# Express distance between two points in given proximity of other points

I have two movable receivers, both receivers get straight line signals from fixed transmitters, where the strength of the signal is given in DBm. Assuming the signal strength as a measure for distance, (not a pure measure, since walls/objects alter the signal strength, but let's not dive into physics) how would I express the distance between the two receivers, provided there are enough transmitters in range?

My guess is that this is a case of trilateration, and that by using circle intersections, we can narrow down the location of the receivers in a simple plane. Next we can express the distance between the two receivers within the plane. • Are the locations of the transmitters known? – copper.hat Oct 30 '13 at 17:17
• No. We only know which transmitters each receiver has in range, and the strength of the respective signals. – ln e Oct 30 '13 at 17:26
• Is this a homework problem? – copper.hat Oct 30 '13 at 19:32

Suppose the distance information is exact. The measurements are translation and rotation invariant, so we can assume that one receiver is at the origin (in $\mathbb{R}^2$) and the other is located at $(x,0)$ (and $x \neq 0$). The goal is to determine $|x|$.

Let $\phi:\mathbb{R} \times \mathbb{R}^2 \to \mathbb{R}^2$ be given by $\phi(x,t) = (\|t\|, \|t-(x,0)\|)$. If a transmitter is at $t \in \mathbb{R}^2$, and the other receiver is at $(x,0)$, then the distance measurements for that receiver are given by $\phi(x,t)$.

We note that if $R$ is a reflection through the $x$-axis, then $\phi(x,Pt) = \phi(x,t)$, so we might as well assume that $t_2 \ge 0$.

We want to determine if the measurements $d_k = \phi(x,t_k)$ uniquely determine $|x|$.

Suppose $\phi(x,t) = d$. Then $d_1^2 = t_1^2+t_2^2$, $d_2^2 = (t_1-x)^2 + t_2^2$. Subtracting gives $x^2-2 x t_1 = d_2^2-d_1^2$, from which we get $t_1 = \frac{1}{2}(x + \frac{d_1^2-d_2^2}{x})$, $t_2 = \sqrt{d_1^2-t_1^2}$.

Let $X = \{ (x,d) | x \neq 0, d_2 >0, d_1 > \frac{1}{2} | x + \frac{d_1^2-d_2^2}{x} | \}$. Note that $X$ is open and non-empty (for example, $(1,(\sqrt{2},1)) \in X$).

Define $t:X \to \mathbb{R}^2$ by $t_1(x,d) = \frac{1}{2}(x + \frac{d_1^2-d_2^2}{x})$ and $t_2(x,d) = \sqrt{d_1^2-t_1(x,d)^2}$.

Then we have $\phi(x,t(x,d)) = d$ for all $x$ such that $(x,d) \in X$.

In particular, if for some $\hat{x}$ we have $(\hat{x}, d_k) \in X$ for all $k$, then we have $\phi(x,t(x,d_k)) = d_k$ for $x$ in some neighbourhood of $\hat{x}$, hence the measurements do not uniquely determine $|x|$.