Finding the unknown coordinates of a point, knowing its difference of distances

[amirahmadian]

Hi everybody,
Assume that I have 3 "reference points", which are some fixed points with known coordinates (known location on x-y plane).Take "A" as an unknown point (I mean a point that I don't know its coordinates). If a system measures the "difference" of distances from A to that "reference" points (I emphasize on "difference", not the distance itself), can it calculate the coordinates of A (Can it find the unknown point)? I mean having 3 equations that represent the differences of distances to the known (fixed) points, am I able to solve the system and obtain a unique solution for the unknown coordinates? If its possible, are there any simple algorithms to solve these equations? (All the points are on the x-y plane).

 

[albbg]

Let's say you have the three points (x1,y1), (x2,y2) and (x3,y3) as reference and (xa,ya) unknown. The unknown are two: xa and ya then you need two linearly independent equations

The distance from A to 1,2 and 3 are:

d1=sqrt[(xa-x1)^2+(ya-y1)^2]
d2=sqrt[(xa-x2)^2+(ya-y2)^2]
d3=sqrt[(xa-x3)^2+(ya-y3)^2]

So we have three differences:

D12 = d2-d1
D13 = d3-d1
D32 = d2-d3

each pair of them are linearly independents, while the third is the linear combination of the other two. We can choose arbitrary the first and the second, then:

D12 = sqrt[(xa-x2)^2+(ya-y2)^2]-sqrt[(xa-x1)^2+(ya-y1)^2]
D13 = sqrt[(xa-x3)^2+(ya-y3)^2]-sqrt[(xa-x1)^2+(ya-y1)^2]

You can solve it using a solver of non-linear system of equations.

 

[amirahmadian]

D12 = sqrt[(xa-x2)^2+(ya-y2)^2]-sqrt[(xa-x1)^2+(ya-y1)^2]
D13 = sqrt[(xa-x3)^2+(ya-y3)^2]-sqrt[(xa-x1)^2+(ya-y1)^2]

You can solve it using a solver of non-linear system of equations.

Thanks for the reply.
But aren't there more than one solution (answer for (xa,ya)) for the above system of equations? Because it seems to be the intersection of two parabolas. You know I need just one answer for the problem.

 

[albbg]

Yes I think you are rigth, but in this case we need an additional information other than the difference of the distance since we saw only two are linearly independents.

 

[amirahmadian]

I've read something about "Multilateration" on the web. I think it does something similar to what I want to. Any information or experience about this technique??

 

[albbg]

No, I'm sorry I've no experience about that method.

 

[flanello]

What happens if you test if the solutions satisfy d1, d2, and d3?
Perhaps only one solutions satisfy all three equations.