Given a set \mathcal{O of k orientations in the plane, two points inside a simple polygon P \mathcal{O   if there is an O -staircase contained in P that connects them . The O -kernel of P{\rm  -}{\rm   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  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 (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 .
Let \mathcal{O 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 sees q if there is an O -staircase contained in P that connects p and q . The 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) {2}) such that the area and/or the perimeter of the \{0^{o},90^{o}\}} \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 .
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