Cross Track Error

Cross track error is the perpendicular distance from the line you meant to be on to the position you are actually at. It is the number behind the deviation bar on every GPS and the course deviation indicator in every aircraft, and it is the single most useful piece of information a plotter produces, because it answers the only question that matters while you are steering: which way, and how far, to correct.

The spherical form used here takes the distance from the start of the track to your position and the difference between two bearings — the bearing along the intended track, and the bearing from the track's start to where you are. Over short distances it reduces to the flat-earth answer, distance times the sine of the angle off. Over long ones the sphere trims it slightly, because the perpendicular from a point to a great circle is itself an arc rather than a straight line.

The sign is the part to get right and the part that is not standardised. In this shard a POSITIVE cross track error means you are to the RIGHT of the intended track, looking along it toward the destination, and a negative one means you are to the left. Some marine plotters use the same convention. Some avionics, and some ARINC-labelled data, report the CORRECTION to steer instead of the error — the same magnitude with the opposite sign — and some instruments are deliberately built to show the deviation bar as the runway or track sitting off to one side, so that you "fly toward the needle" regardless of the underlying sign. Before you trust any of them, put the boat or the aeroplane knowingly to one side of a track and look at what the display does.

Two further cautions. Cross track error tells you nothing about how far along the leg you are, which is why the along-track distance is reported beside it; a large error at the very start of a leg and the same error at the very end are quite different situations. And a cross track error that grows steadily on one side is almost never a steering problem. It is a current or a wind, and the fix is a corrected course to hold rather than a series of turns back toward the line, each of which is undone by the same set within the hour.