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Trilateration is a type of measurement used to determine the location of a point by using the geometry of spheres, circles, or triangles. Unlike triangulation, which uses the measurement of angles to determine location, trilateration uses measures of distance. Global positioning systems (GPS) and land surveys apply this method to provide precise geographical locations on the Earth’s surface.
In two-dimensional geometry, the radii of circles can sometimes be used to find the location of a point. For example, if an unknown point X is 3 units away from point A, it could lie anywhere on a 3-unit radius around point A. When visualized, it is possible to see that the set of possible locations for X forms a circle around point A. If point X is also known to be 4 units away from point B, however, a second circle of possible locations is formed around B with a 4-unit radius. These two circles overlap, intersecting at two points – one of which must be X.
A third measurement allows X to be found. If X is known to be 5 units away from point C, a third circle is formed with a radius of 5. This circle will only intersect with the other two at one of the two points, which we then know must be X.
The above example can be applied to situations where an approximate geographical location must be found. For example, if a person were trying to find his or her location on the ground, he or she might be able to deduce it by finding out three distances to known landmarks or cities. In most cases, however, precise navigation requires that a three-dimensional trilateration method be used.
The main difference between two-dimensional and three-dimensional trilateration is that in the three-dimensional method, spheres are used instead of circles. The radii of four spheres are measured, and location is determined from the intersection points. This is visualized through one of the most common applications of trilateration — a GPS system.
GPS systems work by calculating their distance to satellites orbiting the Earth. If the user of the GPS is standing on the surface of the Earth, we have one sphere with known radius already — the planet itself. The GPS sends precisely timed signals to satellites and receives messages that it uses to determine transit time and, by extension, its distance to each satellite. The distances to three satellites form three spheres, which overlap into two possible locations, one on the Earth’s surface and one in space. For GPS users on the ground, the coordinate on the surface is the correct location.