Let $P$ be a polygon (perhaps with no acute angles inside) and let $L$ be a line segment. The segment may move through the area inside $P$ in straight lines, orthogonal to $L$, or it may pivot on any point on $L$ (while remaining entirely within $P$).

Let $S$ be a legal sequence of pivots and straight motions for $L$ in $P$, and say $S$ covers $P$ if applying the motions $S$ to $L$, passes over every part of the area in $P$. The covered area of $S$ is the cumulative area passed over by $L$.

- How can we compute the minimum number of pivots over all covering sequences $S$?
- How can we compute the minimum covered area over all covering sequences $S$?

Better formulations of the problem are welcome.

For a rectangular polygon, $h\times w$ and a line segment of length $l$, with $h < w$, and $l < h$. So in the best sequence I can think of the number of pivotes is $\lceil h/l \rceil$, and the second question is just a sum of areas of semicircles of radius $l$, plus the rectangular overlap from the last strip.

```
___________________________
| l--> | 1 pivot at each end
| <--l |
| l--> |
| <--l | 2 pivots if h is not integer multiple of l
---------------------------
```