Trilateration

Trilateration is the geometric method of determining the position of a point in space by measuring its distance from known reference points. Unlike triangulation, which uses angles, trilateration uses only range measurements — typically the time-of-flight of signals from multiple sources. It is the foundational positioning principle of GPS, GLONASS, and all satellite navigation systems: a receiver solves for its position as the intersection of spheres centered on each satellite, with radii equal to the measured signal propagation delays multiplied by the speed of light.

The mathematical problem reduces to solving a system of nonlinear equations. With four satellites, the system is overdetermined and can be solved via least-squares minimization, yielding position in three dimensions plus receiver clock bias. The geometry of the satellite constellation — the spatial distribution of the visible satellites — determines the dilution of precision: poor geometry (satellites clustered together) amplifies ranging errors into large position errors, while good geometry (satellites spread across the sky) suppresses them.

Trilateration is not limited to satellite navigation. It is used in acoustic positioning for underwater vehicles, in wireless sensor networks for indoor localization, and in multilateration systems like aircraft secondary surveillance radar. The underlying mathematics — distance geometry and the study of Euclidean distance matrices — connects trilateration to dimensionality reduction, molecular geometry, and the problem of reconstructing configurations from partial distance information.