Twenty-four satellites orbit 20,200 km above Earth, each carrying an atomic clock accurate to nanoseconds. Each one endlessly broadcasts a radio message: “here is where I am, and the exact time I sent this”. The message travels at the speed of light: c = 299,792,458 m/s. Click or tap anywhere on the globe to place a receiver. That’s you, holding a phone.

Your receiver hears every satellite high enough above its horizon — right now, 2 usable of 2 visible. It measures distance by timing signals: distance = c × (time received − time sent). The catch: “time sent” is stamped by an atomic clock, but “time received” comes from your phone’s cheap quartz clock — off from true time by some unknown amount. At light speed, being just 1 µs off makes every distance wrong by 300 m. Solving needs four satellites in view — wait while the constellation moves.

Each signal expands from its satellite as a sphere growing at c (slowed here ~60×; the real trip from orbit takes ~70 ms), drawn in its satellite’s color. When a wave reaches you, its ground ring — where the sphere slices through Earth — passes exactly through your position: the signal has arrived. Each locked satellite’s small TX ghost marks where it was when the message left.

The rings now show the demo’s pseudorange: ρ = true distance + c·b. The shared term is the unknown clock error, so they no longer meet at your position. Insist the clock is fine, and the best three-sphere answer lands at the ✕ — 888 km away. The dashed fourth ring was measured too — just not used yet.

A fourth satellite flips the problem. Four unknowns — x, y, z and your clock bias b — and four equations: ‖satᵢ − you‖ + c·b = ρᵢ. The solver finds the single value of b that makes all four spheres pass through one point (here: -888.8 µs). You get your position and an estimate of your receiver clock offset against GPS time. Real receivers use more than four satellites when available, but four is the minimum that makes the position-and-clock problem solvable.