Before GPS filled every phone and chart plotter, ships and aircraft often found themselves with LORAN: LOng RAnge Navigation. Achain of synchronized ground transmitters broadcasts carefully timed pulse groups. Your receiver does not need absolute time. It only needs the time difference (TD) between hearing the Master and hearing each Slave.
That is enough to locate yourself.
The chain
- Master (M): the reference transmitter. Everything is timed from its pulse.
- Slave X / Y: partner stations on known baselines from the Master.
- Emission delay: how long after the Master each Slave waits before transmitting (baseline travel time plus a fixed coding delay).
- TD: time from Master arrival to Slave arrival at your receiver.
- Line of position: all points with the same TD for one Master–Slave pair. That curve is a hyperbola.
- Fix: the intersection of two lines of position. You need at least two Slaves.
Why a hyperbola?
Radio waves are electromagnetic waves, so in free space they travel at the speed of light (c). Through air they are still extremely close toc. Real LORAN groundwave paths over Earth are a little slower, and receivers use a known effective propagation speed, but the idea is the same: timing is distance.
A constant difference in travel time is therefore a constant difference in distance to two foci (Master and Slave). The set of points with a fixed distance difference is a hyperbola.
Subtract the known emission delay from each measured TD, multiply by that propagation speed, and you recover the path difference.
Two pairs (M-X and M-Y) give two constants, two curves, and one crossing: the fix.
Try it
The demo below slows light way down so the sequence is visible. Press Play, watch the Master and Slaves fire, then see the two lines of position meet at the receiver.
LORAN-C in reality
Real LORAN-C used roughly 100 kHz groundwave pulses, group repetition intervals, and baselines measured in hundreds of kilometres. Receivers tracked envelopes and cycles for finer timing than a single edge. GNSS replaced operational LORAN in many regions, but the geometry (time difference of arrival and hyperbolas) still shows up in multilateration and backup PNT ideas like eLoran.
This post is a teaching sketch: scaled speed of light, a simplified pulse model, and one receiver. The geometry of the fix is the real thing. For more reading, including historical papers, chain notes, and related programs, see my LORAN GitHub repo.