---
title: "Define intersection and resection, derive"
subject: "Surveying II"
chapter: 5
key: "semester_4/surveying_ii:intersection-resection-define-derive"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/intersection-resection-define-derive
site: BCT Engineering
---

# Define intersection and resection, derive

Surveying II · Chapter 5: Orientation

- Asked in: 2073 Magh, 2+6 marks
- Video: [Intersection and Resection: Orientation,  Surveying II](https://www.youtube.com/watch?v=8GFRyKJRMHQ) — N civil engineers
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/intersection-resection-define-derive
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-5

## Question

Define intersection and resection. Derive the equation for any one method of resection for finding the coordinates of that unknown point.

## Short answer

Intersection finds an unknown point by observing angles from known stations; resection finds an unknown station by observing angles from it to known stations. For three-point resection, with observed angles α=APB, β=BPC, compute S=360°−(α+β+B), k=(a sinα)/(c sinβ), then tan((x−y)/2)=((k−1)/(k+1))tan(S/2) to obtain x,y and coordinates.

## Answer

#### Definitions

**Intersection** is the method of finding the coordinates of an unknown point P by measuring the horizontal angles to P from two (or more) stations of known coordinates. The instrument is set up at the known stations, not at P.

**Resection** is the method of finding the coordinates of an unknown station P by measuring the horizontal angles at P to three (or two) stations of known coordinates. The instrument is set up only at the unknown point.

#### Three-point resection (Pothenot's tangent method)

**Problem:** A, B and C are known stations (B is the middle one). The instrument is at the unknown station P, and the angles $\alpha = \angle APB$ and $\beta = \angle BPC$ are observed. Find the coordinates of P.

```text
         B
        /|\
     c / | \ a
      /  |  \
     A   |   C
      \  |  /
       \ | /
        \|/ 
         P      angles at P: alpha (APB), beta (BPC)
```

**Derivation**

From the coordinates, find the sides $c = AB$, $a = BC$ and the angle $B = \angle ABC$ (from the bearings of BA and BC). Let $x = \angle PAB$ and $y = \angle PCB$ be the unknown angles. In the quadrilateral PABC the sum of angles is $360^\circ$:

$$
x + y + \alpha + \beta + B = 360^\circ \quad\Rightarrow\quad x + y = S = 360^\circ - (\alpha + \beta + B)
$$

In triangle APB, by the sine rule, $PB = \dfrac{c\sin x}{\sin\alpha}$. In triangle BPC, $PB = \dfrac{a\sin y}{\sin\beta}$. Equating,

$$
\frac{\sin x}{\sin y} = \frac{a\sin\alpha}{c\sin\beta} = k
$$

Using $\dfrac{\sin x - \sin y}{\sin x + \sin y} = \dfrac{\tan\frac{x-y}{2}}{\tan\frac{x+y}{2}}$:

$$
\tan\frac{x-y}{2} = \frac{k-1}{k+1}\tan\frac{S}{2}
$$

So $\dfrac{x-y}{2}$ is found, and with $x + y = S$ we get $x$ and $y$ separately.

**Coordinates of P**

1. In triangle APB: $\angle ABP = 180^\circ - \alpha - x$, so $AP = \dfrac{c\sin\angle ABP}{\sin\alpha}$ (or $BP = \dfrac{c\sin x}{\sin\alpha}$).
2. Bearing of $AP$ = bearing of $AB$ $\pm\, x$ (sign depends on the side on which P lies).
3. $E_P = E_A + AP\sin(\text{bearing }AP)$, $\;N_P = N_A + AP\cos(\text{bearing }AP)$.
4. Check by computing P from C: $CP = \dfrac{a\sin\angle CBP}{\sin\beta}$ with the bearing of $CP$ = bearing of $CB \mp y$. Both must give the same coordinates.

**Note:** If P lies on the circle through A, B and C (the "danger circle"), then $\alpha + \beta + B = 180^\circ$, $\tan\frac{S}{2}$ is infinite, and the problem has no unique solution. The point P should be chosen away from this circle.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Resection is best defined as:
   A. Finding an unknown point by angles measured from two known stations
   B. Finding an unknown station by angles measured at that station to known stations
   C. Finding the height of a point
   D. Finding a point by measuring its distance only
   Answer: B. In resection the instrument is set up only at the unknown station and sights known stations. (Key point: Definition of resection)

2. In the derivation, why does x + y = 360° - (alpha + beta + B)?
   A. The angles of quadrilateral PABC sum to 360°
   B. The angles of a triangle sum to 180°
   C. Because alpha + beta = 180°
   D. Because B is a right angle
   Answer: A. x, y, alpha, beta and B are the five angles of the quadrilateral formed by P, A, B, C, so they add to 360°. (Key point: Angle sum in quadrilateral)

3. How is the ratio sin x / sin y = k obtained?
   A. From the cosine rule in triangle ABC
   B. By writing PB from triangle APB (c sin x / sin alpha) and triangle BPC (a sin y / sin beta) and equating
   C. From Pythagoras
   D. By taking x = y
   Answer: B. PB is common to both triangles, so equating the two sine-rule expressions gives sin x/sin y = a sin alpha/(c sin beta). (Key point: Equating the common side PB)

4. What happens if P lies on the circle through A, B and C?
   A. The solution is most accurate
   B. alpha + beta + B = 180°, tan(S/2) is infinite and the solution is indeterminate
   C. The bearings become zero
   D. Only the Easting is undetermined
   Answer: B. On the danger circle S = 180° so tan(S/2) is infinite, giving no unique solution; P must be chosen away from it. (Key point: Danger circle)

## Related in Surveying II

- [Resection, two point problem, intersection](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aresection-two-point-problem-ab-p) — Surveying II, chapter 5
- [Intersection vs resection, three point method](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aintersection-vs-resection-three-point) — Surveying II, chapter 5
- [Three point analytical resection, average coordinates of O](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Athree-point-resection-p-w-d) — Surveying II, chapter 5
- [Derive three point analytical resection](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Athree-point-analytical-resection-derivation) — Surveying II, chapter 5, asked 2×
- [Two point resection method](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atwo-point-resection-method-describe) — Surveying II, chapter 5
- [Analytical resection, derive coordinate expression](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanalytical-resection-derive-three-known) — Surveying II, chapter 5

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