Curvature of the Horizon

[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.

 

[PeroK]

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.

 

[Orodruin]

It's more like what is projected onto a circular (or spherical) surface.

Yes, that is what I started by saying:

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.

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.

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 ...)

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.

 

[bdrobin519]

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.

 

[Orodruin]

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.

 

[russ_watters]

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.

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.

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/

 

[russ_watters]

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.

 

[Orodruin]

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.

 

[Baluncore]

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.

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.

 

[Orodruin]

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.

 

[russ_watters]

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.

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.

 

[Orodruin]

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.

But you cannot see that shape, you can only see their projection on the visual sphere. It is all you see. The horizon on ground level is a straight line on that sphere, but it is also a circle - as defined in spherical geometry.

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.

We know that it is a circle on the sphere because the sphere is what we see. We can point out its center on the visual sphere (a point on the sphere that is equidistant - again, on the sphere - to all points on the horizon) - or rather, two centers as all circles on the sphere has. You are talking about the 3D shape of what is seen, but that is not something we actually observe (as long as we are talking only about monocular vision). I am talking about the 2-dimensional shape on the unit sphere, which is a curved manifold. This shape has a well defined extrindic curvature as seen as a subset of the sphere, which is zero for any great circle.

 

[russ_watters]

But you cannot see that shape["the stuff we are looking at"], you can only see their projection on the visual sphere. It is all you see.

You can see or discern certain real information about that actual shape if you know how to interpret the projection correctly and use a camera or your eye correctly.

We know that it is a circle on the sphere because the sphere is what we see.

That's a statement about our visual sphere, not a statement about the horizon/the objects we see. The visual sphere is a sphere, of course, and it seems pointless to me to describe what we see in terms of that geometry. It just isn't what we want to know - It's only an intermediate step between our eyes and the real world. It's part of the way real objects get translated into a 2D shape traced on the back of our retinas, but it's neither the original thing nor the image. Yes, we need to understand it, but it isn't the real world object nor the image that we see - it is only a geometry on which the things we see are projected temporarily.

You are talking about the 3D shape of what is seen.

Mostly, yes. Just to organize it, there are several different "things" being referred to in the thread as what we "see":

1. The real 3D shape of an object in 3D space. We can't literally see it, only discern it from visual and other information.
2. The 2D projection of an object onto the visual sphere. It's what we are looking at, but isn't quite/necessarily what we see.
3. The captured image of the visual sphere by a camera or eye. That's the only thing we literally see.

Obviously, the only thing we actually "see" is the photons that hit the back of our retina and are interpreted by our brain as an image (#3). Or a capture by a camera. We may or may not be able to discern the real world/3d shape (#1) from the information we get in that image or other information we have. #2 is always a sphere, so that on it's own doesn't say anything useful.

Here are two photos of a line (the same line):

I know it's a line because I drew it with a straight edge. That's thing #1. Real life. When held orthogonal to the eye it is a line projected on our visual sphere - so a segment of a circle - (#2) and appears in our vision (#3) as a line, as it does in the top photo. It's an accurate capture of the real object (which in this case is one/two dimensional to begin with). In the second photo it appears as a segment of a circle, because it was captured at the edge of a wide-angle camera(same camera/lens), which introduces a distortion. Both photos represent thing #3 - a captured image of the real world. But one captured it correctly and the other did not. If all you had was the second photo with no other context, you'd have no idea if you were looking at a line or part of a circle.

Now imagine I hold a flat object and look at it edge-on or trace a projection of it on the piece of paper next to my line. What is that object's actual shape in 3D space(#1)? Is it a circle? A flattened turtle? You have no way of knowing the actual shape of this object because I haven't told you and all you see in the image is a line. But if you use your camera correctly, you will see one true thing about it: it is flat. You will see it as a line. That is an accurate(albeit limited) piece of information about the object's shape in 3d space. If you use your camera incorrectly you will see it as a curve.

The horizon is flat when viewed from the surface of the Earth because it is a 2d circle being viewed edge-on, which is why we see it as a line with our eyes and in photos if we use a camera correctly. That is a true (albeit limited) statement about its 3D shape(#1) -- the only thing we know for sure when looking at it from the surface of the Earth. It is not a statement about our visual system (#3) or about the visual sphere (#2).

