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My problem is the following: A square wooden plate is hit with a bullet. There are sensors at the four corners. Length of the sides are known. Speed of sound in wood is known. The moments where the four sensors detect the shock are known while the exact time of hit is unknown. I am supposed to calculate coordinates of the hit point.

So, we have square ABCD. Length of one side is L. Inside this square there is a point P(x,y). We also know the following: [PB]-[PA], [PC]-[PA], and [PD]-[PA]. We will find values of x and y.

Does this problem have a solution? Do I have enough information? If so, can you provide a solution?

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Later I found this Wikipedia article: and this thesis, which simplifies math a lot (to linear equations!) and provides source code (which I tried, it works!), very helpful indeed. – niw3 Oct 31 '13 at 6:06
up vote 0 down vote accepted

You have the differences in the time of arrival of signal. From each one of these differences you can tell the point of hit is on a certain hyperbola. The intersection of two of these hyperbolas will give you a point. The third hyperbola can help choose one of possible answers.

Note one of the properties of hyperbola is that it is the set of points such that the difference of their distances from the two foci of hyperbola is a fixed quantity. In your problem the fixed quantity is the time difference in arrival of signals times the speed of sound in the medium. Your foci are the two corners used in measuring the time difference.

You may want to read about applications of hyperbola.

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For each given $t$ when we assume the bullet hit, we may calculate the difference in time to each corner, and hence calculate a distance to each corner. For each $t$ there is therefore a radius around each corner on which the bullet must lie. We now find a $t$ where all four circles intersect at a single point -- this point is where the bullet hit. It is possible that there is more than one such point, and also possible that there is no such point (if there is error in measurement).

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Suppose sensor $i$ detects the impact at time $t_i$. Let $d_i$ be the distance from the hit point to sensor $i$, and let $v$ be the speed of sound. Then we know that $$ d_i = (t_i - t)v \quad \text{for } i=1,2,3,4 $$ Taking $i=1$ and $i=2$, and subtracting, we get $d_1 - d_2 = (t_1 - t_2)v$. So, the difference between the distances $d_1$ and $d_2$ is a known constant, and this means the impact point must lie on a hyperbola $H_{12}$ having sensor #1 and sensor #2 as its foci. See this page for details.

By similar reasoning, the impact point lies on five other hyperbolas $H_{13}$, $H_{14}$, $H_{23}$, $H_{24}$, $H_{34}$ (with the obvious notation). In theory, if the measurements were exact, the 6 hyperbolas would have a common intersection point, and this is where the impact point would lie. In practice, the measurements would not be exact, and the calculation would need to use some tolerances.

My intuition says that the available data uniquely defines the impact point, but I can't prove it. I suppose it's conceivable that the 6 hyperbolas could have more than one point in common, but I doubt it.

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I see now that Maesumi gave essentially the same solution as mine, and gave it more quickly. But I'll leave my answer here, too, because it has a bit more detail. – bubba Oct 5 '13 at 15:00
Thanks for the detailed answer. I had to choose Maesumi's answer because he was first as you said. – niw3 Oct 6 '13 at 6:06

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