The GPS system

Exhibit: table of content

Mathematics Math21a, Spring 2006
Multivariable Calculus
Oliver Knill, SciCtr 434,

GPS in two dimensions

Assume you sit at a point X and know the positions of three satellites in P,Q,R in the plane. Each satellite has a built in atomic clock and sends periodically signals. The receiver at position X does not contain an accurate clock and can not determine the distance to each satellite. But it can compare the arrival times from two different satellites and therefore find the distance difference to two satellites. The set of points with fixed distance difference to two points P,Q is a hyperbola XP,Q. Similarly, one defines a hyperbola XQ,R. These two hyperbola intersect in a finite set of points which is your location X.

GPS in three dimensions

In three dimensions, the set of points X, whose difference to two points P,Q is a fixed constant, forms a hyperboloid XP,Q. With 4 known points P,Q,R,S, we have two more hyperboloids. Your location is in the intersection of these three surfaces.

GPS Facts

There are 24 satellites in 6 orbital planes at 20'195 km altitude. Each satellite has 12 antennas and weights 1500 to 2000 kilograms. Each makes two complete rotations in 24 hours. The orbits are arranged so that at any time, anywhere on earth, there are at least 4 satellites visible in the sky. The radio signals are sent at frequencies between 1.176 and 1.841 GHz. The GPS system was developed in 1973 by the US department of defense.


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Oliver Knill, Math21a, Multivariable Calculus, Spring 2006, Department of Mathematics, Faculty of Art and Sciences, Harvard University