- Asked 3/31
- 2081 Bhadra · 6 marks
- 2075 Asoj · 6 marks
Mean angles and VCR, station O (A-D)
Short answer
Mean directions: (Direct + (Reversed−180°))/2, reduced to A=0°00′00″. Closing error 10″ distributed equally gives adjusted angles: AOB=60°54′57.5″, BOC=79°55′47.5″, COD=129°29′07.5″, DOA=89°40′07.5″. Missing vertical readings from FL+FR=360°: A rev=239°44′50″, B dir=51°08′20″, C dir=90°44′50″, D rev=182°39′50″, A clos rev=270°59′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 and adjust them if necessary. Also calculate the missing data of the vertical circle reading.
| Instrument | Target Station | Telescope | Horizontal Circle Readings | Vertical Circle Readings |
|---|---|---|---|---|
| O | A | Direct | 00°00′10″ | 120°15′10″ |
| Reversed | 180°00′20″ | ? | ||
| 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″ | 89°00′10″ | |
| 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′10″ | 180°00′20″ | 0°00′20″ | 0°00′15.0″ | 0°00′00.0″ | - | 0°00′00.00″ | |
| B | 60°55′10″ | 240°55′20″ | 60°55′20″ | 60°55′15.0″ | 60°55′00.0″ | -0°00′02.50″ | 60°54′57.50″ | |
| C | 140°50′50″ | 320°51′20″ | 140°51′20″ | 140°51′05.0″ | 140°50′50.0″ | -0°00′05.00″ | 140°50′45.00″ | |
| D | 270°20′10″ | 90°20′20″ | 270°20′20″ | 270°20′15.0″ | 270°20′00.0″ | -0°00′07.50″ | 270°19′52.50″ | |
| A (closing) | 00°00′20″ | 180°00′30″ | 0°00′30″ | 0°00′25.0″ | 0°00′10.0″ | -0°00′10.00″ | 360°00′00″ |
Mean angles and adjustment
| Angle | Mean angle (before adjustment) | Correction | Adjusted angle |
|---|---|---|---|
| AOB | 60°55′00.0″ | -0°00′02.50″ | 60°54′57.50″ |
| BOC | 79°55′50.0″ | -0°00′02.50″ | 79°55′47.50″ |
| COD | 129°29′10.0″ | -0°00′02.50″ | 129°29′07.50″ |
| DOA | 89°40′10.0″ | -0°00′02.50″ | 89°40′07.50″ |
| Sum | 360°00′10.0″ | -0°00′10.0″ | 360°00′00″ |
Closing error = 0°00′10.0″ (second reading on A larger than the first). The sum of the four angles at O must be 360°, and it is 360°00′10.0″; the error of 0°00′10.0″ is small and is therefore distributed equally: correction = -0°00′02.50″ 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″ |
| A (closing) | 89°00′10″ | 270°59′50″ | 89°00′10″ | +0°59′50″ |
Answer: adjusted angles AOB = 60°54′57.50″, BOC = 79°55′47.50″, COD = 129°29′07.50″, DOA = 89°40′07.50″; missing VCR: A (reversed) 239°44′50″, B (direct) 51°08′20″, C (direct) 90°44′50″, D (reversed) 182°39′50″; A closing (reversed) 270°59′50″.
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Mean angles and VCR, station O (reading 179°59′40″) · 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 with missing VCR (68°30′10″) · Surveying I, chapter 7
- Mean angles; VCR and VA from A · Surveying I, chapter 7
- Theodolite traverse ABCDA, booking format with VCR · Surveying I, chapter 7
- Mean horizontal angles, station O (P, Q, R) · Surveying I, chapter 7