Library / Navigation / NAV-03
Navigation
Range and bearing to a point
Straight-line distance and compass bearing from one point to another on a local flat (north–east) map.
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- Result
- ρ [m]
ΔN and ΔE are the north and east offsets from your own position to the point, in a local flat map. The range is the straight-line distance √(ΔN² + ΔE²) and the bearing χ = atan2(ΔE, ΔN) is measured clockwise from north. The same pair describes a waypoint to fly to and a target as a radar or other sensor would report it. The bearing is shown among the derived results.
Straight-line distance and compass bearing from one point to another on a local flat (north–east) map. The same two numbers describe a waypoint to fly to and a target as seen by a radar.
Calculator
- Bearing, clockwise from north53.13 °
- Reciprocal bearing233.1 °
- Bearing in NATO mils (6400 to a circle)944.5 mils
- Range in kilometres5 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.
- A point 3,000 m north and 4,000 m east of you is 5,000 m away on a bearing of 53.13°. This 3-4-5 layout is a convenient check: the range should come out as exactly 5 units for 3 and 4.
- A bearing error converts into a sideways position error equal to the range times the angle in radians. At 5 km a 1° error moves the point by 87.3 m.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ΔE East offset to the point | −5,000 to 5,000 m | 3,000 | 5,831 | 2,831 | +200 m | +3.23 % |
| ΔN North offset to the point | −5,000 to 5,000 m | 4,000 | 6,403 | 2,403 | +200 m | +2.45 % |
Sweep of North offset to the point
| North offset to the point (m) | Range (m) |
|---|---|
| −5,000 | 6,403 |
| −3,750 | 5,483 |
| −2,500 | 4,717 |
| −1,250 | 4,191 |
| 0 | 4,000 |
| 1,250 | 4,191 |
| 2,500 | 4,717 |
| 3,750 | 5,483 |
| 5,000 | 6,403 |
Sweep one variable
Variables and how each one contributes
Range
What the equation gives you. Everything else feeds it.
North offset to the point
The offset towards north in m. Negative values are towards south. Lengthens the range, though by less than the same percentage because the east offset also contributes. Raising it by 10 % (to 3,300) moves the result by +3.71 %, from 5,000 to 5,186 m.
East offset to the point
The offset towards east in m. Negative values are towards west. Lengthens the range, and turns the bearing towards east. Raising it by 10 % (to 4,400) moves the result by +6.51 %, from 5,000 to 5,325 m.
Worked example with the values above
- Squares: ΔN² + ΔE² = 9,000,000 + 1.6×107 = 2.5×107.
- Range: ρ = √2.5×107 = 5,000 m.
- Bearing: atan2(4,000, 3,000) = 53.13°, clockwise from north.
Assumptions and limits
- A local flat Cartesian map with north and east axes, valid over a limited area (not across continents).
- Coordinates in metres in the same frame for both points, with the angle measured clockwise from north.
Common mistakes
- Swapping the arguments: atan2(ΔN, ΔE) gives an angle from east, counter-clockwise, which is not a compass bearing.
- Applying flat-map formulas over long distances, where Earth curvature matters and great-circle formulas are needed.
Statistics and uncertainty
If the bearing is measured to ±1° and the range to ±10 m, the position uncertainty has a cross-range part of 87.3 m and a range part of 10 m. The first grows in proportion to the distance, so at long range angular error dominates, while at short range range error does.
Where it is used
- Waypoint navigation: the distance to go and the course to fly.
- Converting a target position to the range and bearing a radar or sensor would report.
Maths you need
- Pythagoras' theorem: distance from the offsets
- The atan2 function: angle of a point, with the quadrant resolved