Generalized kernels of polygons under rotation

Given a set $\mathcal{O}$ of $k$ orientations in the plane, two points inside a simple polygon $P$ $\mathcal{O}$-see each other if there is an $\mathcal{O}$-staircase contained in $P$ that connects them. The $\mathcal{O}$-kernel of $P$ 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 the $\mathcal{O}$-${\rm Kernel}$ of a polygon $P$ as we rotate the set $\mathcal{O}$ by an angle $\theta$, denoted $\mathcal{O}$-${\rm Kernel}_{\theta}(P)$. In particular, we design efficient algorithms for (i) computing and maintaining $\{0^{o}\}$-${\rm Kernel}_{\theta}(P)$ while $\theta$ varies in $[-\frac{\pi}{2},\frac{\pi}{2})$, obtaining the angular intervals where the $\{0^{o}\}$-${\rm Kernel}_{\theta}(P)$ is not empty and (ii) for orthogonal polygons $P$, computing the orientation $\theta\in[0, \frac{\pi}{2})$ such that the area and/or the perimeter of the $\{0^{o},90^{o}\}$-${\rm Kernel}_{\theta}(P)$ are maximum or minimum. These results extend previous works by Gewali, Palios, Rawlins, Schuierer, and Wood.Comment: 12 pages, 4 figures, a version omitting some proofs appeared at the 34th European Workshop on Computational Geometry (EuroCG 2018