Generalized kernels of polygons under rotation

02/16/2018

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by   David Orden, et al.

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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- Kernel_θ(P). In particular, we design efficient algorithms for (i) computing and maintaining {0^o}- Kernel_θ(P) while θ varies in [-π/2,π/2), obtaining the angular intervals where the {0^o}- Kernel_θ(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^o,90^o}- Kernel_θ(P) are maximum or minimum. These results extend previous works by Gewali, Palios, Rawlins, Schuierer, and Wood.