Curvature of the Horizon

  • #36
DaveC426913 said:
This arc would be of a portion circle/ellipse that would subtend a measurable amount of his view (at that specific distance from the window).
It's a circle. It can't be an ellipse.
DaveC426913 said:
I doubt the radius of that curve would be the same as the actual horizon if he were looking straight down on the Earth, because, looking out the porthole like this, the rest of the horizon wouldn't be below his feet, it would be literally behind him, almost viewable out the opposite porthole.


It's no longer points on a horizon; It's a curve on a plane perpendicular to the observer. That curve is what the observer sees, absent any 3-dimensional/foreshortening clues.

Maybe I've got it wrong. If the OP can't seem to nail down a description of what he's doing, I'm not going to do much better.
You can't see a defined radius, you can only measure a radius. A ball held at a certain distance will appear to have the same radius as the Moon. You need to calculate the distance before you can say how big something is.
 
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  • #37
PeroK said:
It's a circle. It can't be an ellipse.
A circle, seen at an oblique angle is observed to be an ellipse.

The oblique angle is due to the fact that the plane of his image is in front of him.

If he used that sharpie to extend the arc of the horizon as seen through the porthole, he could see the entire curve on the wall under the porthole at his feet.


PeroK said:
You can't see a defined radius, you can only measure a radius. A ball held at a certain distance will appear to have the same radius as the Moon. You need to calculate the distance before you can say how big something is.
A red herring. We're not talking about how big something is, we're talking about a curve, as seen by the observer that subtends a portion of his vision.


We are clearly talking past each other. I don't know what else I can say. Best I can do is have you re-examine these diagrams. (I have eliminated the red lines representing the horizon - that is causing confusion.)

One has the line-of dight aligned with the centre of the Earth; its plane of imaging tangential with the Earth's surface, and the entire horizon visible.

The other has line-of-sight aligned with the horizon; its plane of imaging perpendicular to the Earth surface, and only a portion of the horizon in his field of view (much of it is outside his line of sight - even below/behind him).

Would the two arcs - draw with a blue sharpie - on the planar surface - have the same radius?
1738606210949.png
 
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  • #39
PeroK said:
It is not wrong. A circle, seen at an oblique angle is, in fact, an ellipse. Your reference is a red herring. (It doesn't mean you don't have a valid point, but you haven't made that case with what you wrote.)



A circle the size of a horizon can only be seen if you have a completely undistorted 360 view from edge-to-edge, which humans don't have when looking horizontally at one part of the horizon.

If human is looking at the horizon from the ISS, they will see a curve - even though the other edge of the curve is literally behind him. But he can still sketch out the curve he sees on the plane that is perpendicular to his line of sight - with the caveat that, at some, point, his observed curve will deviate from the true circle due to perspective distortion. I guess what he would draw is a parabola, with the ends reaching the floor.
 
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  • #40
DaveC426913 said:
It is not wrong. A circle, seen at an oblique angle is, in fact, an ellipse. Your reference is a red herring. (It doesn't mean you don't have a valid point, but you haven't made that case with what you wrote.)
Please calculate the eccentricity of the ellipse as a function of altitude.
 
  • #41
1738607607497.png

I ask you look again at the two diagrams in post 37. In the second diagram, the plane on which the curve is drawn is not perpendicular to a line drawn to the centre of the sphere - it is oblique.

Instead, it is perpendicular to the top edge of the horizon.

Therefore, the image on the plane's oblique 2-dimensional surface will be distorted from a circle.
 
  • #42
PeroK said:
Please calculate the eccentricity of the ellipse as a function of altitude.
That is exactly what the OP is claiming he can do. If I had solved that, I would have surpassed the OP.
 
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  • #43
Next thing, someone will claim that a rainbow does not form a perfect circle, because you are looking at a small part of it, or part is underground. No part of a rainbow is ever an ellipse, it is always a circle to the observer.
 
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  • #44
DaveC426913 said:
View attachment 356769
I ask you look again at the two diagrams in post 37. In the second diagram, the plane on which the curve is drawn is not perpendicular to a line drawn to the centre of the sphere - it is oblique.

Instead, it is perpendicular to the top edge of the horizon.

