Enter a traverse as a list of bearings or azimuths with distances and this calculator computes the coordinates of every station, the sum of latitudes and departures, the linear misclosure and the relative precision. Close the traverse with a Bowditch (compass rule) adjustment, or switch to a least squares adjustment and get a standard error and an error ellipse for every adjusted station, plus a goodness of fit figure, alongside the corrected coordinates and the enclosed area.
Each line is bearing or azimuth, distance, with an optional station label in front:
N 67 22 40 E, 130.00 · 112.5, 138.65 ·
1-2, 112.5, 138.65. Leave the label out if you do not need one. Lines starting with
# are ignored.
Leave the closing coordinates blank for a closed loop that returns to the start station, or enter the known coordinates of the final station to check a link traverse.
Unadjusted and Bowditch-adjusted values for every leg. Distances and coordinate differences are in the unit selected above.
| Leg | Azimuth | Distance | ΔN | ΔE | Corr N | Corr E | Adj northing | Adj easting |
|---|
A traverse walks from one station to the next, each leg defined by a direction and a horizontal distance. Resolving each leg into a latitude (the north–south component, distance × cos azimuth) and a departure (the east–west component, distance × sin azimuth) turns the traverse into a chain of coordinate differences, and adding those differences cumulatively to the start coordinates gives the coordinates of every station.
In a closed traverse the chain should finish exactly where it started. It never does, because every angle and distance carries a small error, and the shortfall is the linear misclosure — the straight-line distance between the computed closing point and the known one. Expressing that misclosure as the ratio total length : misclosure gives the relative precision, which is the usual measure of how good the traverse is. A traverse of 400 m that misses its closing point by 40 mm has a precision of 1:10,000.
The Bowditch adjustment, also called the compass rule, distributes the misclosure across the legs in proportion to their length: each leg receives a latitude correction of −ΣΔN × length / total length, and a departure correction of −ΣΔE × length / total length. After adjustment the traverse closes exactly. Bowditch assumes the angular and distance errors are comparable and spread evenly, which makes it the default for most traverses; where the distance measurement is much more precise than the angular work, a transit rule or a least-squares adjustment is more appropriate.
Two cautions. The distances must be horizontal distances reduced to the same surface as the coordinates — slopes must be reduced, and grid distances carry a projection scale factor. And a good precision figure does not prove the traverse is correct: a systematic error such as a misread bearing can produce coordinates that close neatly while being metres from the truth.
The compass rule, better known as the Bowditch adjustment, is the standard way to close a traverse. It assumes the angular error and the distance error matter in proportion to the length of each leg, so the correction a leg receives is proportional to its length: its latitude correction is −ΣΔN × leg length ÷ perimeter and its departure correction is −ΣΔE × leg length ÷ perimeter. A compass rule calculator, which many references call a Bowditch adjustment calculator, therefore needs only three numbers to begin work — the sum of latitudes, the sum of departures and the perimeter of the traverse.
Worked example. Take a four-leg loop 521.02 m long that closes with ΣΔN = +0.038 m and ΣΔE = −0.021 m. For the 130.00 m leg the latitude correction is −0.038 × 130.00 ÷ 521.02 = −0.0095 m, and the departure correction is +0.021 × 130.00 ÷ 521.02 = +0.0052 m. Applied leg by leg the corrections add up to exactly −ΣΔN and −ΣΔE, so the adjusted traverse closes on its target station to the last decimal — which is why the coordinate table above shows the adjusted values closing exactly.
Bowditch is not the only choice. The transit rule spreads the misclosure in proportion to the size of each latitude and departure rather than to the leg length, which suits a traverse whose angles are much stronger than its distances; the Crandall method is a least-squares variant that weights the two kinds of observation separately. Bowditch remains the default because it needs nothing beyond the field distances and is the convention most specifications are written around.
Relative precision is the perimeter of the traverse divided by the linear misclosure, written as a ratio 1:N. A loop 500 m round that misses its closing point by 50 mm has a relative precision of 1:10,000, while the same 50 mm on a 100 m loop would be only 1:2,000. The closure ratio must therefore always be read together with the perimeter: a fixed 20 mm error is excellent on a 2 km traverse and poor on a 50 m one.
