How does GPS know where you are?

Model. A 2D slice through Earth’s centre, you and the satellites. Earth’s radius is 6,371 km and GPS satellites orbit about 20,200 km up, so the orbit circle has a radius of 26,571 km; Earth, the orbit and every distance are drawn to scale (the satellite pictures are not). Each satellite broadcasts the time from its atomic clock. The phone notes when the signal arrives, and the delay Δt gives the distance d = c·Δt, with c = 299,792 km/s. Straight overhead that is 20,200 km ÷ c ≈ 0.067 s (67 ms); a satellite near the horizon is about 25,800 km away, ≈ 0.086 s. Your distance from one satellite puts you on a sphere around it (a circle in this slice).

Why four. The phone measures every delay with its own cheap quartz clock. If that clock is off by b, every distance is off by the same c·b: these “pseudoranges” are ρ_i = |s_i − x| + c·b. In 3D there are four unknowns (x, y, z and b), so the phone needs four satellites. Two spheres meet in a circle and three meet in two points, one of them far out in space, which is ruled out; the fourth satellite pins down the clock. The tool draws the unknown clock error as a band of ±c·b around each distance circle; the yellow blob is where all the bands overlap. In a flat slice three circles could already solve the clock, so the slice stands in for the 3D count rather than proving it. With four satellites the clock is solved, and the error left is set to about 5 m, the typical accuracy of a phone under open sky. 1 µs × c = 299.8 m, which is where “300 meters” comes from.

Fixed values. The film’s phone clock starts 5 ms off (illustrative) and the signals are shown in slow motion, about 32× slower than light. The map on the phone is an AI-generated illustration; its scale (about 800 m across the screen) comes from the scale bar we drew, not from a real city. The photos (the city at blue hour, the phone, Earth and the satellite) are AI-generated illustrations; every number comes from the model.

Not modeled. Signal delays in the ionosphere and troposphere; satellite orbit and clock errors (corrected with broadcast data); relativity (the satellite clocks are set to tick slower before launch, because in orbit they gain about 38 µs a day); Earth’s rotation during the signal’s trip; reflections off buildings; receiver noise; and how the satellites’ spread in the sky multiplies the errors (dilution of precision). Phones only receive GPS signals; many also download help over the cell network to find the satellites faster, but nothing is sent to the satellites.

Sources: GPS.gov, Space Segment (orbit altitude about 20,200 km); GPS.gov archive, GPS Accuracy (smartphones about 4.9 m under open sky); Global Positioning System; GNSS positioning calculation; Error analysis for the Global Positioning System.