Therefore, the image on the plane's oblique 2-dimensional surface will be distorted from a circle.
That circle is NOT part of the horizon. You are always looking at the horizon from the conical centre, never obliquely. Oblique would be off centre. As drawn erroneously in that diagram, imagining that different points on the horizon are different distances away from the observer.

This is sheer nonsense now.
 
  • #45
All right. I concede. I thought I knew where the OP was going, but it is possible I have over-interpreted his idea. Thanks guys for being patient with me.
 
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  • #46
Baluncore said:
Next thing, someone will claim that a rainbow does not form a perfect circle, because you are looking at a small part of it, or part is underground. No part of a rainbow is ever an ellipse, it is always a circle to the observer.
It's an interesting analogy; one I tried to employ, actually, but abandoned. But it kind of highlights where I've been going. A rainbow will always fit with an observer's point of view, with negligible distortion, and can therefore be treated as a circle.

The Earth's horizon, OTOH, can easily extend well-outside the observer's field of view, necessarily distorting it from a circle - as the observer sees it in the 2D plane of his vision.
 
  • #47
DaveC426913 said:
The Earth's horizon, OTOH, can easily extend well-outside the observer's field of view, necessarily distorting it from a circle - as the observer sees it in the 2D plane of his vision.
At some point, depending on your measurement apparatus, you measure only the arc of a circle. Then the circle is constructed from many of these homogeneous arcs.
 
  • #48
DaveC426913 said:
The Earth's horizon, OTOH, can easily extend well-outside the observer's field of view, necessarily distorting it from a circle - as the observer sees it in the 2D plane of his vision.
Those two dimensions are azimuth and elevation in spherical coordinates, it is not a flat x-y plane. The observer, at the centre of those spherical coordinates, is located on axis, at the apex of the tangent cone. The eye is not a flat sheet like an image sensor, or a flat photograph.

The only time the circle of the horizon, (or a rainbow), can be an ellipse, is when a diagram is drawn from the viewpoint of a third person, (an artist not on the conical axis), picturing an observer at the centre of the observer's horizon, with the horizon and rays sketched as seen by the primary observer.
 
  • #49
DaveC426913 said:
All right. I concede. I thought I knew where the OP was going, but it is possible I have over-interpreted his idea. Thanks guys for being patient with me.
Honestly, I think you're 95% of the way there, you just got a small detail slightly wrong (the angle in your second photo, which the OP pointed out in the next post). Otherwise and more importantly the main point does seem to be what the OP is after: The horizon is flat when you look at it level at zero elevation and curved downwards when you look at it from higher elevation.

I was hoping my planetarium software would show this, but unfortunately it gives a limited and cartoonish view.
 
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  • #50
russ_watters said:
Honestly, I think you're 95% of the way there, you just got a small detail slightly wrong (the angle in your second photo, which the OP pointed out in the next post).
Yes. I regret even putting that line in. It wasn't meant to be to-scale. It was a sketch, not meant to be nit-picked.
 
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  • #52
Baluncore said:
Those two dimensions are azimuth and elevation in spherical coordinates, it is not a flat x-y plane. The observer, at the centre of those spherical coordinates, is located on axis, at the apex of the tangent cone. The eye is not a flat sheet like an image sensor, or a flat photograph.
This is what I'm trying to describe:

1738621678588.png


  • The observer is drawing a curve on the wall of the station. It is perpendicular to his LoS, and is oblique to a line to the centre of the Earth, as well as oblique to the plane of the full 360 horizon.
  • He draws the curve of the horizon he sees. It cannot be a circle since the bottom edge of the Earth will never be visible from his vantage point - it is literally behind him (see magenta arrow pointing at the distant windows in this very big room. Parts of the horizon are behind him).
  • Therefore, whatever he draws on the wall is not going to be a circle. Nor do I think it will be an arc of a circle. At best, what he will draw is a parabola. He is unable to see - let alone draw - the entirety of the circle that is the true horizon.
  • But that wall is his working surface, upon which he can use his sharpie to draw and take measurements, including find the finite centre of a portion of that blue curve. (Again - which can't be a circle, since a circle with an infinite radius would have zero curvature.)

I think this is what the OP is alluding to. The ability to treat what the observer sees of the portion of the horizon that he can see - as 2D geometry on that plane - to deduce his altitude.
 
