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Comment by iliis | Hacker News Reader

1) The intersection of three spheres results in two possible points. One of which is most likely somewhere in space or deep below ground. So 3 satellites are actually enough in theory. Altough you get a [much] higher precision with more satellites.

I agree on the other points. They all have an inclination of 55° in regard to the equator. See Wikipedia [1] for details and a more serious illustration

[1] http://en.wikipedia.org/wiki/Gps#Space_segment

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You need four, not three.

You would need three if you had an atomic clock: if you knew the precise time. But in practice you do not. You have a good clock that has an unknown shift with respect to the clocks in the satellites.

So you can measure differences in times, but not absolute times. I.e. you do not know the distance from the gps to the satellites, you know the difference of these distances. So you need a fourth satellite.

This was designed because it is easy to make accurate relative time measurements, but hard to have an absolute time reference.

I know for a fact that some aircraft are able to operate on three by using altimeter information to resolve the ambiguity between the two possible points. It's not as accurate as using four satellites, but it's still pretty damn accurate. In a boat you could get similar results by assuming that you are at MSL.

When you solve 3 simultaneous equations in 4 unknowns there are far more than 2 possible solutions. However, you are correct that if you have a priori information about one of the unknowns (usually the altitude, since it's easier than carrying your own atomic clock) you can solve for the remaining 3.

You can get around that by starting with a few reasonable amputations. aka speed less than 2,000 MPH Acceleration under 3g within 40miles of surface etc. And then refining your estimated local time based on the reasonable solutions and then repeating this a few times with several data points to get a better estimate for location, velocity, clock drift, and time. This tends to works quickly because your location and velocity are fairly constrained and the satellites are moving very quickly. As for a ground station, the velocity is vary constrained and which makes all of this a lot faster and more accurate.

Don’t forget we have a lot more CPU to throw at these problems than they did in 1980.

Hmm, I never heard that, but thinking about it, it makes a lot of sense. Thanks :)

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