How do satellites find you?

The blue dot is not a satellite photograph of you. Satellites broadcast position and timing information. Your receiver estimates travel time, converts it into distance, then asks which location is consistent with those measurements.

Flatten the world to a plane first. One known distance gives us a circle. Every point on that circle satisfies the measurement, so one signal cannot tell us which side of the circle contains the receiver.

Add a second distance. The circles generally intersect at two points. Most possible locations disappear, but two candidates remain. Choosing whichever looks convenient would not be a valid position fix; we need more information.

Here comes the third circle. Only one candidate also satisfies this distance, so the other disappears. Position is determined by combining range constraints. We are measuring distances here, not intersecting three measured angles.

How do we check that dot? Recalculate its distance to every source and compare those values with the measurements. The differences are residuals. Real measurements contain errors, so receivers generally seek a solution that best fits them together.

Open the plane into three dimensions. One range now defines a sphere, and two spheres usually intersect in a circle. Height adds another unknown, so the three-circle conclusion from our flat diagram cannot simply be carried over unchanged.

There is another unknown: the receiver clock is not as precise as the satellite clocks. A shared timing offset biases several measured ranges together. The spheres then fail to meet properly, even if the receiver has not moved.

Count the unknowns: three position coordinates and one receiver clock offset. We need enough independent satellite observations to estimate them together. That is the reason a basic three-dimensional fix is commonly described as requiring at least four satellites.

Watch the spheres adjust together. The model subtracts a shared clock bias from the observed ranges rather than arbitrarily moving one satellite. With a suitable position and offset, the separate constraints become consistent again.

Satellite count is not the whole story; geometry matters too. When sources cluster in similar directions, small range errors can imply large position changes. A wider spread generally supplies more independent constraints on the solution.

Real signals travel through the atmosphere, may reflect from buildings, and rely on imperfect orbit information. Our scaled distance-and-clock model leaves out many of those corrections. It explains the geometry rather than serving as an actual navigation receiver.

Return to that blue dot. Behind it sits a continually updated set of ranges, timing estimates and geometric constraints. From one circle to several spheres and a shared clock correction, you have now seen the basic structure of the calculation.