Traverse Computations

A series of consecutive lines whose lengths and directions are determined from field measurements. Used to establish control points and locate details.

Types of Traverses

Classifications

  • Open Traverse: Does not return to the starting point or close upon a point of known position.
    • Use: Route surveys (roads, pipelines).
    • Check: No geometric check available unless tied to control points.
  • Closed Traverse:
    • Loop Traverse: Begins and ends at the same point.
    • Link Traverse: Begins and ends at points of known position.
    • Use: Property boundaries, construction control.
    • Check: Sum of angles and coordinates must close.

Adjust the traverse distance and azimuth below to immediately calculate and visualize the resulting latitude and departure components.

Latitudes and Departures Simulator

Calculate latitude and departure from a given distance and azimuth.

Distance (m)150.0
Azimuth (°)60.0

Results

Lat=Dcos⁡(θ)=75.00 m\text{Lat} = D \cos(\theta) = 75.00 \text{ m}
Dep=Dsin⁡(θ)=129.90 m\text{Dep} = D \sin(\theta) = 129.90 \text{ m}
N (+Lat)E (+Dep)LatDep

Methods of Traversing

Field Methods

  • Interior Angle Traverse: Measuring the angles inside a closed polygon. The sum of the angles is checked against (n−2)×180∘(n-2) \times 180^\circ.
  • Deflection Angle Traverse: Measuring the angle by which the next line deflects from the prolongation of the previous line (Right or Left). Common in route surveys (open traverses).
  • Azimuth Traverse: Measuring the azimuth of each line directly using a compass or by backsighting and turning the angle. It allows for quick calculation of latitudes and departures without intermediate bearing conversions.

Latitudes and Departures

To plot a traverse or compute coordinates, each course is resolved into two components:

  • Latitude (LL): The projection of a traverse line on the North-South meridian (L=Dcos⁡αL = D \cos \alpha).

Latitude

L=Dcos⁡α L = D \cos \alpha

  • North Latitude: Positive (++)

  • South Latitude: Negative (−-)

  • Departure (DepD_{ep}): The projection of a line on the East-West line.

Departure

Dep=Dsin⁡α D_{ep} = D \sin \alpha
  • East Departure: Positive (++)
  • West Departure: Negative (−-)

Error of Closure

In a theoretically perfect closed loop traverse, the algebraic sum of latitudes (ΣL\Sigma L) and departures (ΣDep\Sigma D_{ep}) should be zero. Due to errors, they are usually not.

  • Linear Error of Closure (LECLEC): The distance from the starting point to the computed end point.

Linear Error

LEC=(ΣL)2+(ΣDep)2 LEC = \sqrt{(\Sigma L)^2 + (\Sigma D_{ep})^2}
  • Relative Error of Closure (RECREC): A measure of precision.

Relative Error

REC=LECΣD=1ΣD/LEC REC = \frac{LEC}{\Sigma D} = \frac{1}{\Sigma D / LEC}

Where ΣD\Sigma D is the total length (perimeter). Expressed as a ratio (e.g., 1:5000).

Interactive Traverse Tool

Visualize a traverse and automatically calculate closure errors and area.

Traverse & Area Tool

Traverse Lines

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Plot

Closure Error: 0.0000 m
Precision: 1:3,109,888,511,975,475
Area: 15000.00 m²
Start123End
* Y-axis is inverted for SVG rendering (North is up)

Balancing a Traverse

Adjusting the latitudes and departures so their algebraic sums become zero (or match the known difference for link traverses).

1. Compass Rule (Bowditch Rule)

  • Assumption: Errors in distance and angle are equal.
  • Application: Used for most surveys where tape and compass/transit are used.

Compass Rule

cL=−ΣL(dΣD) c_L = -\Sigma L \left(\frac{d}{\Sigma D}\right)
cD=−ΣDep(dΣD) c_D = -\Sigma D_{ep} \left(\frac{d}{\Sigma D}\right)

(Correction is proportional to the length of the side dd).

2. Transit Rule

  • Assumption: Angular errors are less than linear errors. It assumes the direction of lines is more certain than their lengths. The theoretical basis is that coordinate magnitudes dictate the error.
  • Application: Used when angles are measured more precisely than distances (e.g., precise theodolite with stadia distance).

Transit Rule

cL=−ΣL(∣L∣Σ∣L∣) c_L = -\Sigma L \left(\frac{|L|}{\Sigma |L|}\right)
cD=−ΣDep(∣Dep∣Σ∣Dep∣) c_D = -\Sigma D_{ep} \left(\frac{|D_{ep}|}{\Sigma |D_{ep}|}\right)

(Correction is proportional to the absolute magnitude of the latitude/departure of the side, rather than lengths as in the Compass rule).

3. Crandall's Method

  • Assumption: All angular errors are completely eliminated (assumed perfect) before adjusting linear distances. It distributes the closure error entirely to the distance measurements based on a least-squares principle.
  • Mathematical Concept: The sum of the squares of the distance corrections, weighted inversely by their expected precision, is minimized (Σ(v2/w)→min\Sigma (v^2/w) \to \text{min}, where vv is the residual and ww is the weight).
  • Application: Used when angular measurements are exceptionally more precise than distance measurements.

4. Least Squares Method

  • Assumption: The sum of the squares of the weighted residuals is minimized.
  • Application: The most mathematically rigorous method for adjusting any traverse or survey network. Best suited for complex networks with redundant measurements. Easily handled by modern surveying software.
Key Takeaways
  • Latitude: Dcos⁡αD \cos \alpha (North-South component).
  • Departure: Dsin⁡αD \sin \alpha (East-West component).
  • Linear Error of Closure (LEC): (ΣL)2+(ΣDep)2\sqrt{(\Sigma L)^2 + (\Sigma D_{ep})^2}.
  • Compass Rule: Adjusts based on side length (assumes equal error probability).
  • Transit Rule: Adjusts based on coordinate magnitude (assumes angle is more precise).
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