What counts as acceptable is set by the specification for the job rather than by a universal number, but the figures quoted most often in practice are 1:5,000 for lower-order work, 1:10,000 for a good total station traverse and 1:20,000 or better for precise control. Two habits matter more than the number itself. Compare the traverse precision ratio with the accuracy the project actually requires rather than with a rule of thumb, and remember that a good ratio proves only that the errors cancel — one misread bearing can still give coordinates that close neatly while being metres from the truth.
Once a traverse has been adjusted, the area of a closed traverse follows from the corrected coordinates alone, with no need to return to the bearings and distances. The coordinate method, also called the shoelace formula, accumulates the cross products of consecutive station coordinates around the figure and halves the absolute total. The older double meridian distance (DMD) method reaches the same answer by pairing each leg's departure with the meridian distance to the midpoint of that leg, and is still taught because the whole table can be worked by hand without a calculator.
Both methods return the plane area enclosed by the adjusted traverse. On a small site held in ground coordinates the result is the true area to better than one part in 10,000; if the coordinates are grid coordinates, the projection scale factor has to be applied before the area is taken, because an area scales by the square of that factor. The enclosed area for the legs you enter is reported with the other results above, in square metres and hectares, or in square feet and acres when the unit is set to feet. For a boundary held as latitude and longitude rather than as a traverse, the land area calculator will calculate acres from coordinates in a single step.
A traverse fixes position in plan; levelling fixes it in height, and the arithmetic of a level book is close enough to a traverse to set out beside it. A differential leveling calculator works from the readings taken at each instrument set-up: a backsight on a point of known height, a foresight on the next point whose height is wanted, and intermediate sights on any further points in between. A backsight foresight calculator then reduces the book by one of two classic methods.
The height of collimation method adds the backsight to the reduced level of the known point to give the collimation, or height of instrument, then subtracts every foresight and intermediate sight from that figure. The rise and fall method calculator works instead on the difference between successive readings: a backsight larger than the following foresight is a rise, a smaller one is a fall, and the rises and falls are accumulated from the first reduced level. Both methods give identical reduced levels, and both are proved by the same arithmetic check — the sum of backsights minus the sum of foresights must equal the last reduced level minus the first, and in the rise and fall method it must also equal the sum of rises minus the sum of falls.
A level run that closes on a bench mark will carry a small misclosure to distribute, and the permissible value is normally written as a multiple of the square root of the distance levelled, for example ±12 mm √km for ordinary engineering work. This page computes traverses rather than level networks, so the formulas above are given for reducing a level book alongside the traverse results. For a single slope rather than a level run, the slope percentage calculator turns a rise and run into percent, ratio or degrees.
The compass rule spreads the misclosure out by a fixed recipe: every leg is corrected in proportion to its length, whatever the quality of the individual measurements. A least squares adjustment calculator does something more careful. It asks which set of station coordinates comes closest to all the observed directions and distances at once, weighting each observation by how precise it is. A line turned with a 5′′ instrument over 60 m carries more weight than a 300 m line shot through a heat haze, so the adjustment is free to push the misclosure towards the weaker observation instead of sharing it out evenly.
The unknowns are the corrections to the northing and easting of every station that is not held fixed, so a four-leg loop with one fixed station has six unknowns. Each observed azimuth and each observed horizontal distance gives one observation equation relating a small change in the coordinates to a small change in the measured quantity: a distance has the partial derivatives ±cos and ±sin, while an azimuth has ±ΔE / L² and ±ΔN / L². Collecting those into matrices and solving the weighted normal equations gives the corrections, and because the equations are non-linear the whole calculation is repeated with the corrected coordinates until they stop moving — three or four passes for a normal traverse.
In that expression A holds the partial derivatives, P is the weight matrix — one over the variance of each observation — and l is the vector of observed-minus-computed values. This is the variation of coordinates form of a least squares traverse adjustment, and the classic Crandall method is the special case of it where the angular and distance weights are chosen for a simple traverse.
Everything the adjustment does depends on the weights, which is why three numbers are needed before it starts. The angle precision is the standard deviation you expect from a direction, in arcseconds — 5′′ for a routine total station, 2′′ or 1′′ for a better one. The distance precision is entered as a constant part in millimetres plus a scale part in parts per million, which is how manufacturers state it: 5 mm + 5 ppm is a normal electronic distance measurement. Get these roughly right and the adjustment behaves well; get them wildly wrong and a good traverse can be pulled about by an observation that was never as reliable as it was declared to be.
