Q15. The ‘fix’ of a plane table from three known points is good if:
📚 Detailed Explanation: Quality of Fix in Three-Point Resection
In the three-point problem, three visible plotted control points are used to fix the unknown instrument station. However, not all configurations of three known points provide an equally reliable fix. The quality of the fix depends heavily on the relative positions of the three known points and the instrument station.
When the three known points form an angular arrangement as seen from the instrument station, the middle point (the one between the other two in angular terms) closest to the instrument gives the most acute intersection angles. Acute, well-separated angular rays from a nearby reference produce a stable, precise fix. Conversely, when the instrument station lies on or near the circumscribed circle (great circle) of the great triangle, the fix becomes indeterminate — this is the “danger circle.” A nearby middle station helps ensure the instrument station is well inside the great triangle and away from the danger circle.
Geometric Reasoning
The precision of an intersection-based fix is greatest when rays cross at angles close to 90°. When all three known points are at similar large distances, the angular spread from the instrument is narrow and the rays are nearly parallel — a poor geometric configuration. Having the middle known point close to the instrument station means the angular difference between sightings to it and to the outer two points is large, producing well-angled ray intersections and a strong fix.
Key Concepts for Students
- Middle station nearest = strongest fix: This is the standard textbook rule. The geometric intuition is that a nearby middle reference point creates well-separated angular rays, maximising intersection strength.
- Avoid the danger circle in field practice: Before setting up at a new station, mentally check whether the instrument station might lie on or near the circumscribed circle of the three control points. If it does, choose a fourth control point or move the instrument slightly off the circle.
- Contrast with the two-point problem: The two-point problem does not have an equivalent danger circle issue because the solution geometry is different. The danger circle applies specifically to the three-point resection (Lehmann / Bessel methods).