The Math Behind GPS

Have you ever opened a mapping app on your phone or tagged your location on social media?

Your phone is able to do this because it can access the Global Positioning System or GPS. This system of satellites helps us figure out where we are and get where we want to go.

Image - Text Version

Shown is a colour photograph of a cell phone attached to the dashboard of a car. Visible on the phone is a map showing directions to a location.

Not only do we rely on GPS in our phones, it is also important for ships, planes, ambulances and so much more. However, many of us don’t know much about the math, science, and history of this technology. How can a system that fits inside a smartphone instantly pinpoint your location?

History of GPS

In the 1950's, researchers developed the first satellite navigation system for military purposes. After several decades of improvements, civilians began to use GPS in the 1980s. The system became fully functional in 1994. The United States government owns and maintains the Global Positioning System that we use in North America.

Scientists from many different fields contributed to the development of the GPS. Gladys Mae West was one of the mathematicians who made this system possible. She created an incredibly detailed mathematical model of the shape of the Earth. This model allows the GPS to accurately locate positions on Earth.

Image - Text Version

Image - Text Version
Shown is a black and white photograph of Sam Smith and Gladys West. Sam Smith is a middle-aged white man in a business suit. Gladys is a middle-aged black woman also in business attire. They are standing in an office in front of a desk covered in papers. Gladys is pointing at something on one of the pieces of paper with a pen in her hand. Sam is looking at what she is pointing at. Beside Gladys is a very large global on a pedestal.

Misconception Alert

Earth is not a perfect sphere. It has lumps and bumps which Gladys Mae West helped to model.

Roger Easton, Ivan Getting, and Bradford Parkinson also played key roles in the development of the GPS. These physicists and engineers developed ways to locate satellites and locations on Earth.

How does GPS work?

Before the development of GPS, people used a variety of different navigation techniques. Sailors often used the position of the stars to know what direction they were going.

Navigation techniques help people find their latitude and longitude. Lines of longitude run from North to South. The line of 0º longitude runs through Greenwich, England. Lines of latitude run from East to West. The line of 0º latitude is the Equator. Knowing both your longitude and latitude gives your position on the globe.

Instead of relying on stars, the GPS uses satellites to figure out longitude and latitude. GPS uses a network of over 30 satellites. These satellites orbit Earth twice a day. Control centers on the ground make sure that the satellites operate properly.

Image - Text Version

Shown is a colour illustration of the Earth in Space. Orbiting the Earth are around 20 tiny satellites. The direction of their motion is suggested using white streaks of colour behind each satellite.

These satellites communicate with receivers. A receiver might be in your smartphone, car, or computer. Receivers get signals from satellites. Signals move at the speed of light. This is about 300,000,000 metres per second).

An important part of the signal is an accurate time. Satellites know the time using incredibly accurate atomic clocksatomic clocks.

Image - Text Version

Shown is an illustration of a satellite and cell phone.
The satellite is in the upper left part of the image. Radiating downward are parallel semi-circular lines that indicate that the satellite is sending down a signal to Earth. The cell phone is in the lower, central part of the image. Radiating upward are parallel semi-circular lines that indicate that the phone is sending up a signal to the satellite. Connected to the phone via a line is a small image of a map pin and a car to indicate that the phone uses the information for map apps and vehicle navigation systems.

 A receiver uses the following formula:

Distance = signal speed x signal time

The signal speed is the speed of light (about 300,000,000 m/s). The signal time is how long it takes for the signal to travel from the satellite to the receiver. Multiplying these two numbers tells us the distance between the satellite and the receiver.

For example, imagine that it takes a signal 0.005 seconds to travel from a satellite from your phone.

Distance = 300,000,000 m/s x 0.005 s = 1,500,000 metres

That means that the satellite is 1.5 million metres or 1500 kilometres from the receiver.

Did you know?

GPS relies on Albert Einstein’s theory of relativity. The time information sent by satellites is adjusted to account for relativity. This makes the system more precise.

However, knowing how far away you are from one satellite doesn’t give you your position. For that, GPS uses trilateration.

Trilateration

Trilateration is a way of locating a point based on how far away it is from three other points. Multiple satellites must connect with the receiver to use this process. The distance between the receiver and several satellites is calculated. Trilateration is a method of using this distance information to find a location.

Misconception Alert

GPS uses trilateration not triangulation. Trilateration involves measuring distances. It is sometimes confused with triangulation which measures angles.

Here’s a simple example of how trilateration works. Imagine that you are standing on a soccer field. Next image that you are trying to find a ball on the field based on the following clues.

1. The ball is 20m from the bottom right corner.
2. The ball is 40m from the center of the field.
3. The ball is 15m from the center of the left goal.

Based on these three clues, can you find the soccer ball?

First, we can draw a circle with a radius of 20 metres around the bottom right corner. We now know the ball is somewhere ON the blue circle.

Image - Text Version

Shown is an illustration of a soccer field. At the bottom right of the illustration is an outline of a blue circle that has its centre at the bottom right corner of the soccer field. The circle is identified as having a radius of 20 metres.

Then, we can draw a circle with a radius of 40 metres around the center of the field. Now, where are the two possible places that the ball might be? Remember, the ball is on the circle.

You probably realized that it’s the two places where the circles cross each other.

Image - Text Version

Shown is an illustration of a soccer field. At the bottom right is an outline of a blue circle that has its centre at the bottom right corner of the soccer field. The circle is identified as having a radius of 20 metres. In the centre of the illustration an outline of a pink circle that has its centre at the centre of the soccer field. The circle is identified as having a radius of 40 metres.

We need to use the third hint to figure out which of these two locations is correct. Once we draw in the third circle, we know exactly where the ball is.

Image - Text Version

Shown is an illustration of a soccer field. At the bottom right is an outline of a blue circle that has its centre at the bottom right corner of the soccer field. The circle is identified as having a radius of 20 metres. In the centre of the illustration is an outline of a pink circle that has its centre at the centre of the soccer field. The circle is identified as having a radius of 40 metres. At the centre right of the illustration is an outline of an orange circle that has its centre at the centre of the goal net area. The circle is identified as having a radius of 15 metres. An image of a soccer ball overlaps the place on the soccer field where the lines of the three circles intersect.

The soccer ball example involves 2D trilateration. Trilateration with satellites involves the same method. However, 3D trilateration involves the intersection of spheres instead of circles.

Image - Text Version

Show is an illustration of 2D and 3D trilateration.
On the left is an image of the Earth with four satellites flying above it. The satellites are coloured green, purple, orange and blue. Each satellite is surrounded by a sphere in the same colour as the satellite. The four spheres overlap slightly.
On the left is an image of three satellites. The satellites are coloured blue, orange and green. Each is centred in a circle that is drawn in the same colour as the satellite. There is one point where the three circles intersect.

 

Watch the video below to learn more.

(Please Note: The following video appears with a thumbnail that typically indicates a 'broken video' on YouTube, but the video works properly.)

As shown in the example above, trilateration only requires three satellites. So why do we need a fourth satellite? The fourth satellite adjusts for errors in the receiver’s clock. This makes GPS even more accurate. Currently, a GPS-enabled smartphone usually knows your location within about 5 metres. 

Using satellite navigation has become a normal part of our daily life. Mapping apps help us find our way around the world. We use GPS for many social activities, such as geotagging posts or geocaching. First responders use GPS to quickly get to the location of an emergency and respond to disasters. GPS also helps planes safely get to their destination. Scientists use GPS to forecast the weather. These are just a few of the ways that we use this technology. 

Next time you use GPS, take a minute to think about what makes this technology possible.