A note on the eye: I haven't studied it in detail, but it seems to me that our optics+brain produce surprisingly flat images across a near 180 degree field of view. In other words, no matter which direction you look (with your peripheral vision) you will see lines as lines and not curves. You don't have to center-up an object to avoid the barrel/fisheye distortion. That's probably because unlike a camera imaging sensor our retina is spherical.

 

[Orodruin]

The eye and brain is far too complex to get a full and proper description of how the brain interprets the image. Regardless of how things turn out on a photo, I believe that the brain/eye system will work closer to the visual sphere than photography, simply because the brain has adapted to living inside a visual sphere.

The geometry of the visual sphere is also relatively simple - something like a great circle will trace as a straight line on it and so that’s what the brain will interpret. Remember that the focus of the eye is also comparatively very small. To get a good image of the surroundings, the eye will trace out relevant features by moving the focus around and the brain will use the information gained to create the full image.

 

[OmCheeto]

I must say, that I very much like this thread, as it has brought up many topics, and is worthy of being broken up into subthreads.

That being said, I've noticed two puzzles, one of which I've solved.

The solved thing being how to test if something is a 'visually' a circle or not.
Russ mentioned earlier that his software couldn't capture what he was looking for, so I used 'Google Earth Pro' and captured the image of the horizon over the Pacific Ocean @ 4.33 km altitude, digitized it, and found a very strange image.

The blue line did NOT look circular at all, so I did some fancy maths using three points, and created a data set(red) for a real circle using the original x coordinates and found that circles do look like parabolas when the x and y axes are stretched in relation to each other.
But anyways, this gave me confidence that Googles images of the horizon are legit.

My 2nd puzzle goes way back to when Baluncore mentioned that 'curvature' is the reciprocal of the radius.
Although I kind of now understand what B was saying, it still struck me as ludicrous that the property of an object changes with the distance from which you are observing it. Are there any other properties like this?

The observer is at the apex of the tangent cone surface. The radius of the horizon is the half-angle of the cone, measured by the observer, as the angle from the axis of the cone to the tangent horizon. The curvature of the horizon is (1 / radius angle), specified in units of inverse angle.

ps. This still strikes me as how the progeny of a lawyer and a mathematician would communicate, and hence, I do not retract my giggle.

 

[Baluncore]

ps. This still strikes me as how the progeny of a lawyer and a mathematician would communicate, and hence, I do not retract my giggle.

Someone had to say it, to mark one extent of the issue.

I find the unit of "horizon curvature", the Naidar, to be quite amusing, as it is the reciprocal of the angle, observed between the nadir and the horizon.

 

[ScarBest]

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.

Your post helped me a lot. I got me thinking quite a bit. I couldn’t shake the idea that there must be some foreshortening, yet I couldn’t see how one would see an ellipse. For a while, I thought you were right. I thought that my paper might be taking the long way around to describing a circular arc. Eventually, I realized that we were both wrong.

The thing is, we don’t see the world the way we describe it with geometry. For example, we see straight lines as curves. If you doubt this, imagine standing beside a tall fence which stretches far to either side. When you look at the part of the fence nearest to you, you see the top of the fence with a horizontal slope. But in your peripheral vision to the sides, you see the slope gradually changing downward. Clearly, you must be viewing the straight line as a curve. Our minds tell us that the edges of buildings are straight, but they actually appear as curves. For more on this, look up Helmholtz’s chessboard.

One of the attached images shows the plan view of a playing field which is marked with a semicircular arc and equally sized rectangles. The other image is a perspective view which would be seen by someone standing at the focus of the circular arc (and thus their eyes would be at the apex of a cone). You can see that in the perspective view, the rectangles gradually get more foreshortened the further you look down the field. And of course, the circular arc is being foreshortened in a similar manner. Since the foreshortening is variable, the resulting curve will be neither circular nor elliptical.

This does not affect my paper. My algorithm does not depend on the shape being an ellipse – that was just my intuition. My algorithm selects points on the horizon circle, calculates their Cartesian coordinates, and then, through a series of coordinate system transformations, shows where the point would appear in a spherical coordinate system with the observer at the origin. Thus, point by point, it constructs the curve that the observer would see.

