Library / Geometry, coordinates, surveying and trigonometry / GEO-33
Geometry, coordinates, surveying and trigonometry
Smaller angle between two bearings
The angle, between 0° and 180°, between two bearings taken from the same station and measured from the same north.
- Ref
- GEO-33
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- α [°]
The angle between two bearings is the difference between them, but it must be taken the short way round the compass. The expression ((b₂ − b₁ + 180) mod 360) − 180 maps the difference into −180° to 180°, and the absolute value gives the angle between 0° and 180°. It handles crossings of north, such as from 350° to 020°.
The angle, between 0° and 180°, between two bearings taken from the same station and measured from the same north. It handles bearings either side of north, such as 350° and 020°.
Calculator
- Difference taken the long way round330 °
- Clockwise turn from b₁ to b₂30 °
- Simple difference b₂ − b₁ (can exceed 180°)−330 °
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.
- Bearings of 350° and 20° are 30° apart, the short way across north when they sit either side of 000°.
- The simple difference of −330° would be wrong whenever it passes through north, which is why the modulus is needed.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b2 Second bearing | 0 to 359.9 ° | 2.097 | 178 | 175.9 | +10% | +2 ° |
| b1 First bearing | 0 to 359.9 ° | 3.993 | 176.1 | 172.1 | +10% | — |
Sweep of Second bearing
| Second bearing (°) | Angle between the bearings (°) |
|---|---|
| 0 | 10 |
| 44.99 | 54.99 |
| 89.97 | 99.98 |
| 135 | 145 |
| 179.9 | 170.1 |
| 224.9 | 125.1 |
| 269.9 | 80.08 |
| 314.9 | 35.09 |
| 359.9 | 9.9 |
Sweep one variable
Variables and how each one contributes
Angle between the bearings
What the equation gives you. Everything else feeds it.
First bearing
The first bearing in degrees clockwise from north. Changes the angle by an equal amount in the opposite direction, until the short way round changes. Raising it by 5 (to 355) moves the result by −16.7 %, from 30 to 25 °.
Second bearing
The second bearing in degrees clockwise from north, measured from the same station and the same north. Changes the angle by an equal amount, until the short way round changes. Raising it by 5 (to 25) moves the result by +16.7 %, from 30 to 35 °.
Worked example with the values above
- Difference: b₂ − b₁ = 20 − 350 = −330°.
- Shift by 180 and wrap: ((−330 + 180) mod 360) = 210°, then subtract 180 = 30°.
- Angle = 30°.
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