[TotesNotReptilian]

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 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. 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?



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, 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.

[Tom Bishop]

As you turned the scene the height of the far end of the terminal arm dropped in height, due to perspective.

[TotesNotReptilian]

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?

[TotesNotReptilian]

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.

[Tom Bishop]

Draw it. Your math assumes an out-of-universe 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.

[Woody]

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.

[TotesNotReptilian]

Draw it.

Wish granted.



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. As long as the camera isn't very wide angle, the distortion should be minimal.

Your math assumes an out-of-universe 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 "out-of-universe", 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.

[Tom Bishop]

Wish granted.

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. 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?

[rabinoz]

Draw it. Your math assumes an out-of-universe 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.

Those buildings seem past the horizon They have certainly have not merged to a vanishing point.

The Diamond Princess (16 miles away seems past the horizon It certainly has not merged to a vanishing point..

Those buildings seem past the horizon They have certainly have not merged to a vanishing point.

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 candle-flame, the most well-founded 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,

[Tom Bishop]

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 in Earth Not a Globe by Samuel Birley Rowbotham for additional information.

[Rama Set]

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.

[TotesNotReptilian]

Wish granted.

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. 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.

[Tom Bishop]

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.

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.

[Tom Bishop]

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?

[TotesNotReptilian]

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.

[Tom Bishop]

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

Any continuous mathematical model like this which predicts infinities should be looked at with scrutiny and demands justification.

[rabinoz]

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.

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.

[Rama Set]

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

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.

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!

[Nostra]

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  !

[andruszkow]

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  !

Actually, please don't.