at all, apart from the fact that they would start to look much smaller than the sun as they recede into the distance. In fact if you could still see these lines 9000 miles in front of you and they ended, there would still be an angle of over 40 degrees between them. Now turn this point of view through 90 degrees an imagine it in the vertical plane. The same would apply.

FE silenced, so let's add some more pressure! Another question for the boffins, how can perspective change every day? To show this simply, take a city like Quito in Ecuador, and the solstices in March and December. the sun rises and sets at the same times on both these days, but using the FE model the sun will be about 28% further away when it disappears! Surely perspective should be consistent?
We will not even go into the cases when the sun sets or rises over the ocean in places where the distance to the sun on the horizon vanishing point could be easily shown as over 3 times the distance comparing winter sunrises/sunsets during winter months to summer months on FE maps. Please explain how perspective changes with season!

FE silenced, so let's add some more pressure! Another question for the boffins, how can perspective change every day? To show this simply, take a city like Quito in Ecuador, and the solstices in March and December. the sun rises and sets at the same times on both these days, but using the FE model the sun will be about 28% further away when it disappears! Surely perspective should be consistent?
We will not even go into the cases when the sun sets or rises over the ocean in places where the distance to the sun on the horizon vanishing point could be easily shown as over 3 times the distance comparing winter sunrises/sunsets during winter months to summer months on FE maps. Please explain how perspective changes with season!
Solstices are in June and December, not March. Also, it would be helpful to show how you came up with your 28% figure. You do have a good point, but lets try to stick to the topic, since this thread is already rather lengthy. Here is a thread that deals with sunset/sunrise times and how they relate to distances. (http://forum.tfes.org/index.php?topic=5237.0) Or, you are welcome to start a new thread.
The actual fact of the matter is that not all parallel lines seem to converge AT the vanishing point of a person, they would only converge along the midpoint of the projected line in the middle of these two parallel lines! If someone was standing in the middle of a flat piece of desert, and in the middle of parallel lines were stretched out in front of him at widths apart of one inch, one foot, one yard, 100 yards, 1000 yards, 5000 yards, 3 miles, and some further outside his vision, the first 3 would appear to touch well within his vanishing point, the 4th maybe near the VP, while the others would only appear to be converging inwards to meet way beyond the VP or will not even be visible yet. Add two parallel lines 32 miles high, but 6000 miles apart, visible just above the horizon (ie the size the sun appears to be) and these lines will not even seem to converge at all, apart from the fact that they would start to look much smaller than the sun as they recede into the distance. In fact if you could still see these lines 9000 miles in front of you and they ended, there would still be an angle of over 40 degrees between them. Now turn this point of view through 90 degrees an imagine it in the vertical plane. The same would apply.
We have already kind of beaten to death what WOULD happen at really long distances. The problem is that Tom Bishop just straight up denies what would happen according to common sense and basic mathematics. His logical reasoning for discounting the math has already been shown to be nonsensical, but he insists the burden of proof is on us to disprove his own theories rather than on him to provide proof positive. I offered to show that the math works using photos, but he dodged that offer as well, so at this point I don't think there is much more to be said.

No, the burden is on you to prove your own theories.

No, the burden is on you to prove your own theories.
I have bad news Tom. You seem to have your fingers permanently embedded in your ear holes. Surgery might be your only option. Oh well, at the risk of repeating myself... I have provided as much proof as can be expected on an anonymous internet forum:
1. I provided logical reasoning. I showed why your infinity arguments were ridiculous.
2. I outlined several ways that you can empirically prove to yourself that the math works for any testable distance.
3. I offered to show that the math empirically works at any testable distance using photographs, but you dodged the offer.
Your only argument remaining is "maybe the math works differently at longer distances". These distances are not testable without appealing to astronomical data that you automatically claim is faked/wrong. Therefore, this argument is unfalsifiable. Therefore, the burden is on you to offer proof positive that the math works differently at long distances, because proof negative is literally not possible. So far, you have offered none. Instead, you settle for trying to cast doubt on other peoples theories, rather than trying to support your own.
Tom, you constantly make claims of having higher ethical standards than the rest of the scientific community. But this is not how honest scientists behave. Honest scientists don't settle for nitpicking other peoples theories and claiming that as proof that their own counter theories are true. Honest scientists try their hardest to find holes in their OWN theories. They gladly accept the burden of proof, rather than constantly casting it off on someone else. You sound more like a defense lawyer, desperately trying to find legal loopholes to allow his serial killer client escape a guilty verdict. That is not how honest scientists work. This is not how honest scientists come to conclusions.
Speaking of honesty, how much longer are you going to ignore this post? (http://forum.tfes.org/index.php?topic=5366.msg104930#msg104930)

No, the burden is on you to prove your own theories.
Proven by many, many times. Which particular one do you have a problem with and please show your proof of an alternative. You are in the minority and have yet to prove any of your beliefs are correct.
Have you checked any satellite dish angles to dermine the transmitter location, simple to do for anyone unsure of their location, as you are.