GPS basics: GPS requires at least 4 satellites

Several times I encountered people misunderstanding how GPS works and believes that GPS requires just 3 satellites. Other misunderstood concept regards how GPS works. Some people believe its triangulation. The real methodology is trilateration. In this post I will summarize how GPS works and show that it requires at least 4 satellites.


How GPS works?

GPS are based on trilateration, which is a method of determining the relative positions of objects using the geometry of triangles. GPS does not use triangulation because it does not measure angles. We can visualize this concept in the following way:
  • When a GPS connects with a satellite, the GPS device estimates the distance between the device and a satellite. This distance define the radius of one sphere. Thus, the GPS device can be located on any point of the sphere’s surface.
  • When the GPS connects to a second satellite, it defines another sphere. The intersection of those 2 spheres define a circle. The GPS can be located anywhere on the surface of that circle.
  • When the GPS connects to a third satellite, it defines another sphere. The intersection of this third sphere defines 2 potential points (in a worst case scenario it may define a line segment). Thus, 3 satellites are not enough to define one single location. The GPS requires a fourth satellite to define one single location.

Image 1. GPS trilateration (Source: Trimble)

Time is the 4th variable

Some people argue that 3 satellites are enough to define one position, because one of the mentioned 2 potential locations is a not reasonable one, or non real one. This idea could be better understood by looking at the maths. When GPS connects with three  satellites, it is possible to define three distance equation with three unknown variables  (the x, y and z coordinates of the point). As the equations are quadratic ones, sometimes  one solution may involve imaginary numbers or very far location. Thus, some people  believe that 3 satellites may be enough. However, is GPS we have a fourth variable.  This 4th variable is time.

Summarizing, GPS estimates the distance to a satellite based on the travel time of a  signal emitted from the satellite. Satellites are emitting signals at given times, and when  the GPS receives the signal it records the time when signal was received. However, the  satellite’s clocks are more accurate than the clock of the GPS device. Thus, there is a  time discrepancy that will affect the measurements and the solution. GPS solves this  problem by applying a correction factor (a 4th variable). Therefore, GPS has to solve  4 variables
  • X coordinate
  • Y coordinate
  • Z coordinate
  • C time correction factor
4 equations (4 satellite connections) are needed to solve the 4 variable.
Image 2. Set of equation solved by GPS

Note. for simplicity I used the term GPS. However, currently the new term is GNSS (Global Navigation Satellite System)

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