A Third Practical Positioning Example

  When one can bounce a signal off an object and measure the time it takes for the signal to travel to and from the object one can get an estimate of the range of the object. One can do this with the "echo" of a radar or sonar pulse and the strength of the returned pulse is improved if the object has a transponder. The strength of a pulse inside a cone follows the inverse square law and this also applies to a reflected pulse so the strength of the returned pulse follows an inverse 4th power law. The intensity with a transponder gives a stronger return signal but one needs power to generate the return pulse.

  Again one can triangulate the unknown position, X, given two positions, P₁ and P₂, and their ranges, r₁ and r₂. To find X one needs the distance of X, c, from the line connecting P₁ and P₂. If the closest point on the line to X is A then the distance, a, is determined as follows.

Using the formulas derived above one can easily find the distances a and c and consequently the positions A and X.