Three things appear that the compass rule cannot produce. The unit weight standard error σ₀ is a goodness of fit test: it compares the scatter actually found with the precision that was assumed, so a value near 1 means the field work matched expectations, while a value well above 1 warns that the instrument was worse than assumed or that a blunder is hiding in the book. The coordinate standard errors and the error ellipse attach a precision to every adjusted station, which is what a specification normally asks for. And the residual list shows how far each individual direction and distance had to move, so an observation that disagrees with all the others is at least visible rather than averaged away.
Least squares is not a magic wand. It cannot repair a systematic error — a mis-set prism constant or a wrongly reduced slope distance will be absorbed quietly and leave good-looking statistics behind. It cannot reliably identify a blunder either: with only two redundant observations in a simple closed traverse the residual pattern is smeared across several legs, so a large residual is a prompt to re-observe rather than proof of which leg is wrong. And it cannot create redundancy that the field work never had. A traverse measured as a single chain and closed on itself has exactly two degrees of freedom however many legs it contains, so the standard errors describe the geometry that was chosen, not every mistake that might have been made. On a traverse that closes well the least squares coordinates usually land within a millimetre or two of the Bowditch ones; the reason to prefer the method is the precision figures and the goodness of fit that come with it, not a dramatic change in the answer.
Traverse computation is the backbone of engineering survey, and this tool covers the checks that are performed on almost every job:
For each leg, multiply the horizontal distance by the cosine of its azimuth to get the latitude and by the sine of the azimuth to get the departure. Add the latitudes and departures cumulatively to the start coordinates to find each station in turn. The sums of latitude and departure for the whole traverse give the misclosure.
It depends entirely on the specification for the job, not on a universal number. A precision of 1:10,000 or better is regarded as good for a total station traverse, and 1:5,000 is commonly acceptable for lower-order work. The tool labels the result so you can see the magnitude, but always compare against the accuracy your project actually requires rather than a rule of thumb.
The Bowditch adjustment, or compass rule, spreads the closure error over the traverse in proportion to each leg's length. Each leg gets a latitude correction of minus the sum of latitudes times the leg length divided by the total length, and similarly for departures. After correction the traverse closes exactly. It assumes angular and distance errors are of similar importance and spread evenly through the work.
Horizontal. A traverse computed on slope distances will not close, because coordinates are a plane representation of horizontal positions. Reduce every slope distance to horizontal before entering it here, and if your coordinates are grid coordinates rather than ground coordinates, apply the projection scale factor first.
Yes. Choose the unit in the input row and the whole calculation is carried out in that unit; the coordinate differences and the misclosure are reported in the same unit. No conversion is applied internally, so do not mix metres and feet within one traverse.
A perfect closure gives a misclosure of zero, an infinite relative precision, and no corrections at all. That is the norm for a traverse computed from already-adjusted data, and the tool reports it without dividing by zero.
The compass rule, or Bowditch, distributes the misclosure in proportion to the length of each leg; the transit rule distributes it in proportion to the size of each latitude and departure. Bowditch suits work where the angular and distance errors are of similar standing, the transit rule work where the angles are much stronger than the distances. Both close the traverse exactly, and neither claims to remove a blunder.
Reduce the level book by the height of collimation or the rise and fall method set out above, then apply the arithmetic check: the sum of backsights minus the sum of foresights must equal the last reduced level minus the first. The traverse table on this page is computed independently of that working, so the two sets of figures can be checked side by side.
It is the adjustment that finds the station coordinates which come closest to all the observed directions and distances at once, by minimising the sum of the squared residuals with each observation weighted by its own precision. A sharp angle or a short, well measured line is trusted more than a weak one, so the misclosure is distributed according to how good each measurement actually is rather than by a fixed proportion such as the compass rule. The calculation is repeated until the coordinates stop moving, because the observation equations are non-linear.
It is more correct rather than dramatically different. On a traverse that closes well the two sets of adjusted coordinates usually agree to within a millimetre or two, because both are distributing the same small misclosure. Least squares becomes clearly better when the angles and the distances are of very different quality, when traverses are linked into a network, or when you need the by-products: a standard error for every adjusted coordinate, an error ellipse, and a goodness of fit test that shows whether the field work matched the precision you assumed.
It compares the scatter the adjustment actually found with the precision you entered in advance. A value near 1 means the observations scattered about the assumed standard deviations as expected; much larger than 1 means the work was weaker than assumed or a blunder is present; well below 1 means the assumed precision was pessimistic or the observations were over-weighted. Multiply the reported coordinate standard errors by it to get the realistic values.
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