GPS satellites are equipped with precise atomic clocks and constantly broadcast their time and location through radio waves. A receiver, such as a smartphone, collects these signals from satellites within its line of sight. By calculating the time it takes for the signal to travel to the receiver, the system computes the distance to each satellite. These distances, combined with satellite positions, allow the receiver to determine its location through a system of equations.
However, a complication arises because the receiver's clock is not as accurate as the atomic clocks on the satellites. Even a discrepancy of one millionth of a second can introduce an error of 300 meters in the location estimate. Thus, the GPS system needs to solve for both the receiver's location and the exact time, a concept rooted in relativity theory as space-time.
If too few satellites are available, the system struggles to provide reliable results, potentially yielding multiple possible locations or failing altogether. Until now, the exact number of satellites required for a unique solution had only been a matter of conjecture.
Five Satellites for Precise Location
Mireille Boutin, a professor of discrete algebra and geometry at Eindhoven University of Technology (TU/e), and Gregor Kemper, a professor of algorithmic algebra at the Technical University of Munich (TUM), have now mathematically proven that five satellites are sufficient to determine the exact position of the receiver in nearly all cases. "Although this was a long-standing conjecture, nobody had managed to find a proof. And it was far from simple: We worked on the problem for over a year before we got there," said Gregor Kemper.
At present, every location on Earth is within sight of at least four satellites at all times. "Roughly speaking, with only four satellites, the probability of having a unique solution to the GPS problem appears to be 50 percent. Proving that statement is one of our next projects," added Kemper. When only three or fewer satellites are in view, GPS navigation fails entirely.
Geometry and Uniqueness
The researchers approached the GPS challenge geometrically, discovering that a receiver's position is not unique if the satellites are positioned on a hyperboloid of revolution of two sheets-a theoretical surface that is open in all directions. While this finding is theoretical, it has practical implications for improving our understanding of location inaccuracies.
Research Report:Global Positioning: The Uniqueness Question and a New Solution Method
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Technical University of Munich
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