Trilateration: how GPS finds a point
Your phone does not ask anyone where it is, and it sends nothing. It just listens to a few satellites, measures how long each signal took to arrive, and works out the one place on Earth where it can be standing.
What you are looking at
- Each circle is a measured distance, not a coverage area. The radius comes from how long the signal took: nothing is nudged to make things line up.
- With two satellites two candidate points are left. They are not placed by eye: they are the solution of intersecting two circles, and that is why a third is needed.
- At the end the same clock error is added to all three radii. Watch how the circles stop sharing a point: at that moment no position is possible at all.
01One satellite does not locate you: it puts you on a circle
If you know you are 20,000 km from a satellite, you know you are somewhere on an enormous sphere. That is far less information than it sounds.
Two spheres leave you a circle; three leave two points, and one of them is absurdly far from Earth, so it rules itself out. That is trilateration, and that is the whole idea.
02What is really measured is TIME
No satellite knows you exist or sends you anything personal. It broadcasts, over and over, a message saying "I am satellite so-and-so, and this left at exactly this time".
Your phone compares that time with its own. The difference, multiplied by the speed of light, is the distance. All of GPS is a stopwatch with a good view.
That is why it works on a hill with no mobile signal, and why it uses no data: your phone does not talk, it only listens.
03And that is why one extra satellite is needed
Light travels 300 metres in a microsecond. If your phone's clock is off by a single microsecond, every one of your distances is wrong by 300 metres at once — which is what happens at the end of the animation.
An atomic clock in your pocket is not viable. So the receiver treats its own offset as a fourth unknown and solves for it with one more satellite: the fourth adds no geometry, it fixes the time.
As a side effect, that gives your phone the most accurate clock in your house.
Where this is used
Measuring how far Chile moved
Fixed GPS stations across the country measure their own position without pause. They are how we know the 2010 earthquake shifted part of the territory several metres west: not an estimate, a subtraction between two positions.
The tractor that sows by itself
With corrections from a ground station, accuracy goes from metres to centimetres. That lets a machine sow and spray without covering the same strip twice: less seed, less chemical, less fuel.
The world's clock
Power grids, mobile networks and stock exchanges sync their clocks to GPS. It is its least visible use and probably its most critical: if GPS goes down, far more than maps stops working.
Mountain rescue
An emergency beacon sends a position computed this way. The difference between searching a valley and searching twenty square metres is, literally, the difference between finding someone or not.
Explaining it to a child
The idea, in plain wordsImagine being blindfolded in a square while three friends, standing in places you know, shout at the same time. If you could measure how long each shout took to reach you, you would know exactly where you are standing.
- In the yard, mark three fixed points with stones. One of you stands somewhere without saying where.
- With a tape measure, measure the distance from the first stone and draw that circle on the ground with chalk. They are somewhere on the whole circle: not much help.
- Do the same from the second and the third. Where the three circles cross there is a single point, and it is exactly where they were standing.
The question to leave them withAnd what if one of the friends shouted a little earlier than you thought?
For the classroom
Ten minutes with the animation, ready to take into a room. Free under CC BY 4.0, like everything else here.
- On the board, mark a point and say: "I am 3 metres from here". Ask them to draw every possible place. A circle appears.
- Mark a second point with another distance. Now two candidates remain. A third leaves only one. So far this is pure geometry and needs no satellites.
- Play the animation and watch the ending: when the receiver's clock runs late, the three circles stop meeting at a point.
- Close with the question: if the three circles no longer meet, how does the receiver know its clock is wrong?
Why does a GPS need four satellites and not three?
Answer: Because there are four unknowns, not three: the three coordinates and the error of its own clock. The fourth satellite adds no geometry, it solves for the clock.
How does the receiver measure the distance to a satellite, if it cannot go there?
Answer: It measures how long the signal took and multiplies by the speed of light. It is the same thing an ultrasound does with sound: distance equals speed times time.
A clock error of one millionth of a second produces how much position error?
Answer: About 300 metres. Light covers 300 metres in a microsecond, which is why the satellites carry atomic clocks: time IS distance.
Many believe the phone "talks to" the satellite. It does not: it only listens. A GPS is a pure receiver, which is why it works for millions of devices at once without saturating. Worth taking apart, because it explains why GPS uses no data.
FísicaEterna classroom guide · CC BY 4.0 · fisicaeterna.cl
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The code behind it
It is written in Remotion, which renders video by writing React. The bar and the number are not drawn next to the figure: they come out of the calculation on the same geometry you are watching. If the drawing lies, the number lies with it.
Licence
This animation is under CC BY 4.0: you may copy, redistribute, adapt and use it commercially, as long as you credit the source.
Suggested attributionTrilateration: how GPS finds a point — FísicaEterna (fisicaeterna.cl), CC BY 4.0