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In an old paper "A New Approach to Geometry of Range Difference Location", there is a mathematical derivation that does not include any mention of the relationship to the underlying problem. In the problem, there are 3 receivers (at $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$) and a transmitter at unknown position $(x_0, y_0)$. We also know the range difference (computed from the time difference of arrival) between each pair of receivers ($\Delta_mn = r_n - r_m$).

The derivation starts by writing the distance (range) from each receiver to the unknown transmitter location as:

$$r_i^2 = (x_0 - x_i)^2 + (y_0 - y_i)^2$$

So far so good. However, the next step only says "Then, $r_j^2 - r_i^2 = \Delta_{ij} (r_i + r_j)$"

The author goes on to show more manipulations to get to the point he is trying to show.

It seems like you can apply the "difference of squares" factorization to $r_j^2 - r_i^2$ to obtain $(r_j-r_i)(r_j+r_i)$ (where $r_j - r_i$ is exactly $\Delta_{ij}$), but what I don't understand is the meaning of $r_j^2 - r_i^2 = \Delta_{ij} (r_i + r_j)$. That is, why is the difference of squared distances from the transmitter to two receivers related to the sum of the distances from the transmitter to the two receivers (besides "because we happened to subtract two things and then find a factorization that happened to contain a term we know)? Does anyone have any insight on this? Perhaps a way to explain this using some kind of triangle theorem in a diagram?

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  • $\begingroup$ Not a bad question, but since the paper is behind a paywall, I'm not sure how many responses you will get. It's also not really clear what you're asking, especially without the context provided by the paper itself. Sometimes there just isn't a particularly intuitive way to think about it. $\endgroup$ – Jason R Apr 6 '13 at 15:40

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