A Geometric Structure Associated with the Convex Polygon

TL;DR

This paper introduces a new geometric structure called $Nest(P)$ associated with convex polygons, revealing its properties, computational aspects, and applications to optimize parallelogram areas within the polygon.

Contribution

The paper defines $Nest(P)$, explores its properties, and develops efficient algorithms for location queries and polygon optimization problems without explicit construction.

Findings

$Nest(P)$ has $ heta(n^2)$ segments parallel to $P$'s edges.

Location queries on $Nest(P)$ can be answered in $O( log^2 n)$ time.

Maximum area parallelogram in $P$ can be found in $O(n log^2 n)$ time using $Nest(P)$ properties.

Abstract

We propose a geometric structure induced by any given convex polygon $P$, called $Nest(P)$, which is an arrangement of $\Theta(n^2)$ line segments, each of which is parallel to an edge of $P$, where $n$ denotes the number of edges of $P$. We then deduce six nontrivial properties of $Nest(P)$ from the convexity of $P$ and the parallelism of the line segments in $Nest(P)$. Among others, we show that $Nest(P)$ is a subdivision of the exterior of $P$, and its inner boundary interleaves the boundary of $P$. They manifest that $Nest(P)$ has a surprisingly good interaction with the boundary of $P$. Furthermore, we study some computational problems on $Nest(P)$. In particular, we consider three kinds of location queries on $Nest(P)$ and answer each of them in (amortized) $O(\log^2n)$ time. Our algorithm for answering these queries avoids an explicit construction of $Nest(P)$, which would take $\Omega(n^2)$ time. By applying the aforementioned six properties altogether, we find that the geometric optimization problem of finding the maximum area parallelogram(s) in $P$ can be reduced to answering $O(n)$ aforementioned location queries, and thus be solved in $O(n\log^2n)$ time. This application will be reported in a subsequent paper.