trilateration

Trilateration is a method for determining the position of a point by measuring its distances to a set of known reference points. In two dimensions, at least three reference points are required; in three dimensions, at least four. Each reference point defines a circle (2D) or a sphere (3D) of possible locations with radius equal to the measured distance. The position is found where these geometric loci intersect. Because real measurements contain errors, the circles or spheres may not intersect at a single point, and the solution is typically obtained by solving the corresponding equations in a least-squares sense.

Mathematically, for known reference points (xi, yi) with distances ri to an unknown (x, y), the equations are (x − xi)² + (y − yi)² = ri². Subtracting equations for pairs of references yields linear equations in x and y, which can be solved. In three dimensions, the same idea uses four spheres: (x − xi)² + (y − yi)² + (z − zi)² = ri², and the resulting linear system provides the coordinates (x, y, z).

Applications include global positioning systems (GPS), where distances are inferred from time-of-flight measurements and may involve an unknown receiver clock bias. Trilateration is also used in cellular localization and Wi‑Fi-based positioning, where signal strengths or timings provide the range measurements.

Limitations include sensitivity to measurement noise, poor geometric arrangement (dilution of precision), and environmental factors such as multipath and obstructions, all of which can reduce accuracy.