# You are here – GPS and geometry

## Introduction

Many of us now rely on a little electronic help in finding our way, making the most of the GPS in our phones and satnavs. GPS shows us the way thanks to some simple geometry and a little help from Einstein.

## Description

Many of us now use a little electronic help to find our way around.  We have satnav in our cars, GPS enabled phones in our pockets, even our cameras can geotag our photos, telling us exactly where we were when we took our holiday snaps.

The Global Positioning System (GPS) was developed by the US military and initially was very expensive and cumbersome, the only other people who used it were specialists such as Arctic explorers.  But as it has become more affordable and more compact, many of us now use this remarkable technology everyday.   And surprisingly, the way GPS works is all down to some simple geometry along with a little help from Einstein.

Listening to satellites

Your phone (or satnav or camera) acts as a GPS receiver listening out for a signal from one of the nearest GPS satellites.  There are 31 GPS satellites, their orbits criss-crossing the Earth  in such a way that there is always at least 4 satellites in the sky above any spot on Earth at any time.

These satellites constantly bellow out their locations to the universe and your phone receives this message, along with the time at which it was sent.  This message is a radio signal and therefore travels at the speed of light.  [How we can calculate where we are in relation to the satellite?]  With this information, your phone performs a simple calculation to determine the distance it is from the satellite.

Distance from satellite = (time it took message to reach us) x speed of light

Suppose it took half a millisecond for the message from the satellite to reach your phone.  The speed of light is about $$3\times10^8 ms^{-1}$$, so a radio signal travels about 300 km in a microsecond.  From this, our phone would calculate it is 150km away from this GPS satellite.

You are here

All your phone now knows is that you are a certain distance, in our example 150km, away from the known position of the satellite.   [Where can we be in relation to the satellite?]

In our 3D space if you are 150km away from a satellite, you are somewhere on a sphere with that satellite as its centre and with a radius of 150km.  The principle of finding your location with this method is the same in two dimensions as in three, and it’s a little easier to picture, and to draw!  In two dimensions you will be somewhere on a circle with a 150km radius that is centred on the satellite’s location.

At any time there will also be at least 3 other satellites overhead.  Your phone uses the messages from these other satellites to calculate that you are also on circles of particular radii centred on each of those satellites.

Suppose your phone has calculated that you are 100km away from a second satellite, so you are somewhere on the circle centred on that satellite with a 100km radius. These two circles must intersect at least once (at your location) – they could either just touch or more likely they overlap and intersect in two places.

To nail down exactly on which of these two places you are standing, your phone needs a third satellite.  These three circles must intersect at at least one place (your location) and in fact they intersect in only one place.  Your phone has calculated that you are here!

Working in 3D

In two dimensions, you only need 3 circles to pinpoint where you are; two circles intersect in at most two points, requiring a third circle to identify which of these is your location.  The phone does these calculations in three dimensions, solving where spheres centred on the satellites intersect.  [How many satellites do you think are needed in three dimensions? How can two, three and four spheres intersect?] In three dimensions at least 4 satellites are needed; two spheres intersect in a circle, this circle intersects the third sphere in two points, and the fourth sphere pinpoints which of these is your location.

Although it is easier for us to picture this as intersecting circles and spheres, your phone does not know where you are by actually drawing circles.  Instead it uses the equations for spheres centred on the satellites and solves these equations to find where the spheres intersect.

Right on time?

In order for GPS to work, your phone needs to know exactly how long it takes the radio signals to travel from the satellites.  It calculates this from the difference between the time the satellite sent the message and the time your phone received it; so it is vital that the clocks on the satellites and the clock in your phone agree.  The smallest error in this time would be hugely magnified when it is multiplied by the speed of light in the distance calculation.

The GPS satellites orbit the Earth twice each day, travelling at more than 14,000 km/h relative to your phone here on Earth.  Einstein’s theory of special relativity says that clocks tick more slowly when moving at such high speeds.  A clock travelling at the speed of a GPS satellite would lose about 7 microseconds a day compared to if it were on the ground.

This is partly counteracted by the fact that the satellite’s high orbit means the gravity due to the Earth’s mass is far weaker than on the Earth’s surface.  Einstein’s theory of general relativity says that clocks tick more quickly the further they are from a massive object; a clock at the distance of a GPS satellite would gain 45 microseconds a day compared to if it were on the ground.

But taking these two effects into consideration the clock on a GPS satellite would still tick faster than clocks on the ground, gaining 38 microseconds a day compared to the time it would show if it were on the ground.    This would make the satellite useless for navigation – your distance calculations would be out by more than 10km within one day of the satellite's launch!

To account for these relativistic effects engineers set the GPS clocks to tick slower on the ground than our normal clocks.  This means that when they are in orbit the GPS clocks will synchronise with the ticking of the clocks on the ground, including the one in your phone.

It is very surprising to find an application of Einstein’s theory of relativity, something that seems so esoteric, in a device that many of us use everyday.  So the next time you find your way using your phone, you’ll know that the geometry of circles, and a little relativity, are showing you the way.

### Demonstration

Props:

• Three loops of strings (it’s nice if they are different lengths) and some chalk.

It’s a good idea to try the demonstration in situ so you can get a feel for where to lay out your string.  It takes up more room than you expect.  And obviously make sure it is ok to draw on the ground with chalk before you run the tour!

When you arrive ask the group to stand in a rough circle.  Start by asking them if anyone used a satnav to get to Oxford today, or a phone to navigate their way to the starting point of the tour.

Before you explain the second section “Listening to satellites” ask if anyone has a GPS-enabled phone or camera with them, and if you can borrow it.  You can then run the demo referring to this phone as it uses GPS to calculate it’s location.

Demo:

Questions are suggested in italics in the text above to help engage the group.

If people have trouble understanding how to convert time into distance in the “Listening to satellites” section, you could write the equation out with chalk on the ground.

In the section “You are here” use the loops of string to represent the distance to the satellite.  For the first satellite, stretch the loop of string out on the ground explaining that it represents our distance from the satellite, marking the position of one end of the loop as the location of the satellite. Ask for a volunteer to act as the satellite, standing on the mark with their foot in the loop.  Can anyone suggest where we can be in relation to the satellite if we know they are a fixed distance away? Then ask for another volunteer to act as the phone and mark out where we could be in relation to the satellite, pulling the loop taut and using chalk in the other end to mark out a circle using the satellite/foot as a pivot.

For the second satellite, stretch the loop out so that one end lies within the first circle, and mark the position of the other end (allowing a little give) as the second satellite.  Again ask for a volunteer to act as the satellite, standing on the mark with one foot in the loop, and ask the phone volunteer to again mark out the circle.

For the third satellite, stretch the loop out so that one end sits at one of points where the first two circles intersect.  Mark the position of the other end (again allowing a little give) as the third satellite.  Then ask for a volunteer to act as the satellite and your phone volunteer to inscribe the final circle.  The three circles should intersect at one point (you might have to help a little with the positioning) and you can ask the phone volunteer to stand on the final spot to indicate they have found their location!

## See this Site in a Tour

Lamb and Flag Passage,
Oxford, UK.
Viewpoint:
This site can actually be anywhere in any city, you just need to be able to draw some chalk circles on open ground.

## Access information

Wheechair/buggy accessible:
No