The following bearings were taken in a closed compass traverse. Calculate the corrected fore bearing of line 'DE' by assuming that the observed bearing of line 'CD' is correct.
| Line | Fore Bearing | Back Bearing |
|---|---|---|
| AB | 80° 10' | 259° 0' |
| BC | 120° 20' | 301° 50' |
| CD | 170° 50' | 350° 50' |
| DE | 230° 10' | 49° 30' |
| EA | 310° 20' | 130° 15' |
🧭 Understanding Traverse Correction
In a closed traverse, the survey lines form a closed loop. Theoretically, the sum of the internal (or external) angles of this polygon should match a fixed geometric value. However, due to unavoidable errors in measurement, there is almost always a "closing error." The standard procedure is to calculate this error and distribute it among all the measured angles to get the corrected values.
The condition that the bearing of line 'CD' is correct means we use it as the starting point for calculating the corrected bearings *after* the angular error has been distributed.
🔬 Step-by-Step Calculation
Step 1: Calculate the Observed Exterior Angles
For a clockwise traverse, the exterior angle at a station is calculated as: Fore Bearing of the Next Line - Back Bearing of the Previous Line.
- Angle at B: 120° 20' - 259° 0' = -138° 40' (+ 360°) = 221° 20'
- Angle at C: 170° 50' - 301° 50' = -131° 0' (+ 360°) = 229° 0'
- Angle at D: 230° 10' - 350° 50' = -120° 40' (+ 360°) = 239° 20'
- Angle at E: 310° 20' - 49° 30' = 260° 50'
- Angle at A: 80° 10' - 130° 15' = -50° 5' (+ 360°) = 309° 55'
Step 2: Calculate the Closing Error
First, find the sum of the observed exterior angles.
Sum = 221° 20' + 229° 0' + 239° 20' + 260° 50' + 309° 55' = 1260° 25'
The theoretical sum of exterior angles for a polygon with 'n' sides is (2n + 4) × 90°.
Theoretical Sum = (2×5 + 4) × 90° = 14 × 90° = 1260°
Closing Error = Observed Sum - Theoretical Sum = 1260° 25' - 1260° = +25'
Step 3: Distribute the Correction
The total correction to be applied is the negative of the error (-25'). Assuming all angles were measured with equal precision, we distribute this correction evenly among the 5 angles.
Correction per angle = -25' / 5 = -5'
Step 4: Calculate the Corrected Fore Bearing of DE
We work forward from the assumed correct line, CD.
1. Correct Back Bearing of CD: The problem states the bearing of line CD is correct. Since the difference between its observed Fore and Back bearings is exactly 180°, we can accept both as correct. Correct BB of CD = 350° 50'.
2. Corrected Exterior Angle at D: Apply the correction to the observed angle.
Corrected Angle D = 239° 20' - 5' = 239° 15'
3. Calculate Corrected Fore Bearing of DE: Use the definition of the exterior angle.
Corrected Angle D = Corrected FB of DE - Correct BB of CD
239° 15' = Corrected FB of DE - 350° 50'
Corrected FB of DE = 239° 15' + 350° 50' = 590° 05'
Since this is greater than 360°, subtract 360°.
Corrected FB of DE = 590° 05' - 360° = 230° 05'