What is the Rate of Perceived Horizon Curvature Change with Altitude?

[PeroK]

Joining a little late but it seems to me the OP is talking about how an observer would see a segment of the horizon in a limited view - as if projected on a plane perpendicular to his line-of-sight. It would still be a circle but, being a different field-of view, it seems the apparent size would be different. (But I may be talking out of my hat. Dammit Jim! I'm a photographer, not a geometer).

Wide-angle shots versus human-eye-esque fields-of-view tend to greatly distort the proportions of objects.
View attachment 356738

In the upper diagram, which you've marked with an X, the red line represents the horizon. In the second diagram the red line does not represent the horizon. The red line below the apex would be hidden behind the horizon. The horizon must be symmetrical in all directions. If it's an ellipse, then what defines the major and minor axes?

 

[Baluncore]

When the observer is looking at the central point of the horizon (from his point of view), he will be viewing the horizon circle at an inclined angle, so the circular arc of the horizon will be seen as an elliptical arc, due to foreshortening of the horizon circle arc in the direction in which the observer is looking.

I do not believe that. The observer is at the apex of a cone, the horizon is tangent with the sphere. The distance to the horizon, from the observer, is the same in any direction. The horizon will always be part of a circle, never an ellipse. To get an ellipse, you must cut a diagonal plane through a cone, but the horizon is a cut perpendicular to the axis, so must remain a circle, when viewed from the apex.

 

[DaveC426913]

In the upper diagram, which you've marked with an X, the red line represents the horizon. In the second diagram the red line does not represent the horizon.

Diagram not-to-scale.

The red line below the apex would be hidden behind the horizon. The horizon must be symmetrical in all directions. If it's an ellipse, then what defines the major and minor axes?

The point I am trying to draw attention to is the axis of the observer's line of sight. (The centre, dotted red arrow), and the observed angle of the arc of the horizon is perpendicular to that.

A camera would show this by projecting the curve onto its focal plane, which is perpendicular. Closing one eye would also achieve the same effect - by removing any clues about distance and foreshortening. @ScarBest: This is why I think your yellow line annotation is incorrect. But I may be misinterpreting you.


In diagram A, the observer's line of sight coincides with the centre of the circle - the horizon equidistant in all directions. And the viewing angle of the horizon is highly acute (magenta) not perpendicular.

In diagram B, it lines up with the horizon's near edge, and is perpendicular to it (magenta). Only a portion of the horizon is visible, most of it being off the bottom of the observer's field-of-view.

 

[PeroK]

Diagram not-to-scale.


The point I am trying to draw attention to is the axis of the observer's line of sight. (The centre, dotted red arrow), and the observed angle of the arc of the horizon is perpendicular to that.

A camera would show this by projecting the curve onto its focal plane, which is perpendicular. Closing one eye would also achieve the same effect - by removing any clues about distance and foreshortening. @ScarBest: This is why I think your yellow line annotation is incorrect. But I may be misinterpreting you.


In diagram A, the observer's line of sight coincides with the centre of the circle - the horizon equidistant in all directions. And the viewing angle of the horizon is highly acute (magenta) not perpendicular.

In diagram B, it lines up with the horizon's near edge, and is perpendicular to it (magenta). Only a portion of the horizon is visible, most of it being off the bottom of the observer's field-of-view.




View attachment 356757

Unless you get into the optics of an eye or camera lens, the horizon is a horizontal cut of a cone and is hence a circle. Even projected to an intermediate horizontal plane, you get a circle.

The right angle in the second diagram makes no sense to me in terms of points on the horizon.

 

[DaveC426913]

Unless you get into the optics of an eye or camera lens,

Exactly so. The OP mentions at least once or post that he is talking about what the observer sees visually.

If the observer looked out the window of his craft, with the horizon vertically centred, and the window perpendicular to his line-of-sight, he could draw an arc right on the window with a sharpie.

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).

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.

The right angle in the second diagram makes no sense to me in terms of points on the horizon.

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.

 

[PeroK]

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.

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.

 

[DaveC426913]

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.

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?

 

[PeroK]

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.

Sorry, this is still wrong:

https://physics.stackexchange.com/q...izon-visually-appear-curved-at-high-altitudes

 

[DaveC426913]

Sorry, this is still wrong:

https://physics.stackexchange.com/q...izon-visually-appear-curved-at-high-altitudes

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.

 

[PeroK]

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.

 

[DaveC426913]

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.

 

[DaveC426913]

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.

 

[Baluncore]

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.

 

[PeroK]

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.

 

[DaveC426913]

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.

 

[DaveC426913]

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.

 

[PeroK]

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.

 

[Baluncore]

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.

 

[russ_watters]

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.

 

[DaveC426913]

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.

 

[russ_watters]

BTW, here's a journal article on the subject @ScarBest should probably get ahold of:

https://opg.optica.org/ao/viewmedia.cfm?uri=ao-47-34-H39&seq=0

And an animation of it, using a piece of software called "Space Engine":

 

[DaveC426913]

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:

• 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.

 

[russ_watters]

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...?

 

[Baluncore]

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.

 

[DaveC426913]

OK, I know when I've been beat. Thanks for humoring me.

 

[Orodruin]

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.

 

[Orodruin]

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 $h\to \infty$.

 

[Orodruin]

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.

 

[PeroK]

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!

 

[Orodruin]

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.