I have radius $r < \frac{1}{2}$ congruent balls with centers randomly distributed uniformly within a region, say, within a unit-radius sphere $S$. I shoot a ray/path through $S$, hoping to intersect many balls. The cost of my probe is the length of the ray within $S$; the payoff is the number of balls intersected. My question is:

. What is the optimal shape of the ray, as a function of $r$?Q0

For example, above shows $100$ balls of radius $r=\frac{1}{10}$. On the left I shoot a (green) straight-line ray of length $2$, and pierce $3$ (red) balls, for a hit/length ratio of $1.5$. On the right I shoot a spiral ray of length $3.57$ through the same distribution of balls (but displayed rotated), and pierce $4$ balls, for a hit/length of $1.1$.

Intuition might suggest that the shape of the ray probe is irrelevant. But this cannot be correct, for a jittery path with deviations much smaller than $r$ is inefficient in that such oscillations increase the path length without increasing the likelihood of intersecting more balls.

It may be possible to answer the following '*No*' without entirely resolving ** Q0**:

. Is the optimal shape in fact a straight-line probe, independent of $r$?Q1

An answer '*Yes*' to ** Q1** answers

**as well.**

*Q0*