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- W3143165719 abstract "Let $mathcal{O}$ be a set of $k$ orientations in the plane, and let $P$ be a simple polygon in the plane. Given two points $p,q$ inside $P$, we say that $p$ $mathcal{O}$-emph{sees} $q$ if there is an $mathcal{O}$-emph{staircase} contained in $P$ that connects $p$ and $q$. The $mathcal{O}$-${rm Kernel }$ of the polygon $P$, denoted by $mathcal{O}$-${rm Kernel }(P)$, is the subset of points which $mathcal{O}$-see all the other points in $P$. This work initiates the study of the computation and maintenance of $mathcal{O}$-${rm Kernel }(P)$ as we rotate the set $mathcal{O}$ by an angle $theta$, denoted $mathcal{O}$-${rm Kernel }_{theta}(P)$. In particular, we consider the case when the set $mathcal{O}$ is formed by either one or two orthogonal orientations, $mathcal{O}={0^circ}$ or $mathcal{O}={0^circ,90^circ}$. For these cases and $P$ being a simple polygon, we design efficient algorithms for computing and maintaining the $mathcal{O}$-${rm Kernel }_{theta}(P)$ while $theta$ varies in $[-frac{pi}{2},frac{pi}{2})$, obtaining the angular intervals where: (i) $mathcal{O}$-${rm Kernel }_{theta}(P)$ is not empty, (ii) $mathcal{O}$-${rm Kernel }_{theta}(P)$ optimizes area or perimeter. Further, we show how the algorithms can be improved when $P$ is a simple orthogonal polygon." @default.
- W3143165719 created "2021-04-13" @default.
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- W3143165719 date "2018-02-16" @default.
- W3143165719 modified "2023-09-27" @default.
- W3143165719 title "Optimizing generalized kernels of polygons" @default.
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