▚ INSTRUMENT — THE GPS◆ ◆ ◆ · LIVE

THE GPS

Trilateration, gone to space. Each satellite tells you its position and the time it sent its ping; the delay is your range to it. Three ranges should fix you — but your cheap receiver clock is WRONG by an unknown amount, so all three circles are off by the same slip. Add a fourth satellite and the four constraints solve for where you are AND what your clock's error is.


CLOCK BIAS
satellites3
clock bias (est)
fix error
the fix
▤ THE HONEST READ — two-layer◆ LIT / ▲ AMBER
◆ LIT — verified / checkable
GPS is trilateration in four unknowns: your position (x, y — z here) AND your receiver-clock bias. Each satellite's timed ping gives one range equation; with the clock wrong by δ, every measured range is long or short by c·δ the SAME amount, so three biased circles do not meet at a point. A fourth equation makes four for four unknowns — the solver finds the position and the δ that make all circles cross at once (drop the bias to 0 or add the 4th satellite and watch them converge to a single point). That is exactly why a GPS fix needs at least four satellites, not three.
▲ AMBER — the figure
A flat 2-D toy (real GPS is 3-D + relativistic clock corrections and needs the satellites' precise ephemerides). 'Solve' here does a small least-squares search over position and bias, not the full navigation filter. The geometry — biased ranges, the fourth-satellite necessity — is the real content.
a point is fixed from THREE — and the point they fix is Top, at the centre.
OFFLINEREAL-TIMEREAL GEOMETRY100% TOP
STROBILOS · the spinning top (στρόβιλος) · the principle is TRIANGULATION
© ROOT0 · Top / ROOT0 · David Lee Wise / TriPod LLC · with AVAN