Q2. In plane table surveying, Lehmann’s rules are associated with:
📚 Detailed Explanation: Lehmann’s Rules & the Three-Point Problem
When a plane table is set up at an unknown station and three visible, pre-plotted control points are available, the surveyor faces the three-point problem: how to simultaneously orient the table and locate the instrument station using those three known points. Lehmann’s rules provide the classic graphical trial-and-error solution to this specific problem.
What Lehmann’s Method Does
The procedure works iteratively. The table is oriented approximately, and three resection rays are drawn from the plotted positions of the three known points. If orientation is imperfect, the rays form a small triangle of error instead of converging to one point. Lehmann’s rules then specify how to choose a new trial position inside that triangle (or outside it, depending on the configuration relative to the “great circle”). The table is moved to the trial position, re-oriented, and the process repeats until the triangle of error vanishes and all three rays meet at one point — the true station location.
• Great triangle: The large triangle formed on the ground (or sheet) by connecting the three known control points.
• Great circle: The circumscribed circle passing through all three known control points.
• Triangle of error: The small residual triangle formed by three misaligned resection rays when orientation is not exact.
Why the Other Options Are Wrong
| Option | Error in reasoning |
|---|---|
| A — All types of resection | Lehmann’s rules apply specifically to the three-point version of resection. Other resection problems (two-point problem) use different procedures such as the auxiliary-point method. |
| B — Resection after compass orientation | Using a trough compass gives a single approximate orientation. Lehmann’s method is an angular refinement technique, not a compass-based one. |
| C — Two-point problem | The two-point problem uses a completely different procedure: the surveyor sights two known points, draws an auxiliary ray on a separate sheet, and solves by back-sighting — no triangle of error is involved. |
Key Concepts for Students
- Lehmann’s = three-point problem only: Memorise this pairing. If a question mentions the triangle of error, great circle, great triangle, or a trial-and-error graphical resection, the answer is always Lehmann’s method applied to the three-point problem.
- How to choose the trial point: Lehmann’s rules state that the true station lies on the same side of each resection ray as the corresponding control point. This quickly narrows down the correct side of the triangle of error to place the next trial point.
- Accuracy advantage: The three-point problem using Lehmann’s method is more accurate than the two-point problem because it uses an additional control point and eliminates orientation error graphically through iteration rather than relying on a single compass or back-sight reading.