Library / Geometry, coordinates, surveying and trigonometry / GEO-35
Geometry, coordinates, surveying and trigonometry
Triangulation from two surveyed bearings
Distance from station A, at the origin, to a target sighted from A on bearing b₁ and from station B, a distance d due east of A, on bearing b₂.
- Ref
- GEO-35
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- s [km]
Station A is at the origin and B is a distance d due east. The target is sighted on bearing b₁ from A and b₂ from B. Solving the two rays with the sine rule gives the distance from A as s = d cos b₂/sin(b₁ − b₂) and the distance from B as d cos b₁/sin(b₁ − b₂). Both must be positive, so the rays cross ahead of both stations. Parallel rays (sin = 0) give no fix.
Distance from station A, at the origin, to a target sighted from A on bearing b₁ and from station B, a distance d due east of A, on bearing b₂. Both rays must run forward, so a negative result or a negative distance from B is rejected.
Calculator
- Target easting from A, s sin b₁3 km
- Target northing from A, s cos b₁3 km
- Distance from station B to the target4.243 km
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Sighting on 45° from A and 315° from B, 6 km apart, places the target 4.243 km from A, at (3, 3) km.
- The two 45° sight lines of the standard example meet at (3, 3) km, a distance of 3√2 ≈ 4.243 km, because they form a right angle.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (km) | Local step | Change in result |
|---|---|---|---|---|---|---|
| d Distance from A to B (due east) | 1 to 50 km | 0.7071 | 35.36 | 34.65 | +10% | +10 % |
| b2 Bearing from B | 275 to 355 ° | 0.6826 | 7.803 | 7.12 | +10% | +61.3 % |
| b1 Bearing from A | 5 to 85 ° | 4.243 | 5.538 | 1.296 | +10% | +0.309 % |
Sweep of Bearing from A
| Bearing from A (°) | Distance from A to the target (km) |
|---|---|
| 5 | 5.538 |
| 15 | 4.899 |
| 25 | 4.515 |
| 35 | 4.308 |
| 45 | 4.243 |
| 55 | 4.308 |
| 65 | 4.515 |
| 75 | 4.899 |
| 85 | 5.538 |
Sweep one variable
Variables and how each one contributes
Distance from A to the target
What the equation gives you. Everything else feeds it.
Distance from A to B (due east)
The distance from station A to station B, due east of A, in km. A longer baseline moves the target further away in direct proportion. Raising it by 10 % (to 6.6) moves the result by +10 %, from 4.243 to 4.667 km.
Bearing from A
The bearing of the target from station A, in degrees clockwise from north. Turns the ray from A and so moves the crossing point. Raising it by 1 (to 46) moves the result by +0.0152 %, from 4.243 to 4.243 km.
Bearing from B
The bearing of the target from station B, in degrees clockwise from north. Turns the ray from B and so moves the crossing point. Raising it by 1 (to 316) moves the result by +1.75 %, from 4.243 to 4.317 km.
Worked example with the values above
- Angle between the rays at the target: sin(b₁ − b₂) = sin(−270°) = 1.
- Numerator: d cos b₂ = 6 × 0.70711 = 4.2426.
- s = 4.2426 / 1 = 4.2426 km.
Assumptions and limits
- A local flat plane with parallel north lines at all stations, and bearings measured clockwise from north.
- Distances are straight-line distances on level ground, in one consistent unit.
Common mistakes
- Measuring a bearing anticlockwise or from east instead of clockwise from north.
- Forgetting to take the result modulo 360° after adding or subtracting.
- Using the angle of depression or a flat-ground assumption when the ground is sloping.
Statistics and uncertainty
Angle errors dominate: a 1° error in a bearing moves a point 1.7 % of the distance sideways. Intersections of near-parallel sight lines amplify this, so keep the angle between the rays as large as practical.
Where it is used
- Navigation, orienteering and dead reckoning from legs and bearings.
- Surveying heights and positions with a compass, clinometer or theodolite.
Maths you need
- Bearing: angle clockwise from north, 000° to 360°
- Sine and cosine: east and north parts of a leg
- Right triangle trigonometry: sine, cosine and tangent