Could DF measure the FoV through the 2D image output of a 3D game?
It can be more or less difficult depending on the imagery and the camera controls (FPS games are typically the easiest since camera movement and camera rotation are free and decoupled), but yes, there are ways to do it.
Yeah, it's just a matter of rotating the camera.
Sort of. This approach is fine in FPS games, but it's imprecise when rotating the camera causes the camera to move (like in third-person shooters) (unless the on-screen objects you're using for reference are very far away or at infinity, so that the impact of the motion is negligible).
Also, if you're using the method of counting how many screens it takes to make a full rotation and then dividing 360 by that number, it's not quite precise if you don't take the warping of perspective projection into account. That won't matter if the number of screens is an integer*, which causes the warping to be evenly accounted for, but otherwise the result can be a bit imprecise. I think I've worked out an iterative algorithm for finding the precise result from the "naive" count, though.
Additionally, an alternate method which ignores the warping of perspective projection can be used if you can rotate at a constant velocity: make a note of what's at the left and right sides of the screen, and divide the amount of time to rotate the middle of the screen from one of those objects to the other by the amount of time to rotate a full circle, and then multiple that result by 360.
Although, especially for FPS games with a small FoV, a somewhat sloppy count and divide can give *reasonably* precise results.
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*Actually, any number of screens that's a multiple of .5 should give a precise result, at least if the projection is centered, because the screen has horizontally symmetric angular sweep.
**Also, as a general note with FoV measurement, you usually want the camera to be facing the horizon (i.e. no vertical tilting); this decouples the vertical height of objects from their horizontal location on-screen, which tremendously simplifies the math.