At what altitude do you see the curvature of the Earth?

Depends on what you mean by ‘see the curvature of the Earth’, what people usually mean by this is actually a misnomer, they are actually seeing their horizon curve around them and not the curvature of the Earth. Of course, your horizon is a product of the curvature of the Earth, but it's technically an indirect observation.

The Earth is an oblate (slightly squished) spheroid - but for practical purposes (to within ~0.3%) you can consider it a sphere. What you see when you look at a sphere is a fraction of that sphere proportional to your height above it (h) and the radius of that sphere (R), the proportion you can see is given by (h/2/(h+R)). This boundry forms a circle which we call the horizon.

You can see the curvature of your horizon on Earth at ANY altitude - just by noting that your horizon curves around behind you. Even at hundreds of miles above the Earth you are still only seeing the curve of this horizon, it just becomes a little easier to see it as a circle as you get higher. This is not literally the same as seeing the curvature of the Earth. Doing that is more difficult because the curvature is very slight until you get great distances away (at which point the features of the Earth are very small) and the greatest curvature is always very oblique to our viewpoint along the edge and obscured by the atmosphere.

You can, however, Measure the curvature by observing mountain peaks at various distances, they appear to shrink with distance faster than perspective. You can do the same thing by measuring the angle of Polaris over the horizon - noting that your latitude is almost equal to the altitude angle of Polaris because you are going around a spheroid.

As far as visually observing the curve of your horizon with the naked eye, a good reference is: http://thulescientific.com/Lynch...

All credit here goes to David K. Lynch.

Conclusion from the paper:

In view of the agreement between the visual observations, measurements of the photographs, and the theoretical curvatures, it seems well established that the curvature of the Earth is reasonably well understood and can be measured from photographs. The threshold elevation for detecting curvature would seem to be somewhat less than 35,000 ft (10.6 km) but not as low as 14,000 ft (4.2 km). Photographically, curvature may be measurable as low as 20,000 ft (6 km).

So that is, roughly, a good answer. However, if you take great care to photograph the horizon right through the center of your lens with your camera very carefully leveled and at high resolution with a very high quality rectilinear lens you can actually view the slight curve of the horizon from much lower altitudes (even just a few hundred meters).

From just 200 meters elevation, on a 4000 pixel wide image at 94.4 degree field of view you should expect about 7 vertical pixels of rise out of the horizon circle (as shown in my calculator link). You can make this slight ‘bump’ more visible by compressing the width of the image to about 10% of the original width and stretching it vertically by a factor of 2–4x.

For example, if we do this to the image here (which also required slight rotation):

The results are (try it yourself!)

And since the horizon is below lens center we’re not seeing lens distortion, which would flatten out the horizon. To control for any lens distortion you can add a wide builders level just below the horizon and observe the resulting hump.

If you width compress the image it shows the vertical relief very clearly.

The rest is my commentary and thoughts on visual photographic analysis…

It would be difficult to tell, in general, from a simple photograph if what you are seeing is actual curvature or lens barrel (aka curvilinear distortion) or pincushion. You have to know how it was shot and cropped and what the FOV is.

If you have an uncropped image and the horizon goes through the exact center, then you can judge that photography as likely fairly accurate regardless of distortion because straight lines aren’t bent by normal lens distortion.

This next image is NOT actual curvature, this is mostly lens distortion from a wide-angle lens with horizon well above center point.

This is actual curvature:

Himawari-8 Real-time Web - NICT

Link to one of the Himawari 8 11000x11000 pixel images

As is this one from a High Altitude balloon, this lens does have curvilinear distortion but I have very carefully selected a frame where the horizon remains below lens center so the lens distortion is actually making the horizon appear flatter! At this altitude, resolution, and field of view we expect about 40 pixels of ‘bump’.

The ISS, at 249 miles (401 km) up, is well positioned to observe the curvature.

But even the ISS only has a view of 1426 miles (2295 km) to the horizon - so is only seeing a fraction of one side of the Earth - this is not the whole hemisphere.

Himwari 8 is 22,239 miles (35790 km) away by comparison so it can view a significant portion of the entire hemisphere. But even that is seeing the same circular horizon you see, just a bigger piece of it due to the distance.

Ever wonder what a 360 degree camera would see pointed down from a few thousand feet up?

that's by AirPano.

And here is an image shot from the ISS cupola where we know the exact lens used and can confirm it is a rectilinear lens both by the type and the lack of distortion of the many straight lines in the image.

And we can render the expected horizon of the Globe from this viewpoint and show that it matches the image: