Along-Track Distance Calculator
Along-track distance is the distance from the start of a great-circle path to the point on the path nearest a third point. Navigators pair it with cross-track distance to express how far along a planned route a target lies and how far off the route it sits. Enter the path start coordinates, the path initial bearing, the third point coordinates and the Earth radius to get the along-track distance, the cross-track distance and the great-circle distance to the point.
Along-track distance formula
d13 = great-circle distance from path start to point
d_xt = cross-track distance
along-track = acos( cos(d13 / R) / cos(d_xt / R) ) * R
The cross-track distance is found first from the haversine angular distance and the bearing difference. The along-track distance is then the great-circle arc on the path from its start to the perpendicular foot, recovered from the right spherical triangle formed by d13, d_xt and d_at.
Worked example
Path start (51.5 N, -0.1 E), path bearing 90 degrees, point (51.6 N, 2.0 E), R = 6,371.0088 km. The distance to the point is about 146.07 km and the cross-track offset is near -34 km. The along-track distance, the distance along the due-east path to the nearest point, is approximately 142.0 km.
Along-track distance: frequently asked questions
What is along-track distance?
Along-track distance is the distance measured from the start of a great-circle path to the point on that path closest to a given third point (the foot of the perpendicular). Together with cross-track distance it locates a point relative to an intended track.
What formula is used?
On a sphere, d_at = acos( cos(d13 / R) / cos(d_xt / R) ) * R, where d13 is the great-circle distance from the path start to the third point, d_xt is the cross-track distance, and R is the Earth radius. This is the standard companion to the cross-track formula in Ed Williams' Aviation Formulary.
Which Earth radius should I use?
This calculator defaults to the IUGG mean Earth radius of 6,371.0088 km, which is editable. A spherical model is accurate to within a fraction of a percent for navigation distances.
Can along-track distance be negative?
Yes. If the closest point on the path lies behind the start point (the third point is reached by extending the path backwards), the along-track distance is negative.
Sources and references
- National Geospatial-Intelligence Agency: Geodesy and geophysics reference.
- National Oceanic and Atmospheric Administration, National Geodetic Survey: geodesy fundamentals.
- Formula: standard spherical great-circle trigonometry (Ed Williams, Aviation Formulary).
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.