I would like to:
start with a regular 2D grid like shown in the picture
chose a line of slope $\tan \alpha$, and project, and a window of acceptance (grey area)
project the points of the grid within this window onto the line.
Find the spacing between the projected points on the line
This last point is really my end goal. The thing that I would like to prove is that, if $\tan \alpha$ is rational, then the points on the line will form a periodic structure.
My attempt so far:
From a previous question, I now know how to project $i$ points onto a line with unit vector $v$:
$$ [p_{p1} \ \ p_{p2} \ \ \dots \ \ p_{pn}]=vv^T[p_1 \ \ p_2 \ \ \dots \ \ p_n] $$
How can I include in this expression the projection window?
I.e. that as I change $\alpha$, more points will fall within the grey area and hence will be projected?
(I can do it numerically, e.g. the picture above done in Mathematica, but I am trying to derive it mathematically and more generally)