Trilateration

Trilateration 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.

Number of reference points

The reason that three points are required 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 almost always intersect at 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. Two points almost always narrow it down to a circle, and three points to two points. 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

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.

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 a 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,

<math>r_1^2=x^2+y^2+z^2<math>,

<math>r_2^2=(x-d)^2+y^2+z^2<math>, and

<math>r_3^2=(x-i)^2+(y-j)^2+z^2<math>,

we can set the first two equal to each other to find

<math>x=\frac{r_1^2-r_2^2+d^2}{2d}<math>.

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:

<math>y^2+z^2=r_1^2-\frac{(r_1^2-r_2^2+d^2)^2}{4d^2}<math>.

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

<math>y=\frac{r_1^2-r_3^2+(x-i)^2}{2j}+\frac{j}{2}-\frac{(r_1^2-r_2^2+d^2)^2}{4d^2j}<math>.

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:

<math>z=\sqrt{r_1^2-x^2-y^2}<math>

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.