Radio triangulation

Radio triangulation estimates a radio transmitter's position by combining directions measured from different known locations. Each radio direction finding observation supplies a bearing line, with the forward direction selecting a ray. Their intersection gives an ideal position estimate. This is one method of radio tracking; detecting a signal at one station does not determine its distance.^[ref]

For two stations at $(0,0)$ and $(L,0)$ in a horizontal Cartesian coordinate system, let their bearings make angles $\alpha$ and $\beta$ from the positive horizontal axis. Away from vertical bearings, their lines satisfy $y=x\tan \alpha$ and $y=(x-L)\tan \beta$. Solving givesFor $L=4\mathrm{km}$, $\alpha =45^{\circ }$ and $\beta =135^{\circ }$, the result is $(2,2)\mathrm{km}$. The intersection must also lie in each observed forward direction; intersecting the infinite lines alone is insufficient.

The denominator exposes a limitation: nearly parallel bearings can make the position estimate highly sensitive to angular error. Observations taken at different times can also refer to different target positions. More stations allow consistency checks or a fitted estimate when noisy lines do not meet at a point. Smithsonian bird-tracking researchers use this combination of bearings to locate animals without needing continuous visual contact.^[ref]