Oh No, What About Elevation?!?

Your goal with this research project is to develop a technique for GPS to find somebody’s position, including their elevation. If you have not read “Trilateration: How GPS Works,” it is recommended you do so before attempting this. Also, check out the Resources page to download Geogebra if you have not already. The 3D graphing tool will be a powerful assistant for this task.

For this activity, it is helpful to know the standard equation for a sphere, where (a, b, c) is the center of the sphere and r is the radius:Screen Shot 2019-06-06 at 6.00.52 PM

  1. Consider the following spheres and find out what shape is formed by their intersection:Screen Shot 2019-06-06 at 6.06.36 PM
  2. Based on your answer to question 1, how many spheres do you think you will need to locate someone in three dimensional space?
  3. In three dimensions, you are given the following information about three stations and your distance from them. Find your location:Screen Shot 2019-06-08 at 1.04.09 PM
  4. Here is the location of a fourth station. Can you determine your location now?Screen Shot 2019-06-08 at 1.04.15 PM
  5. Assuming the elevation of points A, B, and C are 0, it is proposed that the following formulas will find the coordinates of the point of interest:Trilat Formula 2.png Prove/disprove this and check it with your work from problem 3.
  6. Conveniently, in question 3, I gave you three stations that were all at an elevation of zero. How would you deal with the situation where I gave you the following:Screen Shot 2019-06-18 at 10.57.14 AM
  7. Is it possible to adjust the formulas from problem 5 to work for problem 6? If not, can you propose a different strategy that quickly comes up with the coordinates of the point of interest.
  8. Maybe you could not find a method to address problem 7, which is not unexpected. The field of linear algebra offers a technique for doing so, therefore read about matrices and attempt the research task that builds off your work here.