Mathematical proof: Five satellites needed for precise navigation | Science & Technology

How does GPS work?

GPS satellites are equipped with an extremely accurate atomic clock and know their positions at all times. They continually transmit the time and their location using radio waves. A mobile phone or other navigation device receives these signals from all satellites within their line of sight. The difference between the arrival time at the local clock of the receiver and the transmission time recorded by the satellite clock corresponds to the time taken for the signal to travel from the satellite to the receiver (the “time of flight”). Since radio waves travel at the speed of light, the time of flight determines the distance covered by the signal. The satellite positions and the distances are used to calculate the position of the receiver using a system of equations.

This simplified description does not take into account the fact that the local clock in the receiver is not an atomic clock. If it is inaccurate by just one millionth of a second, the calculated position will be inaccurate by at least 300 meters. The GPS problem is the need for the phone or other navigation device to determine the precise time along with the location — known in the theory of relativity as space-time.

If too few satellites are in the line of sight, the system no longer functions reliably and delivers multiple solutions, in other words several different locations where the receiver could be. This can lead to the situation where a phone indicates an incorrect location or no location at all. Until now the number of satellites needed to obtain unique solutions to the GPS problem has only been…