Given this coordinates key of "radio towers" -- Pretend these integers are in feet:
{
tower1 : [18, 0],
tower2 : [6, 0],
tower3 : [12, 6],
tower4 : [0, 0],
tower5 : [18, 18]
}
And given this dataset of "signal strength" -- Assume 1.0 means "I'm zero feet away, right on top of the tower" and 0.0001 means "I'm 50 feet away, that's as far as I can go and still receive from this tower":
['tower1', 0.8388],
['tower2', 0.2738],
['tower3', 0.6245]
How can I infer the estimated coordinates of the "radio receiver"?
An overly-simplified example to indicate what I'm hoping to achieve :
['tower1', 0.0],
['tower2', 1.0],
['tower3', 0.0]
---------------
[6, 0]
Things to consider :
- The potential broadcast strength of all towers is equal
- I made up those example strength values as I typed this so they might not make mathematical sense. So maybe it's easier if they're just variables.
- Assume 1.0 == 0 feet, and 0.001 == 50 feet, then a value of 0.5 would mean "I'm twenty five feet from this tower"
This is kind of a home game/project I'm trying to build with some Arduinos for my kids. If I'm going about it completely the wrong way I'm willing to accept that too.
Another way to word this question is like, if I had a few magnets around the room, where "1.0" would mean I'm touching a particular magnet, and "0.0" would mean I don't feel it at all -- if I'm standing in the room and can "feel" the pull strengths of three magnets as a percentage, how could I use that data to determine where I'm standing in the room?
Essentially, given the data available to me, I'm hoping to find the coordinates of "X" :
Given three distances from three distinct points, it is possible to calculate an unknown point. This wikipedia page has the formulas to do this under the "Derivation" section, and you might have to read through it a bit to ascertain which of the formulas you're looking for and what the variables mean.
The "power" concept you're using isn't well defined though. If you can find some function that yields power when given distance (so that $\text{power} = f(\text{distance})$), you can invert the function and plug in the distances to the formulas from that wikipedia page. If not, you're going to have to reframe your "power" idea.
To take a stab at figuring out a function like that, I would use the function $\text{power} = \frac1{1+\text{distance}^2}$. This function looks like this:
As you can see, power is $1$ when distance is $0$, and it tapers off relatively quickly. The nice thing about this function specifically is that it's easily invertible: $\text{distance} = \sqrt{\frac1{\text{power}}-1}.$