Triangulation – Fun time with maths

David Levinson: I can use her signal to triangulate her exact position in the White House.
Julius Levinson: You can do that?
David Levinson: Yeah. All cable repairmen can, Pops.

Hands up if you remember the above scene from the film "Independence Day". So what is triangulation and how on earth is Jeff Goldblum going to do it?

Well, triangulation (or Trilateration if you want to be more exact)  is the process by which the coordinates of a radio transmitter (a mobile phone in our case) can be determined my measuring the distance of the transmitted signal from three at least different fixed points (cell towers in our case). The  same technique can be used so the position of a GPS signal can be determined by 3 different satellites.

The first thing you should know is that a cell tower can determine the distance between a mobile phone and itself. This is simply achieved by measuring how long it takes for the signal to reach the tower. Since the radio signal travels at a constant speed (the speed of light), the distance [S] between the tower and the phone is always equal to the [speed of light] x [time]. Once the distance is determined everything else is simple geometry. Allow me to explain.

A fixed cell tower measures that a distance from the source of a transmitted signal is 1 mile. With this knowledge we can say that the mobile phone is located somewhere on a circle which center is the tower and which radius is 1 mile.

By calculating the distance of the transmitted mobile signal from a second tower things are getting more bright in the task of pinpointing the phone. Lets assume that the second tower finds that the mobile phone is 1.3 miles away. Now again we can say that the phone is located on a circle with a radius of 1.3 miles and a center which coordinates are the location of the second tower. And now that we know what the distance from both towers is then the location of the phone can only be one of two possible points. These points are exactly where the two known circles intersect.

To find the definite location of the mobile phone all we have to do is follow the same process but this time for a third tower. The third constructed circle will intersect the two previous circles at one point and one point only. That is where the mobile phone is located.

Congratulations, you used triangulation successfully to locate the coordinates of a mobile phone. A few notes before we finish though. Keep in mind that this example assumes a two dimensional space. This of course is not real life. We live in a 3D world. In order to achieve the same result all you have to do is use spheres instead of circles. The math get a bit more difficult but the methodology is the same. Also, keep in mind that in the real world we can't achieve pinpoint accuracy with only three known fixed points (cell towers or satellites). But the more points we use for our calculations then the more accurate our results will be.

If you are interested about the math behind triangulation then please follow the link below.

Wikipedia – Triangulation / Trilateration

Have fun.