Q5. With respect to plane table surveying, the terms ‘triangle of error’, ‘great circle’, and ‘great triangle’ are related to:
📚 Detailed Explanation: Triangle of Error, Great Circle & Great Triangle in Lehmann’s Method
All three terms — triangle of error, great circle, and great triangle — are specific vocabulary that belongs exclusively to Lehmann’s trial-and-error method for solving the three-point resection problem. Understanding each term is essential for exam success.
Fig: When the table is not perfectly oriented, three resection rays from known points a, b, c fail to meet at one point — they form the small shaded “triangle of error.” The light dashed outer triangle is the “great triangle.”
The Three Terms Explained
Great Circle: The circumscribed circle passing through all three known control points a, b, and c. Its position relative to the unknown station p determines whether Lehmann’s trial point should be inside or outside the triangle of error.
Triangle of Error: If the table is not perfectly oriented at the instrument station, the three resection rays from a, b, and c will not converge to a single point. Instead, they form a small residual triangle. Lehmann’s rules guide the surveyor in selecting the true station position inside (or sometimes outside) this error triangle.
Why the Other Options Are Wrong
| Option | Reason it is incorrect |
|---|---|
| B — Bessel method | Bessel’s method is another graphical approach to the three-point problem that uses tracing paper. It does not involve a triangle of error; it solves the orientation directly by rotating an auxiliary trace. None of the three terms belong to it. |
| C — Two-point problem | The two-point problem uses only two known reference points and an auxiliary ray. It has no triangle of error because the three-ray configuration that generates the error triangle simply does not exist in a two-point setup. |
| D — Graphic triangulation | Graphic triangulation refers to the graphical solution of surveying networks by intersection/resection, not a specific method. The three terms in the question are not associated with it. |
Key Concepts for Students
- Memorise the three Lehmann terms as a set: Triangle of error + Great circle + Great triangle = Lehmann’s method for the three-point problem. Any question that mentions all three or any combination of these is asking about Lehmann.
- Lehmann’s rule for the trial point: The correct station position lies on the same side of each of the three resection rays as the respective control point it was drawn from. This quickly eliminates three of the four quadrants formed by the rays and places the true position near the triangle of error.
- The danger circle: When the instrument station lies on the great circle (the circumscribed circle of the great triangle), the three-point problem becomes geometrically indeterminate — a unique solution does not exist. Avoid placing instrument stations near the great circle in practice.