 

[ScarBest]

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 your diagram A, if you rotate the eye so that it is looking down the edge of the cone, you can see that the line of sight would not be perpendicular to the plane of the horizon circle. That's why I claim that the horizon circle would be where I marked it in yellow.

 

[ScarBest]

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":

Thank you. I had already read the Lynch article. He shows how you can detect curvature in a photo taken from 35,000 feet, and he says that he couldn't see a curve visually from a mountaintop, and that the height at which you can see curvature must be somewhere in between. He doesn't construct curves, as my paper does.

The Scott Manley video, however, blows me out of the water. Thank you for pointing that out to me.

 

[Baluncore]

I thought that my paper might be taking the long way around to describing a circular arc. Eventually, I realized that we were both wrong.

Only you are wrong. You are taking the long way to describe a circular arc.

One of the attached images shows the plan view of a playing field which is marked with a semicircular arc and equally sized rectangles. The other image is a perspective view which would be seen by someone standing at the focus of the circular arc (and thus their eyes would be at the apex of a cone). You can see that in the perspective view, the rectangles gradually get more foreshortened the further you look down the field. And of course, the circular arc is being foreshortened in a similar manner.

That is an irrelevant coordinate transformation from planar to spherical. It does not alter the fact, that in spherical visual coordinates, the observed equator will always be a circle. By moving the line of sight, away from the original observer's position, the circle then becomes foreshortened, into an ellipse, when seen from the original observer's position. If we are in different positions, my horizon will look like an ellipse to you. For me, my horizon will always be a circle on my visual sphere.

My algorithm selects points on the horizon circle, calculates their Cartesian coordinates, and then, through a series of coordinate system transformations, shows where the point would appear in a spherical coordinate system with the observer at the origin. Thus, point by point, it constructs the curve that the observer would see.

Because your eyes see things positioned on your visual sphere, your horizon will remain a circle, no matter which point you centre your vision on. The image plane will remain spherical, the horizon will remain a circle.

Consider the azimuth and elevation of points on the horizon. Turning your head left or right will change all azimuths by the same angle. Nodding your head up or down will change all elevations by the same angle. The visible horizon will remain a circle, or a circular arc, at the same distance, independent of your line of sight.

 

[ScarBest]

Only you are wrong. You are taking the long way to describe a circular arc.


That is an irrelevant coordinate transformation from planar to spherical. It does not alter the fact, that in spherical visual coordinates, the observed equator will always be a circle. By moving the line of sight, away from the original observer's position, the circle then becomes foreshortened, into an ellipse, when seen from the original observer's position. If we are in different positions, my horizon will look like an ellipse to you. For me, my horizon will always be a circle on my visual sphere.


Because your eyes see things positioned on your visual sphere, your horizon will remain a circle, no matter which point you centre your vision on. The image plane will remain spherical, the horizon will remain a circle.

Consider the azimuth and elevation of points on the horizon. Turning your head left or right will change all azimuths by the same angle. Nodding your head up or down will change all elevations by the same angle. The visible horizon will remain a circle, or a circular arc, at the same distance, independent of your line of sight.

I claim that you continue to describe things as they are geometrically, rather than how we see them visually. Do you also claim that we see straight lines as straight? We don't. Why should circular arcs necessarily appear circular when straight lines appear curved? To test this, I introduced a straight line into my model. I transformed several points along the line, and from the point of view of the observer, my calculations produced exactly the sort of curve that you would expect the observer to see.

 

[Baluncore]

Why should circular arcs necessarily appear circular when straight lines appear curved?

Because the equator is special. It is everywhere at the same distance from the observer, no matter what our line of sight. It is always mapped as a circle onto our visual sphere.

To test this, I introduced a straight line into my model. I transformed several points along the line, and from the point of view of the observer, my calculations produced exactly the sort of curve that you would expect the observer to see.

How do you know what I would expect the observer to see?

You have not shown the equations, nor the code you use, so we have no way of checking if it is applicable to projection of the equator onto the visual sphere of the observer.

As an aside, there is a special shape-preserving projection from a sphere to a flat plane. I have used the Wulff Net for analysis in structural geology.
https://en.wikipedia.org/wiki/Stereographic_projection#Planetary_science

 

[ScarBest]

Because the equator is special. It is everywhere at the same distance from the observer, no matter what our line of sight. It is always mapped as a circle onto our visual sphere.

