Okay, not really a clue about most of this so bear with me on my terminology. Is there a way to calculate the distance away from the surface of a sphere, that an observer is positioned, using only a picture of part of the sphere's apparent edge, and its known radius? The way I'm thinking about this is in terms of some object with a camera at a given altitude, x, above the earth, takes a picture of the horizon, then extrapolates the arc out to form an entire circle with diameter, a.
Is there a way to work out the value of x just from a picture?
Fig. 1: Earth from Space. Credit: University of Leicester
Would it be possible to calculate the altitude that this picture was taken from with just the information: -Radius of Earth ~ 6.4*10e+6m
I have to imagine there is a relationship because as you (one) gets further away from the earth it appears to be smaller. If we could always see 50% of the Earth this would be significantly easier as it would just require some trig, but we can't always (or ever) see 50% of the Earth just slightly less. There's still got to be some mathematical relationship there though right?
I was thinking that it could be somewhere along the lines of using the visible surface area of the earth as described by Henk001 at https://www.metabunk.org/how-to-calculate-the-visible-fraction-of-the-earth-e-g-1972-blue-marble-apollo-17.t8643/ but I can't seem to get it to work.
TIA
There is no way to do this, because a camera with a small field of view will show the earth as a large circle on the image plane; one with a large field of view will show it as a small circle. You actually need more information: you need to know about the CAMERA, and about the relationship of the image (which is maybe 480 x 480 pixels) to the physical size, like in millimeters, of the physical sensor from which it was captured.