I recently wrote a column on pendulums and another on navigation that included mention of the Global Positioning System (GPS). The latest issue of the American Scientist has an article that connects pendulums with GPS. I thought that was a cool coincidence and the discussion about GPS got me thinking about one of my special interests: mathematics (See Seeking Beauty). Celestial navigation relies on trigonometry and the GPS in my phone uses calculus.
Finding one’s location on the sphere that we live on is an interesting problem in three dimensional geometry. I can understand how the Global Positioning System works by drawing a diagram of the process but it totally eludes me how the mathematics work to actually provide an exact location. Never-the-less, I undertook some research to see if I could learn more.
First, let me describe how GPS works. Wherever your phone is located, the GPS receiver in your phone can receive signals from four global positioning satellites. The satellites know their exact position in the sky and send information from which the distance between your phone and the satellite is computed. This data from each satellite defines a sphere and the point where the four spheres intersect is the location of your phone. The illustration shows both a 3D drawing and a simpler 2D one. Now these calculations are never totally accurate and there is some error, which is corrected for, and the resulting position reportedly has an error of several meters. My experience shows the error to be often within a few feet.
Computing this intersection is really complicated and I’ve provided a few references below that give greater detail. See especially Blewitt. Blewitt goes into detail about all the many, many calculations that go into making GPS work. It is a testament to the step-by-step way in which science and engineering evolve to create and develop our many modern wonders.
References
- From a Swinging Chandelier to Global Positioning Systems, Steven Strogatz, American Scientist, March-April 2021, Volume 109, Number 2, Page 106.
- What is GPS?, National Coordination Office for Space-Based Positioning, Navigation, and Timing, National Oceanic and Atmospheric Administration (NOAA).
- How GPS Works Today, Bright Side, YouTube.
- How GPS Receivers Work – Trilateration vs Triangulation, GIS Geography, 2021. (Image of satellite orbits used above is from this article)
- Trilateration Exercise, National Coordination Office for Space-Based Positioning, Navigation, and Timing, NOAA, 2014. (This is a good classroom demonstration of how GPS works)
- Basics of the GPS Technique: Observation Equations, Geoffrey Blewitt, Department of Geomatics, University of Newcastle, 1997.
- Triangulation vs Trilateration vs Multilateration – for Indoor Positioning Systems, Ipshita Biswas, Path Partner, 2019.
- Cell Phone Trilateration Algorithm, 101 Computing, 2019.
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