  • #53
DaveC426913 said:
This is what I'm trying to describe:
  • Therefore, whatever he draws on the wall is not going to be a circle. Nor do I think it will be an arc of a circle. At best, what he will draw is a parabola. He is unable to see - let alone draw - the entirety of the circle that is the true horizon.
FWIW, I think you are correct that the projected disk of the horizon is not a circle, but I think you were right the first time that it is an ellipse. The circle of the horizon rotates as you get closer and rotate your view toward horizontal. Ultimately you are looking at the circle edge-on when your view height is on the surface.

Anybody have a hula-hoop...?
 
  • #54
DaveC426913 said:
The observer is drawing a curve on the wall of the station. It is perpendicular to his LoS, and is oblique to a line to the centre of the Earth, as well as oblique to the plane of the full 360 horizon.
It was all spherical coordinates. Once you introduce a diagonal image plane, and then view it from off-axis, all circular bets are off. But in that case, the observer is not looking at the horizon, a visitor is looking at an ellipse, artistically drawn on a plate.

The conic sections that may occur are circles or ellipses.
https://en.wikipedia.org/wiki/Conic_section
Parabolas are only possible if the diagonal window is so close to the axis, that it stays within the cone.
 
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  • #55
OK, I know when I've been beat. Thanks for humoring me.
 
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  • #56
PeroK said:
It's a circle. It can't be an ellipse.
The orthogonal projection of a circle onto a plane to which it is not coplanar is an ellipse. This projection however is not orthogonal.

But all this discussion, while perhaps somewhat interesting, is not directly addressing the OP’s question regarding where to publish. Let’s be blunt: It is not publishable. It is basic geometry that might suffice for a high-school project report, not novel research publishable in peer reviewed journals.
 
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  • #57
Baluncore said:
It was all spherical coordinates. Once you introduce a diagonal image plane, and then view it from off-axis, all circular bets are off. But in that case, the observer is not looking at the horizon, a visitor is looking at an ellipse, artistically drawn on a plate.

The conic sections that may occur are circles or ellipses.
https://en.wikipedia.org/wiki/Conic_section
Parabolas are only possible if the diagonal window is so close to the axis, that it stays within the cone.
I don't think a conic section is a good way of describing vision in general. Particularly not if you have the possibility to turn your head around and see the horizon in different directions to create an impression of the view. Instead, the projection onto the unit sphere of viewing directions comes to mind. Evidently, the horizon is described by a circle on this sphere (assuming the planet is a sphere) and the reasonable measure of curvature of that circle is the magnitude ##\sqrt{A^2}## of the curve acceleration ##A = \nabla_{\dot \gamma} \dot\gamma## when the horizon curve ##\gamma## is parametrized by its curve length, which is a geometric invariant. It evaluates to ##\tfrac{h}{R}\sqrt{1 + 2\tfrac{R}{h}}## where ##R## is the radius of the planet and ##h## the height of the observer above the planet surface. This clearly has the correct limiting behaviours of going to 0 as ##h \to 0## and ##\infty## as ##h\to \infty##.
 
  • #58
Now that said, if you were to project onto a plane in that fashion then there are several possibilities for the shape. Assuming that you center the window such that the center (your optical axis) is parallel to the direction towards a point on the horizon, then there are three options:
  1. The plane cuts the cone towards the horizon in an ellipse.
  2. The plane cuts the cone towards the horizon in a parabola.
  3. The plane cuts the cone towards the horizon in a hyperbola.
Case 2 is the intermediate case between 1 and 3 and occurs when the cone angle is exactly 45 degrees. Any larger opening and it is a hyperbola. Any smaller opening and it is an ellipse.
 
  • #59
If the horizon is not seen as a circle, then nothing can be seen as a circle. If you are at the centre of a circle, then all you can see is a boundary in all directions. The only way to see the circle is to assume a vantage point above the centre of the circle. And, if you don't see a circle then, you never will!
 
  • #60
PeroK said:
If the horizon is not seen as a circle, then nothing can be seen as a circle. If you are at the centre of a circle, then all you can see is a boundary in all directions. The only way to see the circle is to assume a vantage point above the centre of the circle. And, if you don't see a circle then, you never will!
Again, it depends what you mean by ”see”. It is obviously a circle on the visual sphere, but if you make the projection on a plane that has been discussed above - it is an ellipse, parabola or hyperbola on that plane.
 
