The explanation about the spheres is slightly inaccurate. With one satellite, you won't get a circle on the earths surface, but a sphere.
GPS is not constrained to earths surface (or the oblate spheroid approximating it), luckily.
GPS is not constrained to earths surface (or the oblate spheroid approximating it), luckily.
With one satellite you get a sphere in 3D space, but if you are on a surface (like that of the Earth), that gets translated into circle.
If you are in a plane in the sky (3D space), then you get a spherical 'location fix'.
If you have lat-long and place it on a pre-canned map, then it's unlikely that most people will be either underground or in the air, and so whatever the surface elevation of the lat-long on the map is, that's most likely (>99% of the time?) where the person is.
At least from a UX/responsiveness perspective, this is probably a good way to do the math. (In the background get a more accurate 3D fix.)