Optimizing generalized kernels of polygons

TL;DR

This paper studies the computation of generalized polygon kernels based on multiple orientations, providing algorithms for their optimization as orientations rotate, with applications to simple and orthogonal polygons.

Contribution

It introduces algorithms for computing and optimizing $ heta$-rotated $ ext{O}$-kernels for polygons with one or two orientations, extending to multiple orientations.

Findings

Algorithms for $ ext{O}$-kernel computation as $ heta$ varies.

Identification of angle intervals where the kernel is non-empty.

Methods to optimize kernel area or perimeter.

Abstract

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 $p$ $\mathcal{O}$-\emph{sees} $q$ if there is an $\mathcal{O}$-\emph{staircase} contained in $P$ that connects $p$ and~$q$. The \emph{$\mathcal{O}$-Kernel} of the polygon $P$, denoted by $\mathcal{O}$-$\rm kernel(P)$, 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 $\mathcal{O}$-$\rm kernel(P)$ as we rotate the set $\mathcal{O}$ by an angle $\theta$, denoted by $\mathcal{O}$-$\rm kernel_{\theta}(P)$. 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 the $\mathcal{O}$-$\rm kernel_{\theta}(P)$ while $\theta$ varies in $[-\frac{\pi}{2},\frac{\pi}{2})$, obtaining: (i)~the intervals of angle~$\theta$ where $\mathcal{O}$-$\rm kernel_{\theta}(P)$ is not empty, (ii)~a value of angle~$\theta$ where $\mathcal{O}$-$\rm kernel_{\theta}(P)$ optimizes area or perimeter. Further, we show how the algorithms can be improved when $P$ is a simple orthogonal polygon. In addition, our results are extended to the case of a set $\mathcal{O}=\{\alpha_1,\dots,\alpha_k\}$.