Why does GPS positioning require four satellites?
I have a question on the GPS positioning algorithm. In all books I've read for 3D positioning we need four satellites, and I don't understand why.
We need to calculate three variables: x, y, z. We know when satellite send the signal to earth and when we receive it we can measure the time the signal travel to earth by checking the shift in PRN generator. For what purpose do we need four satellite?
Just a graphic to add to M'vy's answer.
a high-tech version of triangulation,called trilateration. The first satellite locates you somewhere on a sphere (top left of Figure). The second satellite narrows your location to a circle created by the intersection of the two satellite spheres (top right). The third satellite reduces the choice to two possible points (bottom left). Finally, the forth satellite helps calculate a timing and location correction and selects one of the remaining two points as your position (bottom right).
Multilateration should not be confused with trilateration, which uses distances or absolute measurements of time-of-flight from three or more sites, or with triangulation, which uses the measurement of absolute angles. Both of these systems are also commonly used with radio navigation systems; trilateration is the basis of GPS.
+1 Nice graphic. I have an issue with the quoted text though. Trilateration is not a high-tech version of triangulation. It's a totally different beast.
technically you can leave out the 4th if you assume you at the point which is closer to 6,371 km away from the center of the earth (only works for ground bound devices)
I think this answer is technically incorrect. For GPS to work, your receiver generates the same codes the satellites generate, and it compares the one generated with the one received in order to compute the time difference, and hence distance from the satellite. For this to work you need to know TIME. (An aside, GPS time is very, very precise.) The minimum number of satellites required to work out your position is 4, because you're solving for X, Y, Z and TIME. YOu do get an ambiguous position, either on the Earth's surface or up in space, you can trivially dismiss one of these.
Agreed with @AlexLeith. Though so far this answer is still the most helpful, it is still technically wrong. If you have an accurate clock, you only need three satellites. The fourth satellite is required if you don't have an accurate clock.
This answer down plays the real reason you need a 4th satellite in favor of the bad reason you need a 4th satellite. The fourth satellite is needed for time, not picking one of two positions (one far out in space). @zaTricky and Leith are correct. As written, this answer is misleading at best.