Resection on a Plane Table Near a Magnetic Anomaly
A plane table is set up at an unknown point P from which three control stations A, B and C are visible. The distances PA, PB and PC are roughly 60 m, 140 m and 220 m respectively, and the angle at P subtended by A and B is about 25°. The table is oriented by a magnetic compass because the trough compass needle swings sluggishly and settles 3° away from the direction it held at the previous station 200 m away. The surveyor plans to fix P by the three-point problem using the Lehmann or tracing-paper method. A steel water tank stands 15 m from P. The plotted positions of A, B and C on the sheet form a triangle whose sides are 40 mm, 95 mm and 150 mm at 1:1000 scale.
1. Assess whether the three-point fix will be reliable here, considering the geometry and the orientation.
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Geometry of the fix.
The strength of a three-point resection depends on the triangle of error, which grows when:
- the station P lies near the circumscribing circle of ABC (the "dangerous circle"), or
- the triangle ABC is small on the sheet relative to the drawing error, or
- the angles subtended are very small or very large.
Here the plotted triangle has sides 40, 95 and 150 mm - a long thin triangle. With PA = 60 m, PB = 140 m and PC = 220 m, P lies roughly on a line with the stations, and the angle APB is only about 25°. This is close to the dangerous-circle condition: when P is on or near the circle through A, B and C, the resection becomes indeterminate and the triangle of error collapses or explodes.
Orientation. The trough compass settled 3° from its previous reading 200 m away. Over 200 m the magnetic field should be essentially constant, so a 3° shift is a red flag. The steel tank 15 m from P is the obvious cause: a large steel mass distorts the local field, and the needle's sluggish swing is consistent with a weak, distorted field.
Conclusion. The fix is unreliable on two independent grounds: poor geometry (near the dangerous circle) and unreliable orientation (magnetic disturbance). A three-point fix computed under these conditions may look plausible on the sheet and be wrong by metres.
2. Describe how to carry out the resection properly, and what to do if the geometry cannot be improved.
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Preferred method - Lehmann's rules with a properly oriented table.
- Orient the table without the compass. Sight along a known line, e.g. set the alidade on the plotted line ab and swing the table until the line of sight hits A and B simultaneously (or use a backsight to a distant well-defined station). This removes the magnetic problem entirely.
- Draw the three rays. With the table clamped, sight A, B and C in turn and draw rays from a, b and c respectively. The three rays form the triangle of error.
- Apply Lehmann's rules to place P.
- If P lies inside the triangle ABC, the correct point is inside the triangle of error; if outside, it is outside.
- The correct point lies on the same side of each ray as the corresponding station lies relative to the opposite side of the triangle.
- The distances from the correct point to the three rays are proportional to the distances from P to the three stations.
- Iterate. Move the alidade to the estimated P, re-sight, and repeat until the rays meet at a point. Two iterations usually suffice.
Alternative - tracing paper / Bessel's method.
Prick P on tracing paper, draw rays to A, B and C, lay the tracing on the sheet and rotate/translate until the three rays pass through a, b and c. This is fast and avoids the iteration, but it is only as good as the orientation and the plotted triangle.
If the geometry cannot be improved.
- Add a fourth station. A four-point fix gives a check: the fourth ray should pass through the same point. A residual reveals the error.
- Abandon resection; run a traverse. Measure a baseline from P to a known station and carry coordinates by bearing and distance. This is slower but robust.
- Move P. If the job allows, shift the table 10-20 m to a position with better angular spread (angles between rays near 60-120°) and away from the tank.
Never accept a fix obtained only by compass orientation within 15 m of a steel tank. The 3° needle shift is the instrument telling you the field is not trustworthy.
3. How should the surveyor record the orientation problem and the fix in the field book, and what confidence should be attached to the plotted position of P?
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Field book record.
- Station description. "P: plane table, 15 m east of steel water tank; trough compass sluggish, settled 3° from reading at previous station 200 m away; magnetic orientation rejected."
- Orientation actually used. State the method: "table oriented by backsight along ab to A and B; no magnetic reference used."
- Rays and triangle of error. Sketch the triangle of error with the ray directions and the final position of P marked. Note the number of iterations.
- Check. If a fourth station D was sighted, record the residual: "ray from d missed P by 1.5 mm on sheet = 1.5 m on ground." That number is the real measure of confidence.
- Scale and sheet. Note the scale (1:1000) and that the plotted triangle is thin.
Confidence in P.
- With a thin triangle (40/95/150 mm) and a 25° subtended angle, the resection is geometrically weak. A 1 mm plotting error can move P by several metres on the ground.
- Without a fourth-ray check, the position of P should be described as provisional and not used to control detail within, say, 2 m.
- If a fourth ray was observed and the residual is small (under about 1 mm on the sheet), confidence rises accordingly - but the geometry remains the limiting factor.
What to write in the report. State the method, the orientation source, the geometry problem, the check result, and the recommended accuracy for detail plotted from P. A position that is good to 3 m is perfectly usable for a small-scale location plan and not usable for a boundary or a setting-out point; the record must make that distinction explicit so the next user of the drawing is not misled.