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  • #61
Orodruin said:
Again, it depends what you mean by ”see”. It is obviously a circle on the visual sphere, but if you make the projection on a plane that has been discussed above - it is an ellipse, parabola or hyperbola on that plane.
Okay, I see what you are getting at. As you get closer to the surface, you have to tilt your line of vision to match the zenith angle. As you scan the horizon, the zenith angle does not change. So, it appears circular, from that perspective. But, on each small segment of your vision there must be some sort of elliptical distortion.

However, if you look carefully at something, you can only focus on a single pointlike segment at any one time. To generate a visual image, you must move your line of vision. This should eliminate the elliptical distortion from your perspective. The diagrams above are misleading, as they suggest you could look at a significant portion of the horizon simultaneously. What you see is not what is projected onto a single plane some distance in front of your eyes. It's more like what is projected onto a circular (or spherical) surface.

They also suggest that you would see a significant portion of the horizon as an ellipse. Which is not the case. It would be at most a tiny segment (perhaps a degree or so?).

In any case, if your brain constructs the full image (as it must) from a number of different images, pasted together with the necessary interpolations, then I think the image of the horizon will be circular. I suspect you would need the zenith angle to change from point to point to construct an elliptical image in your mind.

We have binocular vision to take into account as well. That might also counteract the distortions associated with projecting an image onto a single plane

I don't know enough about how a regular camera works to know whether they are designed to eliminate this distortion. But, I can see how a wide-angle lens could give a global distortion. Perhaps that's why panoramic photographs never look quite right.
 
  • #62
PeroK said:
It's more like what is projected onto a circular (or spherical) surface.
Yes, that is what I started by saying:
Orodruin said:
I don't think a conic section is a good way of describing vision in general. Particularly not if you have the possibility to turn your head around and see the horizon in different directions to create an impression of the view. Instead, the projection onto the unit sphere of viewing directions comes to mind.
I then went on to discuss observed curvature based on the geometry on that sphere, where at ground level the horizon is a great circle and therefore a geodesic.

PeroK said:
The diagrams above are misleading, as they suggest you could look at a significant portion of the horizon simultaneously. What you see is not what is projected onto a single plane some distance in front of your eyes.
Not as misleading as you might think. We are used to keeping our heads relatively still and look at different parts of a computer screen for example. This would correspond to the curvature on such a "screen" if you put an empty frame in front of the horizon. But as I said from the beginning, the projection on the sphere of viewing directions seems more reasonable to me.

PeroK said:
They also suggest that you would see a significant portion of the horizon as an ellipse. Which is not the case. It would be at most a tiny segment (perhaps a degree or so?).
That would depend on the distance. Also, as a corollary, if you want to make a space computer game where you actually see planets as circular when drawing them at the edges of the screen, you must draw them as ellipses. I used to play a lot of Elite: Dangerous in VR, where the viewing distance to the screens in the headset is fixed. As I think of it, this must have been taken into account to avoid visual distortions. (Another hint that OP's findings aren't really publishable I suppose ...)

PeroK said:
We have binocular vision to take into account as well.
Considering the distances involved here, I don't think the ##\mathcal O(0.1)## m distance between our eyes will play much of a role.
 
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  • #63
ScarBest said:
I meant the curve that would be visible to the observer. At sea level, the horizon appears to be a straight line. From the International Space Station, there is an obvious curve visible. My paper shows how to calculate and graph the curve that would be visible to an observer at a given altitude.
This brings up questions such as theoretical field of view vs. Environmental field of view. There are many optical effects that can limit the measurements of such a study. If this is a calculated correlation I'm sure any geological study on the curvature of the earth would gladly except your findings as a theorem. If it's more related to field observations and their views related to environmental factors, meteorologists, military and aeronautical groups would have interests.
 
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  • #64
bdrobin519 said:
This brings up questions such as theoretical field of view vs. Environmental field of view. There are many optical effects that can limit the measurements of such a study. If this is a calculated correlation I'm sure any geological study on the curvature of the earth would gladly except your findings as a theorem. If it's more related to field observations and their views related to environmental factors, meteorologists, military and aeronautical groups would have interests.
I can guarantee you that any field of study that would be interested in what OP has done is going to already know about it.
 
