- Asked 3/31
- 2079 Bhadra · 5+1 marks
Mean angles and VCR, station O (reading 179°59′40″)
Short answer
Mean horizontal angles are found by averaging FL and FR readings, reducing directions to zero at A, and distributing the 35″ closing error equally; adjusted angles are AOB=60°55′16.25″, BOC=79°55′41.25″, COD=129°29′01.25″, DOA=89°40′01.25″. Missing vertical readings are A(rev)=239°44′50″, B(dir)=51°08′20″, C(dir)=90°44′50″, D(rev)=182°39′50″.
A theodolite is set over station O to measure direction to stations A, B, C and D. The observed circle readings are as follows. Compute the mean horizontal angle by mean direction method and adjust them if necessary. Also calculate the missing data of vertical circle reading.
| Instrument | Target Station | Telescope | Horizontal Circle Readings | Vertical Circle Readings |
|---|---|---|---|---|
| O | A | Direct | 00°00′00″ | 120°15′10″ |
| Reversed | 179°59′40″ | ? | ||
| B | Direct | 60°55′10″ | ? | |
| Reversed | 240°55′20″ | 308°51′40″ | ||
| C | Direct | 140°50′50″ | ? | |
| Reversed | 320°51′20″ | 269°15′10″ | ||
| D | Direct | 270°20′10″ | 177°20′10″ | |
| Reversed | 90°20′20″ | ? | ||
| A | Direct | 00°00′20″ | - | |
| Reversed | 180°00′30″ | - |
Method. The mean direction of each target is found from the two faces: mean = (Direct + (Reversed - 180°)) / 2. The directions are referred to the first reading on A (taken as 0°00′00″). The second reading on A closes the horizon, so the difference from 360° is the closing error, which is distributed equally.
Mean horizontal directions
| Station | Face | Reading L (D) | Reading R (Rev) | R - 180 | Mean direction | Direction from A | Correction | Corrected direction |
|---|---|---|---|---|---|---|---|---|
| A | 00°00′00″ | 179°59′40″ | 359°59′40″ | 359°59′50.0″ | 0°00′00.0″ | - | 0°00′00.00″ | |
| B | 60°55′10″ | 240°55′20″ | 60°55′20″ | 60°55′15.0″ | 60°55′25.0″ | -0°00′08.75″ | 60°55′16.25″ | |
| C | 140°50′50″ | 320°51′20″ | 140°51′20″ | 140°51′05.0″ | 140°51′15.0″ | -0°00′17.50″ | 140°50′57.50″ | |
| D | 270°20′10″ | 90°20′20″ | 270°20′20″ | 270°20′15.0″ | 270°20′25.0″ | -0°00′26.25″ | 270°19′58.75″ | |
| A (closing) | 00°00′20″ | 180°00′30″ | 0°00′30″ | 0°00′25.0″ | 0°00′35.0″ | -0°00′35.00″ | 360°00′00″ |
Mean angles and adjustment
| Angle | Mean angle (before adjustment) | Correction | Adjusted angle |
|---|---|---|---|
| AOB | 60°55′25.0″ | -0°00′08.75″ | 60°55′16.25″ |
| BOC | 79°55′50.0″ | -0°00′08.75″ | 79°55′41.25″ |
| COD | 129°29′10.0″ | -0°00′08.75″ | 129°29′01.25″ |
| DOA | 89°40′10.0″ | -0°00′08.75″ | 89°40′01.25″ |
| Sum | 360°00′35.0″ | -0°00′35.0″ | 360°00′00″ |
Closing error = 0°00′35.0″ (second reading on A larger than the first). The sum of the four angles at O must be 360°, and it is 360°00′35.0″; the error of 0°00′35.0″ is small and is therefore distributed equally: correction = -0°00′08.75″ per angle.
Missing vertical circle readings
For a theodolite with a zenith-angle vertical circle, FL + FR = 360° for a correct instrument. So the missing reading = 360° - the other face. The vertical angle is (positive = elevation, negative = depression).
| Target | VCR Direct (FL) | VCR Reversed (FR) | Mean zenith distance | Vertical angle (90° - Z) |
|---|---|---|---|---|
| A | 120°15′10″ | 239°44′50″ | 120°15′10″ | -30°15′10″ |
| B | 51°08′20″ | 308°51′40″ | 51°08′20″ | +38°51′40″ |
| C | 90°44′50″ | 269°15′10″ | 90°44′50″ | -0°44′50″ |
| D | 177°20′10″ | 182°39′50″ | 177°20′10″ | -87°20′10″ |
Answer: adjusted angles AOB = 60°55′16.25″, BOC = 79°55′41.25″, COD = 129°29′01.25″, DOA = 89°40′01.25″; missing VCR: A (reversed) 239°44′50″, B (direct) 51°08′20″, C (direct) 90°44′50″, D (reversed) 182°39′50″.
Note: the last pair of readings on A has no vertical circle readings (shown as - in the table), so only the four targets are completed. In this question the first reversed reading on A is 179°59′40″, which gives a closing error of 35″.
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Mean angles and VCR, station O (A-D) · Surveying I, chapter 7 · asked 3×
- Direction theodolite, least count 6″, station D · Surveying I, chapter 7
- Mean direction angles with VCR (125°25′20″) · Surveying I, chapter 7
- Theodolite traverse ABCDA, booking format with VCR · Surveying I, chapter 7
- Theodolite traverse ABCDA with missing VCR (68°30′10″) · Surveying I, chapter 7
- Mean angles; VCR and VA from A · Surveying I, chapter 7
- Mean horizontal angles, station O (P, Q, R) · Surveying I, chapter 7
- Taper turning: set-over and compound angle · Workshop Technology, chapter 6
- RL of hill station P from stations R and Q · Surveying II, chapter 3