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 ---------------------------