If you one stands at the focus of a large horizontally-oriented semicircle, the arc will appear to be very wide horizontally but not very wide vertically - a wide, shallow arc. Yet, the slope of the curve will be horizontal directly ahead of you and vertical at 90 degrees to either side. A circular arc can not do this.

 

[Baluncore]

If you one stands at the focus of a large horizontally-oriented semicircle, ...

Is the observer above a focus of the circle, or above the centre of the circle?
The focal length of a circular arc, is half the radius.
https://en.wikipedia.org/wiki/Focal_length#General_optical_systems

A circular arc can not do this.

I disagree. If you draw a circle on a vertical wall, with its centre at the height of your eyes, the top and bottom arcs are closer to horizontal, while the side arcs of the circle are closer to vertical. But you always view your horizon circle from a point above the centre, and your horizon circle lies entirely on one horizontal plane, being defined by the tangent of the cone with its apex at your eye, and the spherical Earth. You cannot map that horizon plane onto another plane, because your eye must always remain at the apex of the tangent cone.

The image of the horizon circle falls on the retina of your eye, which is a spherical surface. No matter where it falls on the retina, it will remain a circle in spherical coordinates.

There are a couple of astronomical analogies.
The full Moon looks round, even when it is away from the centre of your vision.
When you look at a constellation of stars, the pattern can be specified by the angular separation between pairs of stars. The pattern looks the same, no matter where it falls in your vision, or how it is rotated.
There is no foreshortening of the Moon, or of a constellation, nor of the circle of your horizon.

There is no requirement that you look along the axis of the tangent cone. You can kid yourself that your horizon circle could become an ellipse, but you would need to cheat, by removing your eye from the apex of the tangent cone to do that. If you move your eye from the apex of the tangent cone, the original line ceases to be your horizon.

 

[ScarBest]

Is the observer above a focus of the circle, or above the centre of the circle?
The focal length of a circular arc, is half the radius.
https://en.wikipedia.org/wiki/Focal_length#General_optical_systems

I'm not talking about a lens, I'm talking about a circle, for which the focus is at the center. The observer is at a short distance above the plane of the circle, directly above the center.

I disagree. If you draw a circle on a vertical wall, with its centre at the height of your eyes, the top and bottom arcs are closer to horizontal, while the side arcs of the circle are closer to vertical.

Yes, of course, if you draw the circle on a vertical wall. This is the ONLY case in which a circle will fit. But if the semicircle is on a horizontal surface, and you view it from a short distance above the center, the curve you see will be 180 degrees from side to side, and less than 90 degrees from bottom to the curve, with the slope of the curve horizontal in front of you and vertical to the sides. A curve which APPEARS circular can not do this.

The image of the horizon circle falls on the retina of your eye, which is a spherical surface. No matter where it falls on the retina, it will remain a circle in spherical coordinates.

Not if the shape of the curve entering your eye is not circular, due to foreshortening.

The full Moon looks round, even when it is away from the centre of your vision.

Well, it seems to, but then, straight lines seem straight, even though they're curved.

When you look at a constellation of stars, the pattern can be specified by the angular separation between pairs of stars.

Exactly. Constellations are purely angular. They don't occur on a plane.

There is no foreshortening of the Moon, or of a constellation

The moon is so far away that the angle of your line of sight to the plane of the horizon circle is almost perpendicular when you're looking at the limb (which is the horizon). Any foreshortening is negligible. It makes no sense to talk of foreshortening of the constellations.

There is no requirement that you look along the axis of the tangent cone. You can kid yourself that your horizon circle could become an ellipse, but you would need to cheat, by removing your eye from the apex of the tangent cone to do that. If you move your eye from the apex of the tangent cone, the original line ceases to be your horizon.

As I said, I was wrong about it being an ellipse. It is a distorted image which is neither a circular nor an elliptical arc, due to variable foreshortening, which is due to the degree of foreshortening being a function of the angle that the line of sight makes to the plane of the ground. One does not need to move from the apex of the cone for this.

