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 \(\mathcal {O}\)-sees q if there is an \(\mathcal {O}\)-staircase contained in P that connects p and q. The \(\mathcal {O}\)-Kernel of the polygon P, denoted by \(\mathcal {O}\)-\(\mathrm{Kernel }(P)\), is the subset of points of P which \(\mathcal {O}\)-see all the other points in P. This work initiates the study of the computation and maintenance of \(\mathcal {O}\)-\(\mathrm{Kernel }(P)\) as we rotate the set \(\mathcal {O}\) by an angle \(\theta \), denoted by \(\mathcal {O}\)-\(\mathrm{Kernel }_{\theta }(P)\). 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 the \(\mathcal {O}\)-\(\mathrm{Kernel }_{\theta }(P)\) while \(\theta \) varies in \([-\frac{\pi }{2},\frac{\pi }{2})\), obtaining: (i) the intervals of angle \(\theta \) where \(\mathcal {O}\)-\(\mathrm{Kernel }_{\theta }(P)\) is not empty, (ii) a value of angle \(\theta \) where \(\mathcal {O}\)-\(\mathrm{Kernel }_{\theta }(P)\) optimizes area or perimeter. Further, we show how the algorithms can be improved when P is a simple orthogonal polygon. In addition, our results are extended to the case of a set \(\mathcal {O}=\{\alpha _1,\dots ,\alpha _k\}\).

令\（\ mathcal {O} \）为平面中的k个方向的集合，令P为平面中的简单多边形。鉴于两点p，  q内P，我们说p \（\ mathcal {Ø} \） -看到 q如果有一个\（\ mathcal {Ø} \） -楼梯包含在P连接p和 q。的\（\ mathcal {ö} \）-kernel多边形的P，记为\（\ mathcal {ö} \） - \（\ mathrm {内核}（P）\）是P点的子集，其中\（\ mathcal {O} \） -看到P中的所有其他点。这项工作启动了\（\ mathcal {O} \） - \（\ mathrm {Kernel}（P）\）的计算和维护的研究，因为我们将集合\（\ mathcal {O} \）旋转了一个角度\（\ theta \），由\（\ mathcal {O} \） - \（\ mathrm {Kernel} _ {\\ t​​heta}（P）\）表示。特别地，我们考虑以下情况：集合\（\ mathcal {O} \）由一个或两个正交方向\（\ mathcal {O} = \ {0 ^ \ circ \} \）或\（ \ mathcal {O} = \ {0 ^ \ circ，90 ^ \ circ \} \）。对于这些情况，并且P是一个简单的多边形，我们设计了有效的算法来计算\（\ mathm {O} \） - \（\ mathrm {Kernel} _ {\ theta}（P）\）而\（\ theta \ ）在\（[-\ frac {\ pi} {2}，\ frac {\ pi} {2}）\）中变化，得到：（i）角度\（\ theta \）的间隔， 其中\（\ mathcal {O} \） - \（\ mathrm {内核} _ {\ theta}（P）\）不为空，（ii）角度\（\ theta \）的值， 其中\（\ mathcal {O} \） - \（\ mathrm {内核} _ {\ theta}（P）\）优化面积或周长。此外，我们展示了如何在P时改进算法。是一个简单的正交多边形。另外，我们的结果扩展到集合\（\ mathcal {O} = \ {\ alpha _1，\ dots，\ alpha _k \} \）的情况。