Imagine I take a sheet of paper and use a pin to generate an $N$x$M$ rectangular array of small holes. I then fold the sheet to form a cylinder of radius $r_c$ and height $h_c$, where there are $N$ pinholes around its circumference and $M$ pinholes from the top of the cylinder to the bottom. No edge-effects from the folding process are discernible.
Now, say I pick a coordinate, $C$, in the three-dimensional space inside the cylinder. $C$ is some distance from the bottom of the cylinder, $A$, and some distance from the central-axis of the cylinder, $B$. I then proceed to shine a laser, or thread a very thin string between two pinholes, $(p_1, p_2)$, such that the beam or the string is as close as possible to $C$. Here, the laser or the string can be treated as a one-dimensional chord in the interior of the cylinder.
How do I choose $(p_1, p_2)$ to generate a line containing a coordinate $C*$ as close as possible to my chosen coordinate $C$? In general, how well can I do as a function of the density of the pinhole array and the position of the coordinate in terms of $A$ and $B$? Pressing my luck, in terms of minimizing the (straight-line) difference between $C$ and $C^*$, are there better geometries for the pinholes than a rectangular array?
Update: First of all, thanks to Joseph O'Rourke for the awesome graphic! Secondly, I would be very interested in an analysis of worst-case delta with excluded regions, say, at the top and bottom of the cylinder (as Gerhard Paseman suggested).
Update 2: Joseph O'Rourke states a two-dimensional variant of this problem in Part 2 (P2) of his question "Chord arrangement that avoids confining small or large disks", (http://mathoverflow.net/questions/76980/chord-arrangement-that-avoids-confining-small-or-large-disks).