Using triangulation

Question

I have been presented with 3 known points and the power densities at those points. I need to use those points to find the location of the actual antenna which is generating the signals.

Power Density is calculated through $PD=(power by antenna)/4*Pi*R^2$

and the inverse law also applies, $PD=\frac{1}{R^2}$, where R is the distance between a known point and the antenna.

What is the best method of apporaching this problem using the Mathematica enviroment? Till now I havent been able to tackle this problem mathematically aswell, as I don't know how it would be done.

I did however come across something, which said plotting circles will be the best, and their intersections will give the coordinates? how will this work?

This is the module where I have to write the code:

Triangulate[measure_] := Module[(* Enter your code here*)] 

measure_ contains 3 lists, e.g you would input

Triangulate[{{2000,0,1/2},{0,1000,1},{0,0,1/2}}]

where first two elements are the x and y coordinates, and the last element is the power density at those points.

Any help to get me started will be appreciated, cause at the moment i dont know how to start or tackle this.

Answer 1

I assume (sorry if I'm being wrong) that this is some kind of homework. So I've written an answer as guidance. You'll have to work out some details.

If your problem is three dimensional, you can write for example:

dist[x0_,x1_] := (x0-x1).(x0-x1);
power[x0_,x1_]:= c/dist[x0,x1];
findAnt[{{pow1_,pos1_},{pow2_,pos2_},{pow3_,pos3_}}]:=
                           Solve[power[{x0,y0,z0},pos1]==pow1 &&
                                 power[{x0,y0,z0},pos2]==pow2 &&
                                 power[{x0,y0,z0},pos3]==pow3,{x0,y0,z0}]; 

and find the antenna position for a certain configuration as:

findAnt[{{c/2, {1, 0, 0}}, {c/2, {0, 1, 0}}, {c/2, {0, 0, 0}}}]

(*
-> {{x0 -> 1/2, y0 -> 1/2, z0 -> -Sqrt[(3/2)]}, 
    {x0 -> 1/2, y0 -> 1/2, z0 ->  Sqrt[3/2]}}
*)

Or, if you want to write the function in more a compact form:

findAnt[l:{{_, _}..}]:=Solve[And@@(power[{x0,y0,z0}, #[[2]]] == #[[1]]&/@ l), {x0,y0,z0}]; 

Now you can visualize the possible antenna positions and the measurement points, along with their equipotential surfaces like this:

l   = {{c/2, {1, 1, 0}}, {c/4, {0, 1, 1}}, {c/4, {0, 0, 0}}}; 
sol = findAnt[l];

Graphics3D[{PointSize[.05], Blue, Point[l[[All, 2]]], 
                            Red,  Point[{x0, y0, z0} /. sol], 
           Green, Opacity[.2], Specularity[White, 1], 
          {Sphere[{x0,y0,z0} /.sol[[1]], EuclideanDistance[{x0, y0, z0} /. sol[[1]], #]], 
           Sphere[{x0,y0,z0} /.sol[[2]], EuclideanDistance[{x0, y0, z0} /. sol[[2]], #]]}&/@ 
       l[[All, 2]]}]

Edit:

If the power of the antenna is unknown, you can only solve a 2D problem is still easy to solve:

(*assuming an antenna of unknown power*)
dist[x0_, x1_]  := (x0 - x1).(x0 - x1);
power[x0_, x1_] := c/dist[x0, x1];
findAnt[l:{{_, _} ..}] := Solve[And@@(power[{x0, y0}, #[[2]]] == #[[1]]&/@ l), {x0, y0, c}];

l   = {{1/2, {1, 0}}, {1/15, {0, 3}}, {1/2, {0, 0}}};
sol = findAnt[l];

Graphics[
 {PointSize[.05], Blue, Point[l[[All, 2]]], Red, Point[{x0, y0} /. sol],
  Green, Dashed, 
 {Circle[{x0, y0}/. sol[[1]], EuclideanDistance[{x0, y0}/. sol[[1]], #]],
  Circle[{x0, y0}/. sol[[2]], EuclideanDistance[{x0, y0}/. sol[[2]], #]]} &/@ l[[All, 2]]}]  

And you can show the equipotential curves and the measurement positions for the above solution like this:

Show[Plot3D[{power[{x0, y0}, {x, y}] /. sol[[1]], 
             power[{x0, y0}, {x, y}] /. sol[[2]]}, {x, -5, 5}, {y, -5, 5}, 
             Mesh -> {{1/2, 1/15}}, PlotRange -> {0, .6}, 
             ClippingStyle -> Opacity[0.5],  MeshFunctions -> {#3 &}, 
             PlotStyle -> {{Green, Directive[Opacity[.3], Specularity[3]]}, 
                           {Red,   Directive[Opacity[.4], Specularity[3]]}}], 
     Graphics3D[{PointSize[.03], Blue, 
             Point@Join[#, {power[{x0, y0}, #]}] & /@ l[[All, 2]] /. sol[[1]]}]]

---

Finally, I want to mention a method that is probably out of the scope of your homework, but can take advantage of the utilization of more than three sensors by using minimization techniques on the potential parameters:

model = c/ ((x0 - x)^2 + (y0 - y)^2 + (z0 - z)^2);
(* data is {x, y, z, power} *)
data = {{1, 0, 0, 1/2}, {0, 1, 0, 1/2}, {0, 0, 0, 1/2}};
fit = NonlinearModelFit[data, model, {c, x0, y0, z0}, {x, y, z}];
fit["BestFitParameters"]
(*
 -> {c -> 1.25371, x0 -> 0.5, y0 -> 0.5, z0 -> 1.41684}
*)

And now you can test that the solution really fits the input data:

model /. fit["BestFitParameters"] /. Thread[{x, y, z} -> #] & /@ data[[All, 1 ;; 3]]
(*
->  {0.5, 0.5, 0.5}
*)