Spheres and GPS - How it Works

The Global Positioning System (GPS) consists of 24 earth-orbiting satellites operated by the US Department of Defense. These satellites allow any person who owns a GPS receiver to determine his or her precise longitude, latitude and altitude anywhere on the planet. Until recently, inflated prices and Cold War politics conspired to keep GPS receivers out of the hands of the average civilian. But now, for as little as $100, you can know exactly where you are and where you have been. For anyone who has ever been lost -- while hiking in the woods, boating in the ocean, driving in a unfamiliar city or flying a small airplane at night -- a GPS receiver is a miracle. When you use a GPS receiver, you're never lost!

History of Navigation

Throughout history, it has been easy to determine latitude, but it took a long time to be able to measure longitude. For example, when he reached the New World, Columbus did not know how far west he had gone. He thought he had arrived in India. Because of this he named the inhabitants of this new land "Indians" To his death, Columbus never realized that he had discovered a "New World".

Galileo tried to solve the longitude problem by using his newly invented telescope. But while Galileo's method worked well on the land, it was not practical on the heaving sea. Geographers used his method to redraw maps of the world, and found that earlier maps had greatly underestimated the distances between continents and misplaced the boundaries between nations by large distances. When the king of France saw the revised map of his domain, he complained that he was losing more territory to his astronomers than to his enemies.

Despite advancements in astronomy and science, determining the longitude position of ships at sea remained a puzzle. In 1714, the British Parliament offered a prize of 20,000 pounds to anyone who could devise a practical method of finding longitude at sea to an accuracy within 30 nautical miles. In those days, this was a huge amount of money. It was only in 1762, though, that an English clockmaker named John Harrison, fighting not only extreme technical difficulties but also the prejudice of the upper class of society against "that mechanic," won the prize by designing a chronometer. This is the kind of mechanical device that was refined and used as the standard timepiece until quartz and electronic clocks and watches became available not long ago.


Before gps, people took bearings (compass sightings) on existing locations and triangulated these on a chart to compute a fix on location. With a compass bearing, you can draw a "line" on the sphere through the known location and you know you are somewhere on that "line". Do the same thing to a second point and the two "lines" will intersect. This is your position. If you try a third point it should intersect at the same place the other two lines intersect. Usually however, because of imprecise sightings, it intersects both lines at slightly different points thereby forming a small spherical triangle. You are somewhere inside that triangle but you don't know exactly where. If the triangle is small enough you consider it good enough, otherwise you need to take another sighting. Accuracy is determined primarily on your ability to get and plot an accurate bearing as well as the geometry of the known sites available. This means if the sites are very close together you will get poorer results than if they are further apart.

How GPS Works

The gps receiver uses a slightly different approach. It measures its distance from the satellites and uses this information to compute a fix. How can it measure distance? Well it really measures the length of time the signal takes to arrive at your location and then based on knowing that the signal moves at the speed of light it can compute the distance from knowing the travel time. However, unlike the known sites of the olden days, these sites are moving. The solution to this problem is to have the satellite itself send enough information to calculate its current location relative to your receiver.

Armed with the satellite location and the distance from the satellite we know that we are somewhere on a sphere that is described by the radius (distance) and centered at the satellite location.

By acquiring the same information from a second satellite we can compute a second sphere that cuts the first one at a plane.

Now we know we are somewhere on the circle that is described by the intersection of the two spheres.

If we acquire the same information from a third satellite we would notice that the new sphere will intersect the circle at only two points.

If we know approximately where we are we can discard one of those points and we are left with our exact location on the surface of the globe.

In fact, it works a little too well, according to the Department of Defense. Worried that GPS could be used by an enemy to guide missiles or smart bombs, the defense engineers built errors into the system. GPS satellites send out two signals: an encrypted signal for military use and an unencrypted, less accurate signal for civilians. After the downing of Korean Flight 007 in 1983 -a tragedy that might have been prevented if its crew had access to better navigational tools- President Ronald Reagan issued a directive that guaranteed that GPS signals would be available at no charge to the world. GPS burst into public awareness during the Persian Gulf War in 1991. GPS was used extensively during that conflict, so much so that not enough military-equipped GPS receivers were available. To satisfy demand, the Department of Defense acquired civilian GPS units and temporarily changed GPS transmissions to give civilian receivers access to higher-accuracy military signals. In May of 2000, Clinton announced plans for increased accuracy in GPS service for civilians.

Summary: Despite its high-tech trappings, the central concept behind GPS is as old as celestial navigation. But instead of using stars for triangulation, modern-day mariners use GPS satellites. If you know your exact distance from a satellite, you know your location lies somewhere on the sphere defined by that radius. If you know the distance from a second satellite, you know your location must lie along the circumference of the circle where the two spheres intersect. A third satellite results in two points where all three spheres intersect. One of these points is way out in space. This straightforward technique of measuring various distances and then triangulating your position gives your precise location.