Three Bearing Fix

A three‑bearing fix determines a vessel’s position by measuring the true bearings to three known reference points (landmarks, lights, or radio stations). Each bearing defines a line of position (LOP) that passes through the unknown vessel and the known point. The intersection of the three LOPs gives the most probable location, assuming the bearings are taken simultaneously and the reference coordinates are accurate.

Geometrically each LOP can be expressed as a linear equation in a planar (equirectangular) projection: ((x – x_0)sintheta – (y – y_0)costheta = 0), where ((x_0, y_0)) are the projected coordinates of the known point and (theta) is the bearing measured clockwise from true north. Converting latitude/longitude to (x, y) using a mean latitude preserves distances over the small area typically involved in coastal navigation.

The fix is obtained by solving the simultaneous equations for the three lines. In practice the intersection of the first two lines is calculated, and the third line is used to verify or adjust the solution (e.g., by minimizing the perpendicular distance to the third LOP). The resulting latitude and longitude are then transformed back to geographic coordinates and reported to the navigator.