Along-Track Distance (the Companion of Cross-Track Error)

Once an aircraft or a ship is off its track, two numbers describe where it is: how far off the track, the cross-track distance $d_{XT}$, and how far along it, the along-track distance $d_{AT}$. The second is measured along the great circle from the start of the leg to the point abeam of the present position, and it says how much of the leg is behind you. Together with the distance $d_{AD}$ from the start to the present position they form a right spherical triangle: $d_{AT}$ and $d_{XT}$ are the legs and $d_{AD}$ is the hypotenuse.

On a sphere the distances are tied together through their cosines, after each is divided by the Earth radius $R$ to make it an angle at the centre of the Earth. That gives $\cos(d_{AD}/R) = \cos(d_{AT}/R)\cos(d_{XT}/R)$, and the formula is this relation solved for the along-track leg. The cross-track distance enters as a size only, so it makes no difference which side of the track you are on. When the off-track distance is small beside the distance flown, as in the example below, the along-track distance is only a little shorter than the distance from the start.

The relation is from Ed Williams' Aviation Formulary V1.47, which also gives an arcsine form for very short distances. Its worked example has an aircraft on its way from Los Angeles to New York at N34:30 W116:30, 99.87 nautical miles from the start and 7.45 nautical miles to the right of the track: it has made good 99.59 nautical miles along the track. In radians, 0.02905 from the start and 0.00216747 off give 0.0289691 along.

The page works in lengths with the mean Earth radius of 6 371 008.8 m, so its nautical miles differ from the Formulary's by 0.07 percent, because the Formulary counts one nautical mile for each minute of arc. Entering the example as 185.0778 km from the start and 13.80897 km off gives 184.5621 km along, which is 99.6555 nautical miles. Pair this page with the cross-track error page, which gives the off-track leg from the two bearings, and with the haversine page, which supplies the distance from the start.