Resection: Fixing Position via the Three-Point Problem

Overview

Resection is a method of fixing the coordinates of an unknown station ($P$) by observing horizontal angles to at least three known control points ($A, B, C$). Unlike intersection, where you sight toward the new point from known stations, resection allows you to occupy the unknown point and observe outward 27, 57.

Why This Matters

Resection is the ultimate "set out" tool. If you need to set out a building but your control stations are blocked by piles of dirt or machinery, you can set up anywhere with a clear view of three existing points and fix your position in minutes 57. It eliminates the need to run a traverse to get "on-station."

Theory

Resection works by solving the geometry of two overlapping circles. Each measured angle between two points defines a circular arc upon which point $P$ must lie. The intersection of these arcs uniquely identifies $P$ 58.

Mathematical Principles

The Analytical Solution (Cotangent Method)

Given known coordinates ($E, N$) for $A, B, C$ and observed angles $\alpha$ (between $A, B$) and $\beta$ (between $B, C$) 58:

1. Calculate Grid Bearings and Distances: Find bearings $AB$ and $BC$ and distances $AB$ and $BC$ 58, 59.
2. Calculate Constant $S$:$S = 360^\circ - (\alpha + \beta + \hat{B})$ 59. (Where $\hat{B}$ is the internal angle at $B$ from coordinates).
3. Solve for $\theta$ (Angle $BAP$):$\cot \theta = \frac{\left( \frac{AB \sin \beta}{BC \sin \alpha} + \cos S \right)}{\sin S}$ 59.
4. Final Coordinates: Once $\theta$ is known, the triangle $ABP$ is solved for length $AP$ and bearing $AP$ to get the coordinates of $P$ 59.

Field Workflow

Step-by-Step Example

Problem: Points $A(1234.96, 17594.48)$ and $B(7994.42, 24343.45)$ are sighted. $AB_{dist} = 9551.91\text{ m}$. Observed angle $\alpha = 61^\circ 41' 46.6''$. Solve for $P$ 58.

(Note: Full resection requires point C; this example simplifies to the triangle solution step).

1. Find the Bearing $AB$: From coordinates, $AB = 45^\circ 02' 40.2''$ 58.
2. Solve for $P$ using $\theta$: If $\theta$ (angle $BAP$) is calculated as $52.5545^\circ$:
3. Length $BP$:$BP = \frac{AB \sin \theta}{\sin \alpha} = 8613.32\text{ m}$ 59.
4. Coordinates of $P$: Apply bearing $BP$ and length $BP$ to the coordinates of $B$ to find $P$ 59.

Formula Breakdown

The "Circle of Danger": Mathematically, if $P$ lies on the circle formed by $A, B,$ and $C$, then $S = 180^\circ$. Since $\sin(180^\circ) = 0$, the formula for $\cot \theta$ involves division by zero and becomes undefined 62, 63. Best Geometry: Site point $P$ inside the triangle formed by $A, B,$ and $C$ to avoid this 63.

Practical Tips

• Total Station Advantage: Modern total stations have a built-in "Resection" (or "Free Station") program. You simply select the points from the internal database, sight them, and the instrument displays your coordinates and the standard error ($S_x, S_y$) 64.
• Commensurate Accuracy: Ensure the points you sight are at a similar distance. Sighting one point at $50\text{ m}$ and another at $5\text{ km}$ introduces centring errors that degrade the resection 65.

Common Mistakes

• Sighting in a Straight Line: If $A, B,$ and $C$ lie in a perfectly straight line and $P$ is also near that line, the solution becomes very "weak" and unreliable 62, 63.
• Poor Point Identification: Sighting the "wrong" church spire or a nearby station with a similar name will result in a coordinates failure that is often hard to spot without a fourth point 66, 67.

FAQ

Conclusion

Resection is the surveyor's "escape hatch" for difficult site conditions. By mastering the cotangent formula and avoiding the "Circle of Danger," you can turn any clear patch of ground into a high-precision control station.

References

Schofield, W. (2001). Engineering Surveying. 5th ed. Butterworth-Heinemann. 16, 17.