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    Re: An interesting question
    From: Frank Reed
    Date: 2013 Oct 13, 11:12 -0700

    Andrew, you wrote:
    " the land is so flat that when you stand on a water tower in a town you can actually see the curvature of the Earth."

    This is always an interesting claim. Pilots in the stratosphere can see the curvature of the Earth, though it's pretty minimal up there. There are two possible geometric factors in play here that I can think of.

    Obviously, if you're ten thousand miles off the Earth, the Earth no longer looks like a plane --it's a circle in the sky. Geometrically, the Earth's disk is identical to a "small circle" on the celestial sphere for an observer at that altitude. As we drop to lower altitudes, the radius of the small circle increases until finally right at ground level, the circle has a radius of 90° and it becomes a great circle (the difference between the radius of the Earth's apparent disk and 90° is, of course, the navigator's "dip"). A great circle is visually a straight line. When you look at the Milky Way in the sky, it is a straight "stripe" across the sky, despite the fact that it often appears curved in photos. To detect this type of curvature, we would need to compare a straight line (really a great circle) with the corresponding small circle of the visible horizon. If a great circle coincides with the horizon's small circle at some point, it will progressively deviate from it in angle across the field of view. How much deviation is detectable? This we don't know, but let's suppose that my field of view is 90° wide. I could probably detect a one degree deviation from a great circle visually. That is, if the small circle of the horizon falls away from the great circle touching it at one point fast enough so that it has dropped one degree below the great circle both 45° to the right and 45° to the left of the center of my field of view, then I think it would probably be noticeable as "curved". Anything less than that would be a difficult call. How high would you have to be to get a deviation that large? I can assure you without calculating that it's higher than any "water tower" in the Dakotas and probably way up in the stratosphere. Let's see... it appears to me from a back of the envelope calculation that we need an observer altitude where the dip is about 3° which is about 32000 feet up. Above that altitude, the horizon will show slight, but detectable curvature. To see twice that curvature, you would have to fly four times higher.

    But wait, we've all seen photos from stratospheric aircraft and balloons showing considerable horizon curvature. There have been numerous amateur balloon flights in recent years with digital cameras on-board and the images look like they're way out in space. So what gives? The most obvious factor is a fisheye lens effect (with few exceptions, GoPro Hero cameras are used for these videos, and they have a substantial fisheye effect). If you watch any of the videos from those amateur balloon flights, the camera is usually spinning, and when the horizon is away from the center of the frame, it looks sharply curved. When it cuts right across the center of the frame, there's no apparent curvature. So that part is a camera illusion. The second factor is the lack of an actual horizon. At altitude we see a series of, what amounts to, concentric rings of progressively darker sky in a band many degrees deep below and above the horizon. The first visible darkening occurs at a considerable angle below the actual horizon so its curvature is visible at much lower altitudes. From my own experience starting at about 10,000 feet up this second factor creates an impression of curvature even though it's not the horizon.

    There's another way to detect the curvature of the Earth which could, in fact, be visible from the top of a water tower in the Dakotas and other flat regions. It depends on the presence of a checkerboard pattern of features. This sort of pattern certainly does exist in many part of flat farm country. Small county roads and farms are often arranged in a neatly repeating grid out there. On an actual checkerboard on a true plane, the gridlines would appear to remain parallel into the distance, and individual squares would reduce in size in direct proportion to their distance. But on the curved Earth, the lines would curve toward each other and not meet in the distance, instead disappearing over the curve of the Earth. Also the squares of the grid would decrease in size in a non-linear fashion, foreshortening as they become superimposed on the horizon "zone" (which is actually miles deep). I've never seen *exactly* this geometric condition, but I've seen something similar to it looking down a valley with a very flat bottomland many years ago (long enough that I can't remember whether it was in Greece or in Spain). This type of visual evidence of the Earth's curvature would almost never be seen at sea, though ocean whitecaps, if evenly spaced, can show a similar effect. Since the spacing isn't regular, it's nowhere near as obvious.


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