Imagine I take a photo of the same scene using two cameras, one right after the other, from the exact same spot. The cameras may be different in every way possible. They could have very different lenses, image sensor sizes, and image sensor densities, as well as other properties. I then want to crop down the picture taken by the camera with the "larger" view so that the total stuff in it matches the total stuff in the picture taken by the camera with the "smaller" view - not in terms of how things look, but in terms of what each photo contains and how much of each thing, so that at the end if you ignored depth of field, exposure, focus, etc and just looked at the contents you might say they were the same photo.
I keep reading all these articles about crop-sensors and lenses and that's great and they're giving me a great feel for the causes and effects here but I can't find a guide that describes the precise math of the situation. It seems like the two big inputs here are focal length and sensor size, and those two together will determine the field of view, and is that the key value to make the crop?
In my specific case, I'm working with mobile phone cameras and I can interrogate their specs programmatically. It happens that with iPhones I am directly given a field of view value. Is that all I need for the math? Do I simply crop down the larger field of view image proportional to the smaller? e.g. If one camera has 60 degree FOV and one has 70 degree FOV, do I just crop down the larger field of view image to be 6/7 of its original height and width? This seems correct to me but it doesn't seem to be working and I'm having trouble deciding if I'm doing the cropping wrong or if I'm just going in the wrong direction or not factoring something else in.