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  • #65
Orodruin said:
Again, it depends what you mean by ”see”. It is obviously a circle on the visual sphere, but if you make the projection on a plane that has been discussed above - it is an ellipse, parabola or hyperbola on that plane.
Or a line, which is what we spend almost all of our time looking at in real life.

PeroK said:
I don't know enough about how a regular camera works to know whether they are designed to eliminate this distortion. But, I can see how a wide-angle lens could give a global distortion. Perhaps that's why panoramic photographs never look quite right.
Orodruin said:
I can guarantee you that any field of study that would be interested in what OP has done is going to already know about it....

VR...

It's certainly a known issue which lens-makers/photographers/astronomers/VR game developers work hard to fix (to make the image flat). I think the main issue with panoramas is that they can be too wide of an angle to make a flat projection. Once you get near 180 degrees wide you can't accurately capture it on a flat image and the wider you go the worse the distortion gets on the top and bottom in particular.

Combining these two points: When taking a picture of the horizon and trying to observe the curvature the horizon has to be centered vertically in the camera frame. That's the only place (line) the image is guaranteed to be flat(er; vertical line too, but...), and a curve can be accurately observed. In other words, if you are at the surface of the Earth but tilt your camera down you'll probably get a curve instead of a straight line.

https://www.iphotography.com/blog/what-is-lens-barrel-distortion/
 
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  • #66
Just to roll a little grenade into this thread for funzies....

My first thought at reading the OP was that it could be flat-earth thread. Because we spend almost all our lives too close to the surface of the Earth to see the curvature, which means we have almost no evidence in everyday experience that the Earth is round. When your horizon distance is close to zero the actual shape of the horizon could be anything. The Earth could be a flattened turtle a mile wide and from my deck I'd have no idea because I can't see a horizon further than 200 yards from there.
 
  • #67
russ_watters said:
Or a line, which is what we spend almost all of our time looking at in real life.
On the sphere that is also a circle of radius equal to ##1/4## of the sphere’s circumference (which is also its own circumference. Such circles, the geodesics of the sphere, are called great circles.
 
  • #68
Orodruin said:
It is obviously a circle on the visual sphere, but if you make the projection on a plane that has been discussed above - it is an ellipse, parabola or hyperbola on that plane.
Orodruin said:
On the sphere that is also a circle of radius equal to ##1/4## of the sphere’s circumference (which is also its own circumference. Such circles, the geodesics of the sphere, are called great circles.
I do not follow.
Great circles are the intersection, of the surface of a sphere, with a plane that passes through the centre of the sphere.

If you swim in a calm sea, and look with one eye at sea level, you can see about 100 m to the horizon, assuming the pupil of your eye is about 1 mm diameter.
That horizon would look like a circle, edge-on, which would make it a straight line.
 
  • #69
Baluncore said:
I do not follow.
Great circles are the intersection, of the surface of a sphere, with a plane that passes through the centre of the sphere.

If you swim in a calm sea, and look with one eye at sea level, you can see about 100 m to the horizon, assuming the pupil of your eye is about 1 mm diameter.
That horizon would look like a circle, edge-on, which would make it a straight line.
It is a great circle on the visual sphere, i.e., the directions in which you see projected on a unit sphere with the observer in the center.

If you are at sea level then the plane of the horizon goes through the center of the visual sphere and thus its projection on said sphere is … a great circle.

Great circles are the geodesics (colloquially, straight lines) on the sphere. But they are still circles on the sphere. They have a well defined finite radius.
 
  • #70
Orodruin said:
It is a great circle on the visual sphere, i.e., the directions in which you see projected on a unit sphere with the observer in the center.

If you are at sea level then the plane of the horizon goes through the center of the visual sphere and thus its projection on said sphere is … a great circle.
Sure, our vision is best mapped to a spherical projection. That's why planetarium an imax screens are spheres. But im talking about the shape of the stuff we are looking at.

Orodruin said:
On the sphere that is also a circle of radius equal to ##1/4## of the sphere’s circumference (which is also its own circumference. Such circles, the geodesics of the sphere, are called great circles.
My point was that we know it's a circle because we've seen the curve(from higher elevation), but whe you can't see the curve you don't have any evidence that tells you it's a circle. All you see is the horizon line.
 
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