The Trigonometry of GPS: How the Blue Dot Finds You

We take GPS for granted. Whether we are driving to a new city or tracking a pizza delivery, we expect our location to be accurate within a few meters.

But the Global Positioning System (GPS) is arguably the most sophisticated machine humanity has ever built. It is a constellation of satellites dancing in orbit, constantly shouting the time.

Trilateration (Not Triangulation)

Most people think GPS uses triangulation (measuring angles). It actually uses trilateration (measuring distances).

Imagine you are lost in a desert. You ask a stranger where you are. They say, "You are exactly 100 miles from Las Vegas." This doesn't help much; you could be anywhere on a circle with a radius of 100 miles centered on Vegas.

You ask another person. They say, "You are 200 miles from Salt Lake City." Now you have two intersecting circles. You are at one of the two points where they cross.

You ask a third person. "You are 300 miles from San Francisco." This third circle intersects at only one point. You now know exactly where you are.

GPS satellites work the same way inside 3D spheres. Your phone listens for signals from at least 4 satellites to calculate your 3D position (latitude, longitude, and altitude).

Time = Distance

But how does your phone know the distance to the satellite? The satellite doesn't inherently know where *you* are.

The satellite simply broadcasts a timestamp: "I sent this signal at 12:00:00.000."

Radio waves travel at the speed of light (c), which is approximately 300,000 km/s. If your phone receives the signal at 12:00:00.060, it knows the signal took 0.06 seconds to arrive.

This is why satellites carry Atomic Clocks. If the clock is off by just one millisecond (0.001s), the error is 300 km. You would be in the wrong state.

Einstein's Correction

Here is the wildest part: If engineers built GPS using only classical Newtonian physics, the system would fail within minutes.

Einstein's theories of relativity dictate that time is not constant.

• Special Relativity: The satellites are moving fast (14,000 km/h). Moving clocks tick slower. This slows the satellite clocks down by 7 microseconds per day.
• General Relativity: The satellites are far from Earth's gravity. Clocks in weaker gravity tick faster. This speeds the satellite clocks up by 45 microseconds per day.

The net result is that satellite clocks run 38 microseconds faster per day than clocks on Earth.

38 microseconds times the speed of light equals an error of roughly 11 kilometers per day. Without correcting for Einstein's math, GPS would be useless.

The Engineering Triumph

GPS is a continuous miracle. It requires rocket science to launch the constellation, atomic physics to keep the time, and theoretical astrophysics to align that time with Earth.

And yet, the receiver chip in your phone costs less than $5. It is arguably the most democratized piece of advanced technology in history.