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This question is rather specific, and slightly computational in nature. In the picture below the anode/cathode pairs subdivide the rectangular chamber into ~20 different spaces. This is a gas chamber, and particles come in the end (arrow gives direction) and then stop somewhere within the chamber, in one of the spaces. Given information about the x and y positions of the particles, how would you go about determining the space the particle is in? This would be trivial if the anode/cathode subdividing lines were perpendicular to the chamber walls. They are tilted by 30 degrees, thus some sort of geometric adjustment is necessary.

Do I need to map the points to some new grid? One approach I thought of is taking a point and then reconstructing the line that would pass through it and be parallel to the anode/cathode pair lines. This new line would intersect the chamber wall definitively between two of the real grid lines, thus giving a clear definition of which space the point lies in.

This issue has the feel of my overthinking it. Any help would be appreciated.

enter image description here

2 Answers 2

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Let the bin spacing by $s$ and let $x=0,y=0$ be at the start of bin $0$.

Then the point $(x,y)$ is in bin $$ \left\lfloor \frac{x+\frac {y}{\sqrt{3}}}{s}\right\rfloor $$

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Suppose anode/catode $i\in\{1,\ldots,n\}$ intersects the lower red line at a point with coordinates $(x_{i},0)$ and intersects the upper red line at a point with coordinate $(x_{i}-t,h)$. (So that $t$ and $h$ determine the tilt.)

Then anode/catode $i$ is essentially a line $y=a+b\cdot x$, where $a=\frac{h x_{i}}{t}$ and $b=-\frac{h}{t}$. Since $-dy=b\cdot dx$, given particle landing at $(x,y)$, its corresponding position on the lower red line is $x+y\frac{t}{h}$. With 30 degrees, $\tan{\pi/6}=\frac{t}{h}=\frac{1}{\sqrt{3}}$.