= Radio triangulation
\b[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.<What is Radio Telemetry? by the Smithsonian Migratory Bird Center>[Triangulation: directions obtained at separate receiving locations]{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 gives
$$
x=\frac{L\tan\beta}{\tan\beta-\tan\alpha},\qquad y=x\tan\alpha.
$$
For $L=4\,\mathrm{km}$, $\alpha=45^\circ$ 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.<What is Radio Telemetry? by the Smithsonian Migratory Bird Center>[Radio-tag localization methods and limits of local tracking]{ref}
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