Timing offset and GPS GPS uses a constellation of 24 satellites and their ground stations as reference points to calculate positions accurate to a matter of meters. Suppose we find our distance measurements from three satellites to be 18 000, 19 000, and 20 000 km respectively. Collectively this places the position at either of the two points where the 20 000 km sphere cuts through the circle that is the intersection of the 18 000 and 19 000 km spheres. Thus by ranging from three satellites we can narrow our position to just two points in space. To decide which one is our true position we could make a fourth measurement. However, usually one of the two points is a ridiculous answer (either too far from Earth or moving at an impossible velocity)

and can be rejected without a measurement.

Now apply the above principle of location to a two-dimensional space. Assume that points A, B, and C are reference points with known locations, respectively at (x1, y1), (x2, y2), and (x3, y3), and that the unknown position is 3.0m from point A, 4.0m from point B, and 5.0m from point C.

(a) Suppose that accurate measurements are available. Then the three measurements can be used to determine the position uniquely. Let (x1, y1)¼(0, 3.0),

(x2, y2)¼(4.0, 0), (x3, y3)¼(4.0, 3.0). Find the position.

(b) Now assume that all measurements include a single timing offset that corresponds to an error of 0.5 m. In other words, the position is observed to be 3.5m from point A, 4.5m from point B, and 5.5m from point C. Develop a generic procedure to find the true position.