The intersection of two spheres is generally a circle (but might also be a point or the empty set). That circle will intersect at 3rd sphere in at most two points. This geometric fact is the basis for how GPS (Global Positioning System) receivers work, such as the one in your iPhone.
Imagine a network of satellites in the sky. Each repeatedly sends signals that contain the location of the satellite and the time of transmission. Signals travel at the speed of light (a constant), and reach a GPS receiver at slightly different times. A GPS receiver can tell how far away each satellite is by how long it took for the satellite signal to reach it.
Therefore, if the receiver is R units away from satellite A, then the receiver must lie on a sphere of radius R from A’s position. A second satellite will be enough to determine that the GPS receiver lies on some circle, and a third satellite will narrow the location down to at most 2 points. A fourth satellite (not co-planar with the other 3) is usually enough to fix the receiver’s location!
Many people are under the mistaken notion that GPS receivers transmit data to satellites! In reality, they only receive (which is why they are called receivers) and then mathematics does the rest.
The Math Behind the Fact:
This is the general idea; somewhat more sophisticated mathematics is used to adjust for uncertainties due to miscalibrations in the receiver’s clock or errors in satellite locations. More GPS satellites can also help.
How to Cite this Page:
Su, Francis E., et al. “How GPS Works.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
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