Exploring the Global Positioning System

The Global Positioning System, or GPS, is a satellite-based navigation system operated by the United States Government. It can be used for military or civilian purposes. The GPS uses satellites that broadcast signals that enable ground-based receivers to determine each satellite's location and its distance from the receiver. Using this data the receiver can determine its own location. At the present time, errors are deliberately introduced into the part of the system to which civilians have access. Receivers for civilian use can be purchased for as little as $150.00 in many sporting goods stores and discount stores and by mail order.

The basic idea behind the GPS is very simple. We can explore this idea with some graph paper, clothesline, and a compass. We will also need some open space -- perhaps, a open field or a school playground. Begin by setting up three "base stations" and measuring their locations very carefully. Draw a map showing the base stations and their locations. In the figure below the locations of the three base stations are marked by three colored squares.

Missing drawing

Now place an object someplace in the field and measure the distance from the object to each of the three base stations. Do not measure the location of the object in any other way. This is the way that the Global Position System works. The three base stations are like three satellites orbiting the earth. The GPS receiver is able to measure its distance from each of the satellites.

Missing drawing

Now on your graph paper draw three circles, one centered at each of the base stations and using the measured distance from the station to the object as its radius.

Missing drawing

If your measurements were perfect these three circles would intersect at one point -- the location of the object. Your measurements are very unlikely to be perfect, so the three circles may not quite intersect at one point. This is exactly what happens with the GPS -- its measurements are not perfect -- surprizingly good but not perfect. Choose the point in the middle of the three points of intersection as your best estimate for the location of the object.

Missing drawing

Do some experimentation with various different arrangements for the base stations. In particular, compare the arrangement above with the arrangement below. Which arrangement seems to lead to more accurate determinations of the location of an object? For each of the two arrangements have several different people make the measurements with the same object. How much variation do you see in the results?

Missing drawing

You can work with these measurements algebraically as well as graphically. Let the location of the first base station be given by

(a₁, b₁),

the location of the second base station be given by

(a₂, b₂),

and the location of the third base station be given by

(a₃, b₃).

Let the measured distance from the object to the first base station be r₁, to the second base station be r₂, and the third base station be r₃.

Then the equations of the three circles are

(x- a₁)² + (y - b₁)² = r₁²,
(x- a₂)² + (y - b₂)² = r₂²,

and

(x- a₃)² + (y - b₃)² = r₃².

Now you can find algebraically where the first pair of circles intersect by solving the pair of equations below simultaneously

(x- a₁)² + (y - b₁)² = r₁²,
(x- a₂)² + (y - b₂)² = r₂²,

Similarly, you can find where the second and third circles intersect by solving the second and third equations simultaneously and you can find where the first and third circles intersect by solving the first and third equations simultaneously.

Repeat your work above algebraically. Compare your algebraic results with your graphical results.

This work is copyrighted c 1996 by Carroll College, Helena, MT 59625.

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Department of Mathematics, Engineering, Computer Science, and Physics
Carroll College, 1601 N. Benton Avenue, Helena, MT 59625