Direct and inverse problems
These are the two basic computations of plane surveying. The X axis points north and Y east; directions are given by the grid bearing α, measured clockwise from grid north, 0° to 360°.
- Direct problem: given X₁, Y₁, bearing α and distance d, find X₂, Y₂.
- Inverse problem: given two points, find α and d — e.g. the initial bearing of a traverse or setting-out data.
Direct problem
ΔX = d·cos α, ΔY = d·sin α, X₂ = X₁ + ΔX, Y₂ = Y₁ + ΔY.
Inverse problem
ΔX = X₂ − X₁, ΔY = Y₂ − Y₁; the quadrant bearing r = arctan (ΔY / ΔX) taken as an absolute value; the quadrant from the signs:
| ΔX | ΔY | Quadrant | Bearing |
|---|---|---|---|
| + | + | NE | α = r |
| − | + | SE | α = 180° − r |
| − | − | SW | α = 180° + r |
| + | − | NW | α = 360° − r |
d = √(ΔX² + ΔY²), check d = ΔX / cos α = ΔY / sin α; the back bearing is α ± 180°.
Example
X₁ = 6243.17, Y₁ = 4127.52, X₂ = 6098.44, Y₂ = 4305.91 m. ΔX = −144.73, ΔY = +178.39, r = 50°56′50″, quadrant SE, α = 180° − r = 129°03′10″, d = 229.717 m.