How GPS Works
How a receiver on Earth calculates position from satellite signals — trilateration, atomic clocks, and the relativistic corrections that keep the system honest.
GPS answers one question continuously: given signals from satellites whose positions are known exactly, where am I? The answer is pure geometry executed at atomic-clock precision — and it works everywhere on Earth to a few meters, from a constellation of about 31 satellites orbiting at 20,200 km.
The measurement
Each satellite continuously broadcasts its identity, its orbital ephemeris, and the exact time from its onboard atomic clock. A receiver records the signal's arrival time and compares it with its own clock: the offset between satellite timestamp and arrival time, multiplied by the speed of light, is the pseudorange — "distance" contaminated by receiver clock error. With four satellites, four unknowns (x, y, z, clock offset) are solvable; three would pin the position only if time were perfect, which it never is in a cheap receiver. This is trilateration in three dimensions, and the mathematics descends directly from vectors and matrices.
The hard parts
Three problems dominate engineering. Signal strength: the received signal is about 10⁻¹⁶ watts — beneath the thermal noise floor — and is recovered by correlation with the known spreading code. Atmospheric delay: the ionosphere delays signals by meters, corrected by comparing the two broadcast frequencies. Relativity: the satellites' clocks run fast by about 38 microseconds per day relative to the ground (gravitational blueshift plus special-relativistic time dilation from orbital speed — special relativity and general relativity both), and uncorrected, GPS positions would drift by roughly 10 km per day. The system's designers corrected for it in advance; that correction is one of the most empirically validated tests of relativity in everyday life.
Accuracy
Standard positioning gives several meters; differential corrections, multi-frequency receivers, and augmentation systems (WAAS, RTK) push to centimeters for surveying and agriculture. The constellation is maintained by the US Space Force; GLONASS, Galileo, and BeiDou provide equivalent global coverage, and modern receivers use all of them at once.
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