Standing at B, you want to know your location relative to the reference points P1, P2, and P3. Measuring r1 narrows your position down to a circle. Next, measuring r2 narrows it down to two points, A and B. A third measurement, r3, gives your coordinates at B. A fourth measurement could also be made to reduce error.

Trilateration is a method of determining the relative positions 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. Image File history File links Trilateration diagram File links The following pages link to this file: Trilateration ... Image File history File links Trilateration diagram File links The following pages link to this file: Trilateration ... Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ... A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. ... Triangulation can be used to find the distance from the shore to the ship. ... This article is about angles in geometry. ... The distance between two points is the length of a straight line segment between them. ... A 2D geometric model is a geometric model of an object as two-dimensional figure, usually on the Euclidean or Cartesian plane. ...

Trilateration should not be confused with multilateration or hyperbolic positioning, which use measurements of time difference of arrival, rather than time of arrival, to estimate location using the intersection of hyperboloids. Multilateration, also known as hyperbolic positioning, is the process of locating an object by accurately computing the time difference of arrival (TDOA) of a signal emitted from the object to three or more receivers. ... Multilateration, also known as hyperbolic positioning, is the process of locating an object by accurately computing the time difference of arrival (TDOA) of a signal emitted from the object to three or more receivers. ...

Derivation

A mathematical derivation for the solution of a three-dimensional trilateration problem can be found by taking the formulae for three spheres and setting them equal to each other. To do this, we must apply three constraints to the centers of these spheres; all three must be on the z=0 plane, one must be on the origin, and one other must be on the x-axis. It is, however, possible to transform any set of three points to comply with these constraints, find the solution point, and then reverse the transformation to find the solution point in the original coordinate system.

Starting with three spheres,

,

and

,

we subtract the second from the first and solve for x:

.

Substituting this back into the formula for the first sphere produces the formula for a circle, the solution to the intersection of the first two spheres:

.

Setting this formula equal to the formula for the third sphere finds:

.

Now that we have the x- and y-coordinates of the solution point, we can simply rearrange the formula for the first sphere to find the z-coordinate:

Now we have the solutions to all three points x, y and z. Because z is expressed as a square root, it is possible for there to be zero, one or two solutions to the problem.

This last part can be visualized as taking the circle found from intersecting the first and second sphere and intersecting that with the third sphere. If that circle falls entirely outside of the sphere, z is equal to the square root of a negative number: no real solution exists. If that circle touches the sphere on exactly one point, z is equal to zero. If that circle touches the surface of the sphere at two points, then z is equal to plus or minus the square root of a positive number. In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number. ...

In the case of no solution, a not uncommon one when using noisy data, the nearest solution is zero. One should be careful, though, to do a sanity check and assume zero only when the error is appropriately small. A sanity test or smoke test is a brief run-through of the main functionality of a computer program or other product. ...

In the case of two solutions, some technique must be used to disambiguate between the two. This can be done mathematically, by using a fourth sphere and determining which point lies closest to its logically- for example, GPS systems assume that the point that lies inside the orbit of the satellites is the correct one when faced with this ambiguity, because it is generally safe to assume that the user is never in space, outside the satellites' orbits.

Error model

When measurement error is introduced into the picture, things become a little more complicated. If we know that the distance from P to a reference point lies in a range (a closed interval) [r₁, r₂], then we know that P lies in a circular band between the circles of those two radii. If we know a range for another point, we can take the intersection, which will be either one or two areas bounded by circular arcs. A third point will usually narrow it down to a single area, but this area may still be of significant size; additional reference points can help shrink it further, but as the area shrinks more measurements quickly become less useful. In three dimensions, we are instead intersecting spherical shells with thickness, similar to bowling balls. In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...

This new model emphasizes the importance of choosing three points that are in very different directions — if the points are relatively close together and all far from the point being located, it will take very precise measurement to find the point using trilateration.

See also

• Multilateration - position estimation using measurements of time difference of arrival at (or from) three or more sites.