A working draft of resources and reports from an NSF-sponsored project intended to strengthen the role of mathematics in Advanced Technological Education (ATE) programs. Intended as a resource for ATE faculty and members of the mathematical community. Comments are welcome by e-mail to the project directors: Susan L. Forman or Lynn A. Steen.
Global Positioning System (GPS) is a satellite-based system of signals that enables specially designed receivers to calculate precise positions on the surface of the earth. The satellite system, operated by the U.S. Department of Defense, became operational in 1995. Four satellites are located in six circular high-altitute (20,200 km) orbits spaced 60 degrees apart and inclined at 55 degrees from the Equator. Each satellite has a period of 12 hours and transmits on two radio frequencies its position and the exact time. Typically, four or five satellites are visible at any time from any point on the surface of the inhabited part of the earth.
Hand-held battery-powered receivers, now widely available, use signals from these satellites to calculate the receiver's three coordinates of latitude, longitude, and altitude. A marvel of engineering and mathematics, GPS is now widely used for business (in agriculture, surveying, transportation), emergencies (for 911 calls and other rescue operations), driving (in cars), and recreation (by motorists, boaters, and hikers).
Some uses of GPS require merely the data from one location, which can then be transfered to a map to identify the position of the receiver. Other uses (for example, in Precision Agriculture) involve the collection of data from many different points in order to construct a map or feed information to a Geographic Information System where the position data are related to other information. Typical receivers also indicate the positions (altitude, azimuth, and identification number) of the satellites from which signals are being received. Thus GPS receivers can also be used to identify and track the satellites as they traverse their orbits.
If there were no sources of error, the receiver's position could be calculated by simple algebra and spherical trigonometry from two satellite signals. The distance of the receiver from each satellite is determined by the time the signal takes in transit from the satellite to the receiver. Two such distances (measured from different satellites) are sufficient to determine the position of the receiver on the two-dimensional surface of the earth; three satellites give enough data to calculate altitude as well.
This calculation, called trilateration, is akin to triangulation except that it uses the lengths of three lines (rather than the angles from three points) to locate the receiver. Mathematically, it involves solving three simultaneous equations that are based on the Pythagorean formula in three dimensions. It also requires knowledge of conversion factors to transform the satellite information into actual latitude, longitude, and altitude. These calculations are within the scope of standard algebra and geometry courses--and also within the capabilities of a computer chip.
GPS receivers use various strategies for reducing these inevitable errors. To correct for discrepencies between the receiver's and the satellites' clocks, all GPS receivers employ data from four satellites instead of three. The additional data make it possible to solve a system of four simultaneous equations in which the unknowns are the three coordinates of position (latitude, longitude, and altitude) and the error in the receiver's clock. (If only three satellites are visible, the receiver omits calculation of altitude and reports only sea-level data for latitude and longitude.)
Errors due to atmospheric conditions and satellite position are not unique to the receiver, but affect everyone in the same region the same way. So in certain areas (e.g., airports, harbors, farming country), ground stations are built whose positions are known precisely. Each station receives GPS data from the overhead satellites, calculates position from them, and compares the results with its known correct position. It then broadcasts a signal that conveys this error to nearby GPS receivers designed to receive and utilize this second, error-correcting signal.
Under normal circumstances, without error-correcting efforts, the GPS system produces position information that is accurate to approximatley 100 meters. The error-correcting systems now in use reduce errors to under 10 meters--quite adequate for most civilian uses.
Kopytoff, Verne G. "18 Wheels, G.P.S., and Radar." New York Times, March 4, 1999, D1, D7.
Trimble GPS Information
GIS, GPS, and Remote Sensing (Florida)
Associate of Technology in GIS and GPS (Kansas)
Terrain Analysis (New York)
Copyright © 1999.
Last Updated: October 12, 1999.
Susan L. Forman or
Lynn A. Steen.