Latitude and departure calculator

A course is a direction and a distance, which is a perfectly good way to describe a line and a useless way to add two of them together. Latitudes and departures are the fix: they turn every course into a pair of numbers that lie along fixed axes, and numbers along a fixed axis can simply be added. Every traverse computation, every coordinate, and every closure figure in plane surveying rests on that one move.

What a latitude and a departure are

The latitude of a course is its north–south component: how far the line carries you along the meridian. The departure is its east–west component: how far it carries you across the meridian. Together they are the course resolved onto the two coordinate axes, with the latitude going into the northing and the departure into the easting.

Both are signed. A latitude is positive north and negative south; a departure is positive east and negative west. The signs are the whole point — without them the components could not be added, and adding them is the only reason to compute them at all.

The formulas, and why cosine goes with the northing

In ordinary mathematics an angle is measured counter-clockwise from the positive x axis, and the x component takes the cosine. Plane surveying measures the azimuth clockwise from north, and north is the FIRST coordinate. The reference axis has moved, so the function that pairs with it moves too: cosine now belongs to the northing and sine to the easting. Nothing deep is happening — the same trigonometry is being read against a different axis — but the swap is invisible in the formula and catastrophic in the answer.

The check that settles it is the cardinal directions, and it takes five seconds. Put in an azimuth of 90°, which is due east. The latitude must come out zero and the departure must come out the whole distance. Cosine of 90° is zero and sine of 90° is one, so the formulas above give exactly that. Try 0°, due north, and the components swap. If your spreadsheet disagrees on either case, the functions are transposed, and every latitude in the column is a departure.

Signs by quadrant

The two signs follow from the quadrant, and the quadrant follows from the azimuth. When directions are entered as quadrant bearings the letters state the signs outright, which is one of the reasons bearings survive in legal descriptions.

The pattern is that the meridian letter gives the sign of the latitude and the departure letter gives the sign of the departure, which is exactly what the two letters are for. Reading a bearing this way makes the sign table redundant, which is the best kind of table.

How the totals become a closure

Walk a closed traverse and you return to where you started. Every step north is eventually undone by a step south, and every step east by a step west, so the latitudes must sum to zero and so must the departures. That is not an approximation or a convention; it is what closing means.

Real observations do not sum to zero. What the two columns come to are the closure in latitude and the closure in departure, and they are the two components of a single error vector — the gap between where the traverse ended and where it should have ended.

The direction of that error vector is diagnostic, which is why the running table on this page stops at the totals rather than adjusting them. A misclosure whose azimuth matches one course points at that course, and at its distance in particular. A misclosure at right angles to a course points at that course, and at its direction. Distributing the error before looking at it throws that information away, and spreads a single blunder evenly over five courses that were fine.

Once the closure has been examined and accepted, the compass rule or the transit rule distributes it, and the adjusted components run around the figure to produce coordinates. That is the traverse closure calculator, and these are the numbers it starts from.

Rebuilding a course from its components

The reverse operation turns a pair of components back into a direction and a distance, and it is the same computation as an inverse between two coordinate pairs, because a latitude and a departure are just the deltas of an inverse written without the coordinates around them.

The argument order deserves the emphasis. The mathematical atan2 takes the y argument first and returns a counter-clockwise angle from the positive x axis. Feeding it the departure first and the latitude second returns a clockwise angle from north, which is the azimuth. Getting this backwards produces a direction that looks entirely plausible, is wrong, and is wrong in a way that a single spot check will not always reveal — a course at 45° gives the same answer either way.

This latitude is not geographic latitude

This is a genuine collision of terms and it is worth naming plainly. Geographic latitude is a position on the earth: an angle, measured in degrees north or south of the equator, that says where you are. The latitude of a course is a length: a distance in feet or meters, along the north–south axis of a local plane coordinate system, that says how far one line carries you. They share a name, they share no units, and they answer different questions. The word survives in the traverse sense because it is old — it goes back to the era when the north–south direction was simply called the latitude direction — and because every computation sheet, textbook and state survey manual uses it that way. A departure is the corresponding east–west length, and it has no geographic homonym at all, which is why nobody ever asks about that one.