Compass, Theodolite and Traversing

Angles and directions locate points horizontally. A traverse is a series of lines whose lengths and directions are measured to build a control framework.

Bearings

• Whole circle bearing (WCB): 0°–360° clockwise from north.
• Quadrantal (reduced) bearing: N or S, angle (0–90°), E or W. Example: WCB 210° = S 30° W.
• Back bearing = fore bearing ± 180°.
• Included angle between two lines = difference of their bearings (taken properly).
• True bearing = magnetic bearing ± declination (+ for east declination).
• Local attraction (from nearby iron) is detected when the fore and back bearings of a line differ by other than 180°; correct from a station free of it.

Theodolite

• Measures horizontal and vertical angles to seconds of arc.
• Face left and face right observations, averaged, eliminate collimation, trunnion axis and index errors.
• Methods: repetition (accumulate an angle several times) and reiteration (several angles from one station).

Traverse computations

For each line of length L and bearing θ:

• Latitude = L cos θ (northing +, southing −).
• Departure = L sin θ (easting +, westing −).

For a closed traverse, Σ latitudes = 0 and Σ departures = 0 ideally. In practice:

• Closing error e = √((ΣL)² + (ΣD)²), with direction tan⁻¹(ΣD/ΣL).
• Relative precision = e/perimeter (for example 1 in 5000).
• Angular check: sum of interior angles = (2n − 4) × 90°.

Balancing:

• Bowditch's (compass) rule: correction to a line ∝ its length (when angles and distances are equally precise).
• Transit rule: correction ∝ the line's latitude or departure (when angles are more precise than distances).

Area from coordinates

A = ½ |Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)| (the shoelace formula). Areas of irregular boundaries: trapezoidal rule or Simpson's rule (odd number of ordinates).

Worked example

A closed traverse has ΣL = +0.30 m and ΣD = −0.40 m with a perimeter of 2500 m.

• e = √(0.09 + 0.16) = 0.5 m; precision = 0.5/2500 = 1 in 5000.

Practice