How GPS Uses Mathematics to Find Your Location

When you open a map app and a blue dot appears exactly where you are standing, your phone has just solved a set of simultaneous equations using signals from satellites more than 20,000 km away. The Global Positioning System (GPS) is one of the best examples of school mathematics, distance, speed, time and coordinate geometry, working on a planetary scale. Here is how it does it.

The GPS Satellite Network

GPS is operated by the United States and became fully operational in 1995. It uses a constellation of around 30 working satellites orbiting at an altitude of about 20,200 km, each completing an orbit roughly every 12 hours. The orbits are arranged so that from almost anywhere on Earth, at least four satellites are above the horizon at any time.

Other countries run similar systems: Russia's GLONASS, Europe's Galileo and China's BeiDou. Modern phones listen to several systems at once, which is why positioning has become so accurate.

Each satellite carries extremely precise atomic clocks and continuously broadcasts two key pieces of information:

• exactly when the signal was sent, and
• exactly where the satellite was at that moment.

Step One: Distance From Time

Radio signals travel at the speed of light, \( c \approx 3 \times 10^8 \) m/s. If your receiver knows when a signal left the satellite and when it arrived, it can compute the distance using the most familiar formula in physics:

$$ \text{distance} = \text{speed} \times \text{time} $$

Worked Example 1: Signal Travel Time

A satellite is 20,200 km directly overhead. How long does its signal take to reach you, and how big an error does a timing mistake of one microsecond cause?

Step 1: Convert the distance to metres.

$$ 20{,}200 \ \text{km} = 2.02 \times 10^{7} \ \text{m} $$

Step 2: Find the travel time.

$$ t = \frac{d}{c} = \frac{2.02 \times 10^{7}}{3 \times 10^{8}} \approx 0.0673 \ \text{s} $$

So the signal takes about 67 milliseconds.

Step 3: Error from a 1 microsecond timing mistake.

$$ \Delta d = c \times \Delta t = (3 \times 10^{8}) \times (1 \times 10^{-6}) = 300 \ \text{m} $$

A clock error of just one millionth of a second shifts the measured distance by 300 m. This is why timing is everything in GPS.

Step Two: Trilateration

Knowing your distance from one satellite tells you only that you are somewhere on a sphere around it. Distance from a second satellite narrows you down to the circle where two spheres meet. A third sphere cuts that circle at two points, and one of them is usually far out in space or deep inside the Earth, so it can be rejected. This method is called trilateration (not triangulation, which uses angles).

It is easiest to understand in two dimensions, using circles on a flat map.

Worked Example 2: Finding a Position in 2D

Three transmitters are at \( A(0, 0) \), \( B(10, 0) \) and \( C(0, 10) \), with distances in kilometres. A receiver measures its distances as

$$ d_A = 5, \qquad d_B = \sqrt{65}, \qquad d_C = \sqrt{45} $$

Find its position \( (x, y) \).

Step 1: Write a circle equation for each transmitter.

$$ x^2 + y^2 = 25 \qquad (1) $$

$$ (x - 10)^2 + y^2 = 65 \qquad (2) $$

$$ x^2 + (y - 10)^2 = 45 \qquad (3) $$

Step 2: Subtract (1) from (2) to remove the squared terms. Expanding (2) gives \( x^2 - 20x + 100 + y^2 = 65 \). Subtracting (1):

$$ -20x + 100 = 40 \quad\Rightarrow\quad x = 3 $$

Step 3: Subtract (1) from (3). Expanding (3) gives \( x^2 + y^2 - 20y + 100 = 45 \). Subtracting (1):

$$ -20y + 100 = 20 \quad\Rightarrow\quad y = 4 $$

Step 4: Check in equation (1).

$$ 3^2 + 4^2 = 9 + 16 = 25 \ \checkmark $$

The receiver is at \( (3, 4) \). Subtracting one equation from another turned three curved equations into simple linear ones, which is exactly the trick GPS software uses in three dimensions.

Why You Need Four Satellites, Not Three

Satellites carry atomic clocks costing a fortune, but your phone has a cheap quartz clock that may be off by a fraction of a second. As we saw, even a microsecond matters. GPS solves this cleverly: it treats the receiver's clock error \( \Delta t \) as a fourth unknown, alongside the coordinates \( x, y, z \).

For each satellite \( i \) at known position \( (x_i, y_i, z_i) \), the receiver forms an equation

$$ \sqrt{(x - x_i)^2 + (y - y_i)^2 + (z - z_i)^2} = c\,(t_i + \Delta t) $$

where \( t_i \) is the measured travel time. Four unknowns need at least four equations, so at least four satellites. In practice receivers use eight or more and find the best-fitting solution by least squares, an idea from statistics, solved step by step with an iterative method related to Newton's method.

From Coordinates to Latitude and Longitude

GPS first calculates your position as \( (x, y, z) \) coordinates with the origin at the centre of the Earth. Software then converts these into latitude, longitude and height using a mathematical model of the Earth's shape called WGS 84, which treats the Earth as a slightly flattened ellipsoid rather than a perfect sphere. Map apps then use spherical trigonometry to compute distances and directions between places.

Einstein in Your Pocket

GPS is also a daily test of Einstein's theories of relativity.

• Special relativity: the satellites move at about 3.9 km/s, so their clocks run slow by about 7 microseconds per day.
• General relativity: the satellites are farther from Earth's gravity, so their clocks run fast by about 45 microseconds per day.

The net effect is that satellite clocks gain about 38 microseconds per day. If engineers ignored this, the error would grow by roughly

$$ (38 \times 10^{-6} \ \text{s}) \times (3 \times 10^{8} \ \text{m/s}) \approx 11{,}400 \ \text{m} \approx 11 \ \text{km per day} $$

To prevent this, the satellite clocks are deliberately set to tick slightly slower before launch, and corrections are built into the system.

Did You Know?

• Your phone does not send anything to GPS satellites; it only listens. That is why GPS works for billions of users at once.
• Ordinary phone GPS is typically accurate to within about 5 m in open sky, while survey-grade receivers using correction signals can reach centimetre accuracy.
• Tall buildings can reflect signals and cause "multipath" errors, which is why your blue dot sometimes jumps around in crowded city centres.

Key Takeaways

• GPS finds distances using \( d = c \times t \), so tiny timing errors cause large position errors.
• Trilateration intersects spheres (or circles in 2D); subtracting equations turns them into linear ones.
• Four satellites are needed because the receiver's clock error is a fourth unknown.
• Coordinates are converted to latitude and longitude using an ellipsoid model of the Earth.
• Relativity shifts satellite clocks by about 38 microseconds per day, which must be corrected.

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