I'll address the accusation against NASA presently, but let's start with that curve. Does the horizon really stay flat and level no matter how high up you go? And, on a globe Earth, what, exactly, should we expect to see?
Let's start with understanding the difference between seeing something in a photograph and seeing something with our own eyes. There are lots of differences between the way your eyes work and the way a camera works, but the one most germane to Earth's curvature are field of view.
The total field of view that the human eye can take in is, on average, about 180 horizontal degrees. You only can focus on the center two degrees or so, and you are better at detecting edges and motion in the periphery and better at seeing color near the center.
A camera, on the other hand, has a field of view that depends on the size of the sensor in the camera and the focal length of the lens that focuses light on the sensor. A typical DSLR with a 18mm lens has a field of view of around 75 degrees. Add the fact that the sensor is flat, unlike your retina, and you are going to have a certain amount of distortion in a photograph, especially at short focal lengths, which is what you need to have a wide field of view.
Keep these things in mind whenever anyone shows you a photograph of the horizon and says "It looks flat to me." Also keep it in mind when someone shows you photo with a curved horizon. Without some data to go with that horizon, you have no way to judge.
But, come on, from 1000 feet up in the air, or 20.000 feet, or 50.000 feet, every photograph of the horizon should show a curve, right? Well, maybe. Or maybe not. It's not about how high you are; it's about how much of the horizon you are able to see.
Not that altitude has nothing to do with it, but when you consider how small even miles of altitude is compared with the circumference of the Earth, the main contribution of altitude is to increase the distance to the horizon, and increase the amount of horizon you can see.
Let's take one of the flat-Earthers' current favorite pictures. This was shot from a captured V2 rocket launched by the US military in 1946:
Okay, the horizon looks pretty flat. Or maybe it looks a little curved. Frankly it's kind of hard to tell. But the flat-Earthers say that, since it was taken from an altitude of 65 miles, it should have a very distinct curve. Let's see if that's actually true. For that we need some facts, and we really only have a few.
We know 65 miles, and we know that the photos were taken with a 35mm motion picture camera, which means that the frame was 24mm across. But we don't know the focal length of the lens.
I've seen a version of this picture with overlays by flat-Earthers that claim that the width of the horizon is 720 miles, but in fact, as far as I know, there is no data to support this. The distance to the horizon is 720 miles, and I think that's where the confusion comes in.
In order for the horizon to be about that wide, the camera would have to have been equipped with a 28mm lens. That's not completely out of line, so let's go with that.
So, here I am, taking the flat-Earthers at their word, and admitting the the horizon doesn't look all that curved. Am I giving in? No, I'm just investigating. Let's take a look at how curved a 720-mile horizon would look.
What portion of the 25,000-mile circumference of the Earth is 720 miles? About 2.88 percent of the total. That's a circular section less of less than 11 degrees. So how curved should 11 degrees out of a circle look?
From a mathematical perspective, the difference between the center of the arc and the edges will be 2.75 percent of the width of the horizon. That's a small number. But can we visualize this? Well, I'm willing to give it a try.
Using an online graphing calculator, I created a circle and a 11-degree angle. I zoomed in until that section of the arc filled most of the screen, and then I cropped it down to just that portion. Here's what it looked like:
Admittedly, more curvy than the picture, but not that much, and as I said, I don't really know how much of the horizon the photo shows.
And then there's this:
Now, that looks pretty curved. Same mission, same rocket, same camera, same lens. What's going on? I can't be sure, because I don't have all the data, and unlike many flat-Earthers, I'm not willing to say that I know for certain what's happening. But I have my suspicions.
For one thing, the first picture is far grainier than the second, even though they are taken from the same roll of 35mm film. That makes me think that the first one is rather severely cropped. But it could just be the way the print was made.
The practical upshot is that the Earth is a big place. Assuming a sphere, you have to get pretty far up to see the curvature with your wide-field eyes, and really, really far up to see it with a relatively narrow-field camera lens. You can use a very wide lens, of course, but then the distortion caused by projecting the image of a spherical lens onto a flat sensor will make the curves appear in all the wrong places.
These photos, in other words, prove nothing.
Ah, but what if we don't assume a sphere? Isn't that the problem, assuming from the outset that we live on a sphere? Yes, that is a problem. For the flat-Earthers.
Because all I ever hear from the flat-Earth crowd is "the horizon looks flat." What I don't see, and what I should see if this was even remotely a serious investigation, is a model for what these photos should look like if the world were, in fact, a giant disc. I see lots of map projections, and computer models of how the sun and moon are supposed to work, but I don't see a rendering of what a camera should see from, say, 65 miles up if the Earth were a disc.
It's not enough to say "I just see a flat horizon." If you want an idea that's outside the mainstream to be considered at all, you have to provide evidence. You have to make predictions about what we should see and experience if the world was flat, not just find apparent incongruities and declare victory.
Flat-Earthers often complain that, because they've only been at this a short while, they don't have the resources to do the deep research necessary to come up with an accurate model. Aside from the fact that the current flat-Earth "movement" has progressed little since Rowbotham in the 1860s, there are abundant resources available to anyone with access to a computer for creating models, making predictions, and comparing the predictions to actual observations. That's science. Real science.
But instead I see flat-Earthers posting pictures, or commenting on nearly every aerial photograph there is, that it "still looks flat."
And that's lame. Just lame.
ADDENDUM: An excellent paper on this subject by David K. Lynch of Thule Scientific has been brought to my attention. If you're serious about this, you should give it a read and a lot of thought. You can download a PDF here.
Another ADDENDUM: I realized that I had not addressed Eric DuBay's assertion that the curve is only shown in NASA CGI images. Of course, Earth had it's first full-length picture taken in 1968 by the crew of Apollo 8, before the advent of realistic computer-generated imagery. It was shot on 70mm film with a Hasselblad camera. DuBay is making, as he so often does, an assertion for which he has no evidence. He proclaims the imagery fake because he needs it to be, not because he has any evidence of fakery.
Remember that the title of DuBay's book starts with the words "200 Proofs." There really aren't 200, and they really aren't proof, and that shows you the low standard to which he holds himself. I hope you will keep that in mind.