 

[Baluncore]

But if the semicircle is on a horizontal surface, and you view it from a short distance above the center, the curve you see will be 180 degrees from side to side, and less than 90 degrees from bottom to the curve, with the slope of the curve horizontal in front of you and vertical to the sides. A curve which APPEARS circular can not do this.

Why can it not do that? When the circle of your horizon crosses directly in front of you, then it must also sweep around behind your head. Where the image falls on your retina, you can see it remains a circular arc. The part behind you is not being imaged on your retina, so you cannot see that it also remains a part of the same circle, unless you turn your head.

As I mentioned a few posts back, the way the brain maps the image from the retina is stereographic, so it is shape preserving. A circular arc will remain circular, whichever way you look at it.

 

[ShadowKraz]

Ok. Let me see if I have this straight in my brainthingy.
I stand on a planet the size of the Earth but it is featureless aside from an elevator stretching from the surface to a geosynchronous station in orbit (Thank you, Arthur C. Clarke). When I stand in the elevator car (transparent for our purposes) at ground level and scan the horizon (where land meets atmosphere), I perceive it as a flat line in any direction I look. I press the up button (because down would be pointless...) and watch where land meets atmosphere as I travel up to the station. As the elevator car rises, the line between surface and atmosphere as I perceive it begins to appear curved; the higher I go, the more curved I perceive it to be. So... at what rate does that perceived curve change and what calculation(s) do I need to measure that perceived curve?

 

[Baluncore]

Your elevator car is a transparent sphere, with your head at the centre. The sphere is marked with circles of negative latitude, from the Equator, to the south-pole at the bottom. As the elevator car rises, the Earth horizon falls from the equator of your sphere, rapidly at first, then more slowly towards the south-pole of the sphere. The angular depression of the horizon will vary from 0° towards 90°. At each point in time, if you look straight downwards, you will see the horizon as a circle about the south-pole of your car.
How do you wish to perceive that circle.

 

[ShadowKraz]

Your elevator car is a transparent sphere, with your head at the centre. The sphere is marked with circles of negative latitude, from the Equator, to the south-pole at the bottom. As the elevator car rises, the Earth horizon falls from the equator of your sphere, rapidly at first, then more slowly towards the south-pole of the sphere. The angular depression of the horizon will vary from 0° towards 90°. At each point in time, if you look straight downwards, you will see the horizon as a circle about the south-pole of your car.
How do you wish to perceive that circle.

Sorry, why am I looking straight down? The OP's question, and mine, is about looking at the horizon and noting the perceived change of its curvature as one rises, not about looking straight down.

 

[Baluncore]

Sorry, why am I looking straight down?

You are looking straight down because you need to know that you will always be looking at a true circle on a spherical projection. The curvature you perceive will depend on how you choose to perceive that circle, projected onto your retina. The circle's curvature will be measured by its angular radius, measured from the south-pole.

Since your eye is spherical, and the circular Earth's horizon is plotted on a sphere, you will always see a circle, or part thereof, never an ellipse. The ellipse will only appear if you map your retina to a flat rectangle in your imagination.

 

[ShadowKraz]

You are looking straight down because you need to know that you will always be looking at a true circle on a spherical projection. The curvature you perceive will depend on how you choose to perceive that circle, projected onto your retina. The circle's curvature will be measured by its angular radius, measured from the south-pole.

Since your eye is spherical, and the circular Earth's horizon is plotted on a sphere, you will always see a circle, or part thereof, never an ellipse. The ellipse will only appear if you map your retina to a flat rectangle in your imagination.

I already know that I will be looking at a circle or section thereof, that's inherent and established already, as well as how the eye works. Don't know why you're bringing an ellipse into this; I never mentioned one.
But thank you for your reply.

 

[Baluncore]

As the elevator car rises, the line between surface and atmosphere as I perceive it begins to appear curved; the higher I go, the more curved I perceive it to be. So... at what rate does that perceived curve change and what calculation(s) do I need to measure that perceived curve?

You are asking for the curvature, which is usually defined as the reciprocal of the radius.

The radius of the horizon circle, measured as an angle, y, away from the south-pole of the projection, will be: y = arcsin( R / ( R + x ) ); where R is the radius of the Earth and x is your height above the Earth's surface.

You can solve that for the rate of change by finding the derivative y' .