Optimizing generalized kernels of polygons

TL;DR

The paper addresses computing and maintaining the kernel of a polygon under rotation of a predefined orientation set, introducing the notion of $\mathcal{O}$-convexity ($\mathcal{O}$-staircases) and the $\mathcal{O}$-Kernel$$(P)$$ as the region from which all points are reachable via monotone $\mathcal{O}$-staircases. It presents algorithms to compute, for $\mathcal{O}=\{0^\circ\}$ and $\{0^\circ,90^\circ\}$, the angle intervals $\theta\in[-\frac{\pi}{2},\frac{\pi}{2})$ where the rotated kernel is non-empty and to optimize the kernel’s area or perimeter, with runtimes ranging from $O(n)$ to $O(n^2\alpha(n))$ depending on polygon class. The work generalizes to $k$ orientations and provides specialized, faster results for simple orthogonal polygons, where the kernel structure is constrained by staircases and dents, allowing linear-time existence and optimization, as well as an $O(kn)$ bound for $k$ orientations. These methods rely on a combination of angular intervals, dualization/envelope techniques, and event-based partitioning of the rotation parameter, along with geometric constructions like the floating rectangle for the two-orientation case. The contributions advance the understanding of visibility under rotating directional constraints and enable efficient optimization of kernel area/perimeter in both general and orthogonal polygons, with potential applications in robot navigation and guarding problems under rotating constraints.

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 $θ$, denoted by $\mathcal{O}$-$\rm kernel_θ(P)$. 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 $\mathcal{O}$-$\rm kernel_θ(P)$ while $θ$ varies in $[-\fracπ{2},\fracπ{2})$, obtaining: (i)~the intervals of angle~$θ$ where $\mathcal{O}$-$\rm kernel_θ(P)$ is not empty, (ii)~a value of angle~$θ$ where $\mathcal{O}$-$\rm kernel_θ(P)$ 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 $\mathcal{O}=\{α_1,\dots,α_k\}$.

Figures (16)

• Figure 1: A $\{0^\circ\}$-staircase which is not a $\{0^\circ,90^\circ\}$-staircase (left) and a $\{0^\circ,90^\circ\}$-staircase (right).
• Figure 2: Two examples of $\{0^\circ\}$-${\rm Kernel }_{\theta}(P)$ for $\theta=0$. In the left example, the strip $S(P)$ is supported by a lowest reflex minimum $p_N$ and a highest reflex maximum $p_S$. In the right example there are no reflex minima and, therefore, the strip $S(P)$ is supported by the highest (convex) vertex $p_N$ and the highest reflex maximum $p_S$.
• Figure 3: A rotating $\{0^\circ\}$-${\rm Kernel }_{\theta}(P)$ for $\theta=0$ (left), $\theta={\frac{\pi}{8}}$ (middle), and $\theta={\frac{\pi}{4}}$ (right).
• Figure 4: In red, dualization of the angular interval corresponding to the vertex $p_5$ (left) in the primal, which in the dual translates into a segment on the line $\ell(p_5)$ (right). In blue, the angular intervals of the vertices $p_7,p_1,p_2$ in the upper chain of the convex hull in the primal (left), translate into the lower envelope of the arrangement in the dual (right).
• Figure 5: The four triangles $A_1(\beta)$, $A_2(\beta)$ (in green), and $B_1(\beta)$, $B_2(\beta)$ (in red).

Theorems & Definitions (45)

• Definition 1
• Definition 2
• Definition 4
• Lemma 5: SRW
• Corollary 6: SRW
• Corollary 7
• proof
• Definition 8
• Lemma 9
• proof