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.

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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