A geometric method for determining absolute or relative locations by measuring distances from three or more reference points.

Trilateration

Trilateration is a fundamental geometric technique used to determine the precise location of a point by measuring its distance from three or more known reference points. Unlike triangulation, which uses angles, trilateration relies purely on distance measurements.

Basic Principles

The core concept of trilateration works by creating spheres (in 3D) or circles (in 2D) of possible positions around each reference point:

First measurement creates a sphere of possible locations
Second measurement narrows it to a circle
Third measurement typically resolves to two points
Additional measurements can eliminate remaining ambiguity

Applications

Global Navigation

GPS systems rely heavily on trilateration
Satellites serve as reference points
Signal timing determines distances
Requires minimum four satellites for accurate positioning

Indoor Positioning

Scientific Applications

Seismology for earthquake epicenter location
Radio astronomy for source positioning
Acoustic positioning systems

Mathematical Foundation

The mathematical basis involves solving systems of equations representing spheres:

(x - x₁)² + (y - y₁)² + (z - z₁)² = r₁²
(x - x₂)² + (y - y₂)² + (z - z₂)² = r₂²
(x - x₃)² + (y - y₃)² + (z - z₃)² = r₃²

Where (x,y,z) is the unknown point, (xᵢ,yᵢ,zᵢ) are reference points, and rᵢ are measured distances.

Error Sources and Mitigation

Common sources of error include:

Signal propagation delays
Multipath interference
Measurement noise
Geometric dilution of precision

Historical Development

The concept of trilateration has evolved from:

Ancient surveying techniques
Development of radio navigation systems
Modern satellite-based positioning

Related Techniques

Multilateration: Using time differences rather than absolute distances
Hyperbolic navigation: Based on constant difference of distances
Dead reckoning: Alternative navigation method

Future Directions

Emerging applications include:

Autonomous vehicle positioning
Augmented reality systems
Internet of Things device localization
Indoor navigation solutions

The fundamental principles of trilateration continue to find new applications as technology evolves, particularly in contexts requiring precise positioning without direct line of sight to reference points.