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TrilaterationTrilateration is a method of determining the relative position of objects using the geometry of triangles in a similar fashion as triangulation. Unlike triangulation, which uses angle measurements (together with at least one known distance) to calculate the subject's location, trilateration uses the known locations of two or more reference points, and the measured distance between the subject and each reference point. To accurately and uniquely determine the relative location of a point on a 2D plane using trilateration alone, generally at least 3 reference points are needed. The reason for this lies in the geometry of circles. If you know the distance of a subject point from some fixed reference point, then that point could exist anywhere on a circle of that radius from the reference. If you know that it is also a certain distance from a second reference point, then it also exists somewhere on a circle of that radius from the second reference point. These two circles intersect at precisely two points, and the subject could be at either point. The distance between the subject and a third reference point introduces a third circle into the diagram, and all three circles intersect at one point only: the position of the subject, relative to the three reference points. Of course, this assumes that the subject and the reference points all exist on the one plane, meaning that there are only 2 dimensions involved. For 3D space, 4 reference points are needed and the subject point exists on the surface of spheres instead of circles. Apart from those differences, the technique is still the same. In practical use, the minimum number of reference points may not be required to disambiguate the subject's location. For example, if the subject is known to be on land, or on the surface of the Earth, and one of the candidate locations is at sea or in space, that point may be disregarded. On the other hand, the stated number of reference points may not be enough if the geometry is singular, e.g., in the plane all three reference points and the subject are on one straight line. Hyperbolic positioning systems such as DECCA use a variant of trilateration: what is being measured is the difference in distance from the subject to two synchronized reference stations (called master and slave), placing the subject (using an unsynchronized clock) on a hyperbolic curve on a nautical chart. Two intersecting curve bundles, i.e., three reference stations, a master and two slaves, are needed minimally for successful positioning. Also the GPS satellite positioning system is based on hyperbolic positioning, but in three dimensions: four satellites (orbital "reference stations") are commonly sufficient for obtaining a fix. The unknowns solved for are, besides the positioned receiver's three coordinates, its clock offset (thus one can use the GPS system also for precise time dissemination!). Only when also integer-wavelength ambiguities are solved for in real time, is a fifth satellite (or more) welcome.
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