Demo — Geolocation Explorer#
One bearing is a line; cross two and you have a fix. But the quality of that fix is set almost entirely by geometry — the angle at which the bearings cross — not by how good the receivers are. This demo lets you drag the receivers and the emitter and watch the error ellipse breathe.
The idea#
Each receiver gives a line of position with an angular uncertainty band. The fix lives where the bands overlap, and the shape of that overlap is governed by the cut angle: near \(90^\circ\) gives a tight, near-circular error; a shallow cut smears it along range.
Interactive demo#
Walkthrough#
Start with a good cut. With the emitter abeam the receiver pair, the two bearings cross near \(90^\circ\) and the error ellipse is small and round. Note the CEP readout.
Push the emitter downrange. As it moves far from the baseline, the cut angle shrinks, the ellipse stretches along the line of sight, and CEP grows — the same receivers now do far worse.
Drag onto the seam. Move the emitter onto the receivers’ baseline. The cut collapses, range becomes unobservable, and the ellipse and CEP blow up along the line of sight — the geometry has failed, not the hardware.
Widen the baseline. Drag one receiver out perpendicular to the line of sight to reopen the cut toward \(90^\circ\) and watch the ellipse tighten again.
Add a third receiver. Tick Add R3 (LS fix). Three bearings almost never meet at a single point, so the fix becomes the least-squares position that best fits all three — the overdetermined case from the reading. The ellipse tightens, and a third receiver placed off the pair’s line can rescue a cut the pair alone could not open.
Magnify the ellipse. Magnify ellipse ×10 is on by default — the dashed copy is the same ellipse, same centre, same tilt, radii ten times larger. Drag the emitter and watch it breathe; at map scale the true ellipse is only a few pixels across.
Read the PDOP tile. One number for the geometry: small is good, and it climbs as the cut closes. Hit the Seam (collinear) chip and it reads ∞. Tick R3 on and it comes back finite.
Turn on the DOP map. The shading is the PDOP a hypothetical emitter would get at each point, for the receivers exactly where they sit — light is good, red is poor. The seam shows as a red ridge running out both ends of the baseline. Tick Add R3 (LS fix) and watch the third bearing break the ridge.
Turn on the fix cloud. Three hundred fixes are solved from bearings drawn with the same ±0.5° noise. The CEP tile then reports the share that landed inside the CEP circle — read it against the 50% the estimate promises.
Turn on the threat ring. Drag a receiver toward the emitter. The ellipse tightens as the range falls, and the receiver’s marker turns red the moment it crosses inside the ring — the accuracy is bought with exposure.
Turn on ghost fixes. A second emitter appears with its own bearings. Two emitters, two receivers, four crossings — two real, two ghosts — and from bearings alone the processor cannot tell which is which. Tick Add R3 (LS fix): the third bearing runs through the two real fixes and misses each ghost by kilometres, and the ghost labels say by how much.
Key observations#
Geometry sets the error, not receiver quality. This is the whole lesson — the receivers never change in the demo; only the geometry does.
A shallow cut smears error along range. Cross-range stays decent; downrange blows up.
Wide baselines hold a strong cut at long range — exactly the standoff geometry Project 2 exploits to geolocate from outside the threat rings.
DOP is geometry, and geometry is a map: the DOP map shows where a formation can locate well before anyone takes a bearing.
A fix you never associated is a guess. The ghosts cross at a perfectly good angle and sit on perfectly good bearings — only a third, independent look shows they belong to no emitter at all. That is why the reading calls association the one step that cannot be skipped.
Source#
MATLAB bundle · L14_TwoReceiverCrossFix.m↓
The companion script crosses two noisy bearings into a fix, repeats it thousands of times to map the error cloud, then moves the emitter downrange to show the same receivers degrade as the cut angle shrinks.