# Trilateration

Trilateration is the use of distances (or "ranges") for determining the unknown position coordinates of a point of interest, often around Earth (geopositioning). When more than three distances are involved, it may be called multilateration, for emphasis.
The distances or ranges might be ordinary Euclidean distances (slant ranges) or spherical distances (scaled central angles), as in true-range multilateration; or biased distances (pseudo-ranges), as in pseudo-range multilateration.
Trilateration or multilateration should not be confused with triangulation, which uses angles for positioning; and direction finding, which determines the line of sight direction to a target without determining the radial distance.

## Terminology

Multiple, sometimes overlapping and conflicting terms are employed for similar concepts – e.g., multilateration without modification has been used for aviation systems employing both true-ranges and pseudo-ranges. Moreover, different fields of endeavor may employ different terms. In geometry, trilateration is defined as the process of determining absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. In surveying, trilateration is a specific technique.

## True-range multilateration

True-range multilateration (also termed range-range multilateration and spherical multilateration) is a method to determine the location of a movable vehicle or stationary point in space using multiple ranges (distances) between the vehicle/point and multiple spatially-separated known locations (often termed "stations"). Energy waves may be involved in determining range, but are not required.
True-range multilateration is both a mathematical topic and an applied technique used in several fields. A practical application involving a fixed location occurs in surveying. Applications involving vehicle location are termed navigation when on-board persons/equipment are informed of its location, and are termed surveillance when off-vehicle entities are informed of the vehicle's location.
Two slant ranges from two known locations can be used to locate a third point in a two-dimensional Cartesian space (plane), which is a frequently applied technique (e.g., in surveying). Similarly, two spherical ranges can be used to locate a point on a sphere, which is a fundamental concept of the ancient discipline of celestial navigation — termed the altitude intercept problem. Moreover, if more than the minimum number of ranges are available, it is good practice to utilize those as well. This article addresses the general issue of position determination using multiple ranges.
In two-dimensional geometry, it is known that if a point lies on two circles, then the circle centers and the two radii provide sufficient information to narrow the possible locations down to two – one of which is the desired solution and the other is an ambiguous solution. Additional information often narrow the possibilities down to a unique location. In three-dimensional geometry, when it is known that a point lies on the surfaces of three spheres, then the centers of the three spheres along with their radii also provide sufficient information to narrow the possible locations down to no more than two (unless the centers lie on a straight line).

## Pseudo-range multilateration

Pseudo-range multilateration, often simply multilateration (MLAT) when in context, is a technique for determining the position of an unknown point, such as a vehicle, based on measurement of the times of arrival (TOAs) of energy waves traveling between the unknown point and multiple stations at known locations. When the waves are transmitted by the vehicle, MLAT is used for surveillance; when the waves are transmitted by the stations, MLAT is used for navigation (hyperbolic navigation). In either case, the stations' clocks are assumed synchronized but the vehicle's clock is not.
Prior to computing a solution, the common time of transmission (TOT) of the waves is unknown to the receiver(s), either on the vehicle (one receiver, navigation) or at the stations (multiple receivers, surveillance). Consequently, also unknown is the wave times of flight (TOFs) – the ranges of the vehicle from the stations divided by the wave propagation speed. Each pseudo-range is the corresponding TOA multiplied by the propagation speed with the same arbitrary constant added (representing the unknown TOT).
In navigation applications, the vehicle is often termed the "user"; in surveillance applications, the vehicle may be termed the "target". For a mathematically exact solution, the ranges must not change during the period the signals are received (between first and last to arrive at a receiver). Thus, for navigation, an exact solution requires a stationary vehicle; however, multilateration is often applied to the navigation of moving vehicles whose speed is much less than the wave propagation speed.
If $d$ is the number of physical dimensions being considered (thus, vehicle coordinates sought) and $m$ is the number of signals received (thus, TOAs measured), it is required that $m\geq d+1$. Then, the fundamental set of $m$ measurement equations is:
    TOAs ($m$ measurements) = TOFs ($d$ unknown variables embedded in $m$ expressions) + TOT (one unknown variable replicated $m$ times).
Processing is usually required to extract the TOAs or their differences from the received signals, and an algorithm is usually required to solve this set of equations. An algorithm either: (a) determines numerical values for the TOT (for the receiver(s) clock) and $d$ vehicle coordinates; or (b) ignores the TOT and forms $m-1$ (at least $d$) time difference of arrivals (TDOAs), which are used to find the $d$ vehicle coordinates. Almost always, $d=2$ (e.g., a plane or the surface of a sphere) or $d=3$ (e.g., the real physical world). Systems that form TDOAs are also called hyperbolic systems, for reasons discussed below.
A multilateration navigation system provides vehicle position information to an entity "on" the vehicle (e.g., aircraft pilot or GPS receiver operator). A multilateration surveillance system provides vehicle position to an entity "not on" the vehicle (e.g., air traffic controller or cell phone provider). By the reciprocity principle, any method that can be used for navigation can also be used for surveillance, and vice versa (the same information is involved).