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Survey traverse closure

latitude and departureclosure errorprecision ratioback azimuthtraverse adjustment

Closing a traverse: latitudes and departures per leg, the misclosure they expose, and the precision ratio that decides whether it passes.

Latitude of a Traverse Leg

Lat=Lcos⁡α\text{Lat} = L\cos\alpha

North-south component of a traverse course, from the measured slope-corrected distance and the azimuth of the line.

Departure of a Traverse Leg

Dep=Lsin⁡α\text{Dep} = L\sin\alpha

East-west component of a traverse course, taken from the measured horizontal distance and the azimuth of the line.

Traverse Closure Error

Ec=(ΣLat)2+(ΣDep)2E_c = \sqrt{\left(\Sigma\text{Lat}\right)^{2} + \left(\Sigma\text{Dep}\right)^{2}}

Linear misclosure of a closed traverse, combining the residual sums of the latitudes and departures as a right triangle.

Traverse Precision Ratio

N=PEcN = \frac{P}{E_c}

Expresses traverse accuracy as one part in N, dividing the total perimeter walked by the linear closure error.

Back Azimuth

αb=α±180∘\alpha_b = \alpha \pm 180^{\circ}

Reverses a direction by adding or subtracting one hundred eighty degrees, keeping the result inside the zero to three sixty range.

Area of a Three-Sided Parcel by Coordinates

A=12[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]A = \tfrac{1}{2}\left[x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right]

Shoelace area of a parcel from the coordinates of its three corners, listed counter-clockwise so the result comes out positive.

How they fit together

Break each leg into its north-south component (latitude) and its east-west component (departure), and a closed traverse should bring both sums back to zero. They never quite do. The residuals are the misclosure, their magnitude is the closure error, and dividing the perimeter by that error gives the precision ratio a survey is judged on — 1:5000 for ordinary boundary work, tighter for control.

A good ratio is necessary but not sufficient: latitude and departure errors can partly cancel, so a traverse containing one badly misread angle can still close acceptably while every point inside it is wrong. Check the individual legs, not just the total. Note also that the area formula here is written for a three-sided parcel specifically, not the general shoelace sum over any number of vertices.