To determine the true difference of elevation between two points A and B (eliminating both curvature and collimation errors), the level should be placed:
Correct Answer: B. At the exact midpoint of AB
📚 Detailed Explanation: Level at Exact Midpoint Eliminates Both Curvature and Collimation Errors
Why B (Exact midpoint of A and B) is correct: When the level is placed at the exact midpoint between two staffs, both curvature and collimation errors are eliminated simultaneously. This is the key principle of the two-peg test and of accurate differential levelling over long distances.
Why the Midpoint Position Is Special
| Error Type | At Midpoint | Reason |
|---|---|---|
| Curvature and refraction error | Eliminated | Both staff readings are at the same distance from the instrument; the curvature/refraction error affects both readings by the same amount and cancels in the difference |
| Collimation error | Eliminated | Same distance to both staffs → collimation error (which is proportional to distance) affects both by the same amount; cancels in the difference |
| Height difference measured | True difference | Both systematic errors cancelled; result = true height difference, not apparent |
At the midpoint (distance D/2 to each staff):
Error in far reading = k × (D/2) [same for both staffs]
True height diff = (Staff B reading – Staff A reading) + (error_A – error_B)
= (Staff B – Staff A) + (k·D/2 – k·D/2)
= (Staff B – Staff A) + 0
= True difference ✓
Error in far reading = k × (D/2) [same for both staffs]
True height diff = (Staff B reading – Staff A reading) + (error_A – error_B)
= (Staff B – Staff A) + (k·D/2 – k·D/2)
= (Staff B – Staff A) + 0
= True difference ✓
Two-Peg Test: The two-peg test exploits this principle. The instrument is first set at the midpoint to get the true height difference, then moved to one end to detect the collimation error.
- Level at exact midpoint between A and B → true height difference (both curvature and collimation errors cancel).
- This is the principle behind: (1) balancing BS and FS distances, and (2) the two-peg test.
- In precise levelling, equal BS and FS distances are required at every setup for this reason.