geometry - Find a straight tunnel 2

You have the same plugging task as in Find a straight tunnel , but now the problem requires a bit more imagination:

Bob has a field, which is a regular polygon with $N$ sides and perimeter $P$. There is a tunnel, which is just under the surface, but invisible - unless you dig. It is known that the tunnel goes under the field (at least touches it at one point), it is straight and infinitely long (in both directions).
Bob wants to find the tunnel. To do this he has a plow and can dig along some lines with it. If Bob plows and crosses the tunnel he will find it. Bob doesn't want to plow too much so he first figured a strategy out, which guarantees to find the tunnel by plowing at most $L$ meters.
What can be the smallest $k = L/P$ ratio, given $N$ can be any number from 3 to $\infty$?

Bob is allowed to plow only inside of his field. He can take the plow out of the ground and move it over the ground without plowing.

Important to note that now you can plow only inside and you can chose the field, which is best for you (i mean to chose $N$).

I tried extreme cases:
$N=\infty$ - it is circle and the best path is along the perimeter, so $k_{min}=1$.
$N=3$ - it is triangle, I belive the best case is a path along medians, then $k_{min}=1/\sqrt{3} \approx 0.5774$.
The best case should be somewhere inbetween.