GPS needs 4.
It is solving a 4D problem, not a 3D problem. Time is also a variable.
Do I have this right?
Satellites have (multiple) atomic clocks, which are accurate, but receivers don’t. If we used properly set atomic clocks on receivers we could get locations just on the intersections of spheres of the transmission as follows:
2 satellites in line of sight gives 4 possible locations, 2 above and 2 below the satellites. So 2 possible locations (assuming you aren’t in space).
3 satellites gives 2 possible locations, 1 above and 1 below. So really only one possible location.
However, since the receiver clock is not accurate, a 4th satellite is needed to remove clock bias. And presumably additional satellites increase accuracy.
What I’m not sure about is whether 2 satellites to give position, plus 1 to remove clock bias would work, assuming you knew what country you were in? Or whether 4 is actually the minimum to remove bias?
It seems to me that removing timing error is probably more important than removing a hypothetical location thousands of miles from where you really are.
Almost.
Assuming you have the atomic clock, then 2 satellites will produce the intersection of 2 spherical shells (i.e. the surface of a sphere).
This gives an entire ring of locations.
You can get an approximate 2D fix based upon you being on the surface of Earth which would give 2 points.
Assuming the satellites are far enough away from directly overhead and you are close to the surface, the position on Earth would be quite accurate.
However the height would be completely unknown. You could be standing on the surface or in a plane at 35 000 ft.
Likewise, if you just had one, you would have the entire spherical shell which could be simplified to a ring on Earth's surface.
But then yes 3 satellites would then give 2 locations, one point below the satellite, one point off in space, allowing a 3D fix.
With the extra satellite for a clock correction, you are better off than the corresponding case without the extra satellite, the easiest way is to treat it like the previous case and think about what happens as the time changes.
With just 1 satellite, you have nothing, you must exist somewhere in space, presumably on Earth.
With 2 satellites, you first treat it as the 2 satellites. This technically gives you a ring, which is simplified to 2 points. But allowing it to span time will give you a kind of bowl shape, which would be a line on Earth (arguable better or just as bad).
But with 3 satellites, you first treat it as 3, giving you a point (as the one off in space is discarded). You then let the clock play out until that point hits Earth. That gives you a 2D fix.
Hypothetically you do still get 2 points, one on each side of Earth. The one on the far side of Earth can be discarded by noting that the satellites wouldn't be visible there.
So it can actually be used to get a 2D fix. But again, that is based upon you being close to Earth's surface and the satellite not being close to overhead.
Your altitude would still be unknown as a small variation in time would move your point off Earth's surface (or below it).
In order to get a 3D fix, without clock bias, you need 4 satellites.
To get a 2D fix, you can think of it as treating the centre of Earth as a transmitter with a known distance to it.