How do we find the quickest path to our next destination or meeting spot? For many, global positioning systems (GPS) have become an essential part of daily life. Yet until now, the exact number of GPS satellites required to pinpoint the location of a mobile phone or navigation device was often debated. Researchers have now shown that most of the time, five or more satellites are needed for accurate location determination. Currently, we usually only have access to four satellites.
Typically, GPS can locate us with precision to within a few meters. However, we’ve likely all faced situations where the error margin swells to hundreds of meters or the location shown is simply inaccurate. One reason for this could be limited satellite connectivity or unfavorable positions of the satellites.
How is GPS used?
GPS satellites are fitted with highly accurate atomic clocks and always know their locations. They send out signals with the time and their position via radio waves. A mobile phone or navigation device picks up these signals from all satellites within its line of sight. The difference between the time recorded by the satellite clock and the time noted by the device’s local clock indicates how long it took for the signal to reach the receiver (referred to as “time of flight”). Since radio waves travel at the speed of light, we can use this time to calculate the distance the signal traveled. The locations of the satellites and these distances help solve a system of equations to find the position of the receiver.
This basic explanation doesn’t account for the fact that the receiver’s clock isn’t as precise as an atomic clock. If it is off by a mere millionth of a second, the position could be off by at least 300 meters. The challenge is that the device has to figure out the exact time while also determining its location—a concept addressed in the theory of relativity as space-time.
If there are too few satellites in sight, the GPS system can’t reliably function and may produce multiple potential locations for the receiver. This can result in a device showing a wrong location or failing to find one altogether. Until recently, it was only a theory about how many satellites were needed for clear GPS solutions.
Five satellites yield accurate location
Mireille Boutin, a discrete algebra and geometry professor at TU/e, and Gregor Kemper, an algorithmic algebra professor at TUM, have provided mathematical proof that five or more satellites can typically yield an exact position for the receiver. “This long-suspected truth hadn’t been proven until now. It was quite complex: we spent over a year grappling with this,” notes Gregor Kemper. At any given moment, every location on Earth can connect to at least four satellites. “With only four satellites, it seems there’s about a 50 percent chance of finding a unique solution to the GPS dilemma. Proving that statement will be one of our future endeavors,” adds Kemper. With three or fewer satellites in view, GPS navigation will not function effectively.
Understanding geometry and uniqueness
The researchers derived their proof by examining the GPS issue geometrically. They discovered that the receiver’s location can’t be distinctly identified if the satellites are positioned on a specific type of curved surface called a hyperboloid of revolution of two sheets, which opens up in all directions. This theoretical finding also aids in understanding the causes of positioning inaccuracies in practical terms.
