Library / Geometry, coordinates, surveying and trigonometry / GEO-32
Geometry, coordinates, surveying and trigonometry
Endpoint of a bearing traverse
Easting of the end of a path made of up to three straight legs, each given as a distance and a bearing clockwise from north, starting at (e, n).
- Ref
- GEO-32
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- E [km]
Each leg of a path adds d sin b to the easting and d cos b to the northing, when its bearing b is measured clockwise from north. Adding the legs one after another from a start point gives the end point; the easting is shown here and the northing in the derived rows. Set a distance to zero to leave a leg out.
Easting of the end of a path made of up to three straight legs, each given as a distance and a bearing clockwise from north, starting at (e, n). The northing, n + Σ d cos b, is in the derived rows. Set a leg length to zero to leave it out.
Calculator
- End northing, n + Σ d cos b−1 km
- Straight-line distance from the start5 km
- Total distance travelled7 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.
- From (1, 2) the legs finish at (5, −1), 5 km in a straight line from the start.
- Due east adds only to the easting and due south only subtracts from the northing; the 090° and 180° legs of the standard example finish 4 km east and 3 km south.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (km) | Local step | Change in result |
|---|---|---|---|---|---|---|
| e Start easting | −10 to 10 km | −6 | 14 | 20 | +0.4 km | +8 % |
| d1 Leg 1 distance | 0 to 20 km | 1 | 21 | 20 | +10% | +8 % |
| b1 Leg 1 bearing | 0 to 360 ° | −2.978 | 5 | 7.978 | +10% | −0.985 % |
| b2 Leg 2 bearing | 0 to 360 ° | 2.016 | 7.984 | 5.967 | +10% | −18.5 % |
| d2 Leg 2 distance | 0 to 20 km | 5 | 5 | 2.665e-15 | +10% | 0 % |
| n Start northing | −10 to 10 km | 5 | 5 | 0 | +0.4 km | 0 % |
| d3 Leg 3 distance | 0 to 20 km | 5 | 5 | 0 | +10% | 0 % |
| b3 Leg 3 bearing | 0 to 360 ° | 5 | 5 | 0 | +10% | 0 % |
Sweep of Leg 1 bearing
| Leg 1 bearing (°) | End easting (km) |
|---|---|
| 0 | 1 |
| 45 | 3.828 |
| 90 | 5 |
| 135 | 3.828 |
| 180 | 1 |
| 225 | −1.828 |
| 270 | −3 |
| 315 | −1.828 |
| 360 | 1 |
Sweep one variable
Variables and how each one contributes
End easting
What the equation gives you. Everything else feeds it.
Start easting
The starting easting, in km. Moves the end point east by the same amount. Raising it by 0.5 (to 1.5) moves the result by +10 %, from 5 to 5.5 km.
Start northing
The starting northing, in km. It does not change the end easting; it sets the end northing. Does not change the end easting. It moves the end northing by the same amount.
Leg 1 distance
The distance of leg 1, in km. Moves the end point along the bearing of leg 1. Raising it by 0.5 (to 4.5) moves the result by +10 %, from 5 to 5.5 km.
Leg 1 bearing
The bearing of leg 1, in degrees clockwise from north. Turns leg 1, which changes the easting by d₁ cos b₁ per radian. Raising it by 5 (to 95) moves the result by −0.304 %, from 5 to 4.985 km.
Leg 2 distance
The distance of leg 2, in km. Moves the end point along the bearing of leg 2. Raising it by 0.5 (to 3.5) moves the result by +0 %, from 5 to 5 km.
Leg 2 bearing
The bearing of leg 2, in degrees clockwise from north. Turns leg 2, which changes the easting by d₂ cos b₂ per radian. Raising it by 5 (to 185) moves the result by −5.23 %, from 5 to 4.739 km.
Leg 3 distance
The distance of leg 3, in km. Leave it at 0 if the path has two legs. Moves the end point along the bearing of leg 3. Raising it by 0.5 (to 0.5) moves the result by +0 %, from 5 to 5 km.
Leg 3 bearing
The bearing of leg 3, in degrees clockwise from north. Turns leg 3, which changes the easting by d₃ cos b₃ per radian. Raising it by 5 (to 5) moves the result by +0 %, from 5 to 5 km.
Worked example with the values above
- Leg 1 adds 4 × sin 90° = 4 km east; leg 2 adds 3.6739×10−16 km; leg 3 adds 0 km.
- Total easting change: 4 km.
- E = 1 + that = 5 km, with the northing −1 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