Nearly everyone with a phone these days has a Global Positioning System (or GPS) device right in their pocket--something that can tell them their location within about 50 feet. Most people know that these devices operate in conjunction with satellites in orbit, but the details that go into this calculation reveal some of the intricacies of geometry.
It's tough to think about 3D images in your head, so I've changed the format of this post a bit--I'll be predominantly using images to illustrate how each subsequent satellite signal narrows down your location.GPS satellites orbit the Earth and constantly emit signals that can be picked up by your deviceLet's start by thinking about the problem: the Earth roughly a sphere, which is a three dimensional (3D) object. If it were a perfect sphere and we lived on the surface then our problem would be a bit simpler, because the surface of a sphere is a simpler 3D object (it's really just a 2D sheet wrapped around onto itself). We have plenty of different elevations on the real Earth, though, and what's more we'd like to use our GPS devices on airplanes. In that case we care about our elevation, which brings us back to determining our location anywhere in 3D space. GPS satellites are always emitting signals, as depicted in the cartoon above. These satellites are constantly broadcasting two pieces of information: their location in the sky relative to the Earth, and the time that they emitted the signal. These signals can be picked up by GPS receivers on Earth. At any given time your receiver is likely watching 9 or more GPS satellite signals, so it has a list of signal locations and times those signals were sent.
The signals being sent are electromagnetic waves (just like radio waves, or microwaves, or light, but with a different wavelength), so they always travel at the speed of light.So when your device receives a signal it can subtract the time the signal arrived from the time the signal was sent (which was transmitted as
Here comes an important part: your GPS device now knows how far away it is from this one satellite, but it
If two satellites in two different places each sent signals then they have different location spheres, and together they know the signal must have originated from
Note that it’s possible for the two spheres to touch at exactly one point,
which would then give you the correct answer from only two satellite signals, but that requires you to be
exactly
in between the two satellites. This likely won’t ever happen, and if it did it would only be for a moment—remember the satellites are orbiting the Earth this entire time, in a number of different directions. Note also that it’s possible
for two spheres to intersect at every point, but only if they have exactly the same center—this could only happen if
two satellites crashed into one another, and in that case they probably wouldn’t be sending anymore signals.
Three satellites pins you down to two pointsNow add in a third satellite, like in the figure below. This time we’re interested in the intersection of this new sphere with the circle of possible solutions from the intersection of the
first two satellite signals. If the sphere just touches the edge of the circle, there’s only one solution and we’re done, but again this is unlikely. Most of the time there will be
It’s possible for there to be
Finally, a fourth satellite determines your location exactly
A fourth satellite will be able to determine which of the two possible points is correct, because its signal will only touch one of them, as you can see in the figure below. At this point your GPS device knows your location!
The idea of how many dimensions an object occupies (or, how many numbers it takes to describe it's location) is an important one in both mathematics and physics. As we were adding more satellites for our receiver to detect we were decreasing the possible locations by cutting down dimensions, like when the second satellite took the possibilities from a spherical surface (2D sheet wrapped around on itself into 3D) to a circle (1D line wrapped around on itself into 2D). When we narrowed our search down to two points, how many dimensions do you think one of those points had? Hint: a 3D object can be described by it's length, width, and height, for example--which of these does a point have? Look at the images here for the answer. Advanced physics—GPS and Einstein's theory of relativity
Many people have heard of Albert Einstein's theories, and think of them as complex and hard to grasp. This isn't really the case, and I have a planned post describing these fascinating ideas in relatable terms, but here I just want to mention their interesting consequence for GPS systems. Einstein’s theory of General Relativity says clocks that experience different amounts of gravity will experience time slightly differently. There is just a tiny bit less gravitational field up where the GPS satellites orbit, and so these satellites will disagree slightly with ground-based clocks on how fast time is passing. They account for this by sending a slightly modified signal for their timestamp. Einstein's other theory, Special Relativity, matters as well: this one says clocks moving faster experience time differently than slow moving clocks. This is certainly the case for GPS satellites in orbit--they are moving around 14,000 kilometers/second faster than Earth-based receivers! The combination of these two effects causes GPS satellite clocks lose about 35 milliseconds each day compared to Earth-based clocks. That’s a small amount, but the speed of light is 300,000,000 meters/second, or about a foot per nanosecond—so being off by 1 millisecond = 1,000,000 ns can ruin your measurement by 1000 feet. In other words, only a few minutes of non-relativity-corrected satellite signals would render GPS useless! |

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