Latitude Longitude Distance and Initial Bearing

Distance between coordinates

A great circle is the intersection of a sphere with a plane through its centre. The shorter arc joining two positions is the shortest path on the sphere. The central angle in radians is multiplied by the radius you choose. On a sphere of radius 2 km, a 90 degree arc along the equator has length π km.

Reading the bearing

The initial bearing is measured clockwise from true north. East is 90 degrees, south is 180 and west is 270. This direction changes as you follow the route. Antipodal points have multiple equally short paths, so their bearing is undefined. Earth is ellipsoidal, making this spherical calculation an approximation. Use an appropriate datum and geodesic method for precise surveying, navigation or setting out.

Checking coordinates and the model

Check coordinate order in the source data. Some files place longitude first. Here, north and east are positive, south and west negative. Do not read degrees, minutes and seconds as a single decimal degree number. Convert first and ensure both points use the same reference system.

For a geometry check, choose an illustrative sphere of radius 10 km. Start at zero latitude and longitude, and end at zero latitude and longitude 60°. Both points lie on the equator. The central angle is 60° = π/3 radians and arc length is 10π/3 ≈ 10.471976 km. The initial bearing is east, or 90°. This radius is for the example, not an Earth radius.

Changing radius scales distance proportionally without changing the central angle. The result is the shorter surface arc, not a straight distance through the sphere, flight profile or road distance.

Bearing refers to true north. Magnetic declination needed for a compass is not calculated. Reversing the endpoints does not generally mean merely adding half a turn to the original initial bearing. An indeterminate bearing at coincident or antipodal points is not an instrument fault. Complete this preliminary calculation with an appropriate geodesic method before navigation decisions.