Traverse closure and compass-rule adjustment

A closed traverse has to come back to where it started. It never quite does, and the size of the gap is the honest measure of the work. This calculator takes the courses as they come off a field book or out of a deed, computes the latitudes and departures, tells you how far the traverse missed by, and distributes that miss so the figure closes.

What the calculator does, step by step

1. Reads each course. A direction is either a quadrant bearing (N 26-10-00 E) or an azimuth clockwise from north (26-10-00). A line that cannot be read is reported by line number rather than skipped.
2. Resolves each course into its latitude — the north–south component, D·cos α — and its departure, the east–west component, D·sin α.
3. Adds the latitudes and adds the departures. On a perfectly closed traverse both sums are zero; what they actually come to are the closures in latitude and departure.
4. Combines them into the linear misclosure, e = √(ΣLat² + ΣDep²), and expresses it as the ratio 1 : (ΣD / e).
5. Distributes the misclosure by the chosen rule, then runs the adjusted latitudes and departures around the figure to get closed coordinates.
6. Computes the enclosed area from those adjusted coordinates.

Latitude and departure

The azimuth here is measured clockwise from north, which is the plane-surveying convention and not the mathematical one. That single fact decides which trigonometric function goes with which component, and getting it backwards is the most common error in hand-rolled coordinate geometry.

The check is the cardinal directions. On an azimuth of 90° the course runs due east: its latitude must be zero and its departure the whole distance. If your formula gives that backwards, you have the functions swapped.

Misclosure and precision

The two closures are the components of a single error vector. Its length is the linear misclosure, and its direction — reported here as the error azimuth — points from where the traverse actually ended back to where it should have ended.

The compass (Bowditch) rule

The compass rule assumes that angles and distances were observed with comparable care, so error accumulates with the length of a course. Each course therefore takes a share of the correction proportional to its own length.

That assumption holds for an ordinary taped or EDM traverse run with a theodolite or a total station, which is why the compass rule is the default here and the usual choice in practice.

The transit rule, and when to prefer it

The transit rule distributes each closure in proportion to the size of the component itself rather than to the length of the course. It is the right choice only when the angular work is markedly better than the linear work — an unusual situation with modern instruments, where the distances are usually the stronger observation.

Switch between the two rules on a traverse of your own and watch what happens to a course that runs nearly due north. The compass rule gives it a full share of the departure correction because it is a long course; the transit rule gives it almost none, because its departure is almost nothing. Neither answer is wrong — they encode different beliefs about where the error came from.

What to do when a traverse fails

Adjusting a traverse that misses by more than the specification allows does not fix it — it spreads a blunder evenly over work that was fine. Before adjusting, check the angular closure against (n − 2)·180° for the interior angles. If the angles close and the traverse does not, the problem is a distance.

• A misclosure whose direction matches one course's bearing points at that course's distance — a transposed digit, or a tape length missed.
• A misclosure roughly perpendicular to a course points at that course's direction — an angle blunder at one end of it.
• A misclosure of about twice a single course length usually means a course was entered with its back bearing instead of its forward one.
• A misclosure that shrinks to nothing when one course is removed means the blunder is in that course, and no adjustment rule will find it for you.

The area that comes with it

The area reported here is computed from the ADJUSTED coordinates, which is the only defensible order of operations: an area computed from unadjusted coordinates belongs to a figure that does not close, and is therefore an area of nothing in particular. If you want to see the same parcel worked by the double-meridian-distance method as well, paste the adjusted coordinates into the area calculator.