Generalized kernels of polygons under rotation

Given a set O of k orientations in the plane, two points inside a simple polygon P O-see each other if there is an O-staircase contained in P that connects them. The O-kernel of P is the subset of points which O-see all the other points in P . This work initiates the study of the computation and maintenance of the O-Kernel of a polygon P as we rotate the set O by an angle θ, denoted O-Kernelθ(P ). In particular, we design efficient algorithms for (i) computing and maintaining {0}-Kernelθ(P ) while θ varies in [− 2 , π 2 ), obtaining the angular intervals where the {0◦}Kernelθ(P ) is not empty and (ii) for orthogonal polygons P , computing the orientation θ ∈ [0, π 2 ) such that the area and/or the perimeter of the {0◦, 90}-Kernelθ(P ) are maximum or minimum. These results extend previous works by Gewali, Palios, Rawlins, Schuierer, and Wood.