[Part 6] Why are 4 GPS satellites necessary? | GPS mechanics you can read in 5 minutes | Explained as clearly as possible in the universe

Last time, I explained that you need at least 4 GPS satellites to be "visible" to measure your position with GPS, but what is that actually like?
I mentioned that while it can be difficult to get 4 satellites with just GPS, it seems easy enough with GNSS, which combines satellite positioning systems from various countries.

If you want to know more details, check out the previous article.

And this time, as promised, I will answer the question: why are 4 satellites necessary in the first place?

📏 You can determine distance by receiving radio waves from GPS satellites

First, before explaining why 4 are needed, there is a necessary prerequisite.
That is that the distance between the GPS satellite and the receiver (you) is known.

So, first, let's talk abouthow to measure distance.

Actually, the reason we can know the distance to a satellite is thanks to a very important rule.
That is,the speed at which radio waves travel is almost constant.Therefore, if you know the time it takes for the
radio waves transmitted from the satellite to arrive, you can easily determine the distance.

For example, suppose the speed of radio waves is 1 kilometer per second.
The GPS satellite transmits a message saying, "It is exactly 12 o'clock now."
If the receiver receives that radio wave at 12:00:02, it means the radio wave took 2 seconds to arrive.
Since the radio wave traveling at 1 kilometer per second arrived in 2 seconds, you can see through simple calculation that the distance between the satellite and the receiver is 2 kilometers.

In reality, the speed of radio waves is almost the same as the speed of light, which is about 300,000 kilometers per second.
GPS satellites are flying 20,000 kilometers above, so it takes about 0.07 seconds for the signal to arrive.

✌️ When thinking in 2 dimensions, two GPS satellites are necessary

Now, we know the distance to the GPS satellite!
Next is the method to measure the position from there.
Since we are in a state where we know the distance to each satellite, in a word, it is "trilateration."
That would be the end of it, but I will explain it clearly.

To make the explanation easier to understand, let's think in 2 dimensions (a plane).
What happens if there is only one GPS satellite?

If the distance to satellite A is known to be 2 kilometers.
You know you are somewhere on a circle with a radius of 2 kilometers centered on satellite A.

However, you cannot narrow it down to a single point with just this, right?

Then,what if there are two GPS satellites??
If the distance to another satellite B is 3 kilometers, it means you are somewhere on a circle with a radius of 3 kilometers centered on satellite B.
This means you can narrow it down to two points, which are the intersections of the two circles centered on satellite A and satellite B, respectively.

Furthermore, one of the points is in an impossible location in outer space, so it is excluded.
As a result, it is perfectly determined as a single point.

🧊 When thinking in 3 dimensions, are three GPS satellites enough??

When thinking in a plane (2 dimensions), we understood that 2 satellites are necessary.
Then, what about in 3 dimensions?

The answer is simple: in addition to 2D horizontal (width) and vertical (height) information, you need 'depth' information, so you simply need one more satellite.
So, that means you can determine your position with 3 GPS satellites.
That's it!!

Wait??
Didn't we need 4 GPS satellites?
What's going on??

⌛ Why is a 4th GPS satellite necessary?

That's right. You need 4 GPS satellites, not 3.
Actually, there was one thing missing to derive your position using the method I've explained so far.
What could that be?

The correct answer is the
“time it takes for radio waves to arrive”
needed to know the distance between the GPS satellite and the receiver!!

I'm sure you're thinking, 'Huh??'
So, I'll explain it in an easy-to-understand way.

To measure the distance between a GPS satellite and a receiver, you need to measure the time it takes for 'radio waves,' which travel at an incredibly fast 300,000 kilometers per second, to leave the GPS satellite and arrive at the receiver.
If the time is off even a little bit, the value will be completely different, so not even the slightest discrepancy in time between the GPS satellite and the receiver is allowed.

For this reason, GPS satellites use an incredibly high-precision clock called an “atomic clock”.
How accurate is it? It won't lose a second in 300,000 years.
(In reality, the precision varies considerably depending on the type of atomic clock, such as cesium atomic clocks, rubidium atomic clocks, and hydrogen maser atomic clocks.)
On the other hand, they are large and extremely expensive.

So, what about the receiver?
A receiver is, in other words, a smartphone or a car navigation system.
It seems a bit impossible to equip a smartphone with an atomic clock, right?
In reality, the clock in a GPS receiver is quite accurate for a general electronic device. However, even then, it only has a monthly error of about 1 second (it only drifts by a maximum of about 1 second per month).
It cannot be compared to an atomic clock.

So, how much the receiver's clock is off?
is unknown.
That's a problem...

But don't worry!
If you have 4 GPS satellites, you can derive the time discrepancy of the receiver.

I think it would be confusing if I showed you the mathematical formulas here, so I'll explain it as simply as possible.
The 'unknown (values you want to know)' are the receiver's position, which is the coordinates (x, y, z), and the time discrepancy (t), making 4 values in total.
And to find 4 'unknown values,' you need to solve a system of equations consisting of 4 equations.
If you have 4 combinations of GPS satellites and receivers, you get 4 equations, right?
So, having 4 GPS satellites is enough.

Did you understand??
What? Still confusing?

Well, just accept it as: 'Since the smartphone clock isn't accurate, you need 4 satellites' (lol).

📝 Summary

So, this time I explained why 4 GPS satellites are necessary.
It turned out to be quite a theoretical topic...
However, the theoretical talk will continue a little longer.

Because this explanation was just the groundwork to answer the question that came up in the previous article:
'Does it matter which 4 satellites I see?' (sweat)

So, next time, I'll explain why it's better to have many satellites.
Look forward to it~~!!