Q4. Consider the following methods used to establish a point in the field:
Method 1: Point O is plotted by measuring only distances MO and NO.
Method 2: Point O is plotted by measuring only angles ∠NMO and ∠MNO.
Which of these methods also represents the principle of trigonometrical levelling?
📚 Detailed Explanation: Angular vs Linear Position Fixing & the Link to Trigonometrical Levelling
This question requires understanding how a point’s position can be fixed geometrically, and then recognising which technique shares its underlying principle with trigonometrical levelling.
Analysing Method 1 (Distance-based)
Method 1 fixes point O by measuring the two distances MO and NO from two known baseline endpoints M and N. This is a linear approach: position is determined by the intersection of two arcs of known radii. It is the principle behind chain surveying and triangulation by linear measurement. No angles are computed; the position is fixed purely by measured lengths.
Analysing Method 2 (Angle-based)
Method 2 fixes point O by measuring only the two angles ∠NMO and ∠MNO at the two baseline stations M and N. The position of O is then computed mathematically using trigonometry. No physical distance to O is measured. This is the principle of triangulation and, crucially, of trigonometrical levelling.
Trigonometrical levelling determines heights (elevations) of remote or inaccessible points by measuring vertical angles from known stations, then computing vertical distances mathematically — never measuring the slope distance directly to the object. The same logic applies: position is determined from angles alone, not from physical linear measurements to the target. Method 2 matches this angular-computation paradigm exactly.
Comparison Table
| Feature | Method 1 | Method 2 |
|---|---|---|
| Measurement type | Linear distances only | Angular measurements only |
| Survey principle | Chain surveying / distance intersection | Triangulation / trigonometrical levelling |
| Physical access to O needed? | Yes (to measure MO, NO) | No (angles measured from M and N) |
| Computation method | Geometric arc intersection | Trigonometric calculation from angles |
Key Concepts for Students
- Trigonometrical levelling uses angles, not distances: The central idea is that height or position is computed from measured angles and a known baseline, without physically measuring the full distance to the target. Method 2 shares this exact principle: fix O from two angles, not from distances to O.
- Intersection method in plane table = Method 2: The intersection method of plane table surveying is also based on sighting a target from two known stations and drawing rays — a graphical realisation of Method 2. This is why intersection is used for plotting inaccessible points (hills, cliff edges, far bank of a river).
- Do not confuse with chain survey: Chain survey (Method 1 principle) requires physical access to the target point to measure distances. Method 2 / trigonometrical levelling / intersection method are specifically designed for targets you can see but cannot physically reach.