Optimal shape for stabbing balls in $\mathbb{R}^3$

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:

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

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

An answer 'Yes' to Q1 answers Q0 as well.

• The expected number of balls the ray intersects should be the volume of an $r$-neighborhood of the path. – HenrikRüping Feb 15 '15 at 3:18
• Great question, but you may want to rethink the title... – Jonathan Beardsley Feb 15 '15 at 4:38
• I think you want to rule out a very short chord somehow. – Douglas Zare Feb 15 '15 at 6:17
• As Douglas Zare suggests, a tangent line has infinite expected hit/length ratio. – S. Carnahan Feb 15 '15 at 10:19
• @DouglasZare & SCarnahan: Good point re boundary effects. – Joseph O'Rourke Feb 15 '15 at 12:11

From HenrikRüping's comment, we want to consider the volume of an $r$-neighborhood of the topological arc $\gamma$.
Hotelling proved that the volume of an embedded tubular neighborhood of an arc only depends on the length. The $D^2 \times \gamma$ can fail to be embedded due to local or nonlocal self-intersections, both of which decrease the volume. As long as it does not come back near itself or have curvature greater than $1/r$, the $r$-neighborhood of the arc has volume $\pi r^2 |\gamma| + \frac{4}{3} \pi r^3$.