Finding Largest Rectangles in Convex Polygons

We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with $n$ vertices. We give exact algorithms that solve these problems in time $O(n^3)$. We also give $(1-\varepsilon)$-approximation algorithms that take time $O(\varepsilon^{-3/2}+ \varepsilon^{-1/2} \log n)$.Comment: The time bound to approximate the maximum-perimeter rectangle is improved. Christian Knauer becomes coautho

Maximum-Area Triangle in a Convex Polygon, Revisited

We revisit the following problem: Given a convex polygon $P$, find the largest-area inscribed triangle. We show by example that the linear-time algorithm presented in 1979 by Dobkin and Snyder for solving this problem fails. We then proceed to show that with a small adaptation, their approach does lead to a quadratic-time algorithm. We also present a more involved $O(n\log n)$ time divide-and-conquer algorithm. Also we show by example that the algorithm presented in 1979 by Dobkin and Snyder for finding the largest-area $k$-gon that is inscribed in a convex polygon fails to find the optimal solution for $k=4$. Finally, we discuss the implications of our discoveries on the literature

Largest Inscribed Rectangles in Geometric Convex Sets

This paper considers the problem of finding maximum volume (axis-aligned) inscribed parallelotopes and boxes in a compact convex set, defined by a finite number of convex inequalities, and presents an optimization approach for solving them. Several optimization models are developed that can be easily generalized to find other inscribed geometric shapes such as triangles, rhombi, and tetrahedrons. To find the largest axis-aligned inscribed rectangles in the higher dimensions, an interior-point method algorithm is presented and analyzed. Finally, a parametrized optimization approach is developed to find the largest (axis-aligned) inscribed rectangles in two-dimensional space.Comment: 33 pages, 15 figure

Finding all Maximal Area Parallelograms in a Convex Polygon

Polygon inclusion problems have been studied extensively in geometric optimization. In this paper, we consider the variant of computing the maximum area parallelograms (MAPs) and all the locally maximal area parallelograms (LMAPs) in a given convex polygon. By proving and utilizing several structural properties of the LMAPs, we compute all of them (including all the MAPs) in $O(n^2)$ time, where $n$ denotes the number of edges of the given polygon. In addition, we prove that the LMAPs interleave each other and thus the number of LMAPs is $O(n)$. We discuss applications of our result to, among others, the problem of computing the maximum area centrally-symmetric convex body inside a convex polygon, and the simplest case of the Heilbronn triangle problem.Comment: The conference version of this paper was published in CCCG 201

Largest triangles in a polygon

We study the problem of finding maximum-area triangles that can be inscribed in a polygon in the plane. We consider eight versions of the problem: we use either convex polygons or simple polygons as the container; we require the triangles to have either one corner with a fixed angle or all three corners with fixed angles; we either allow reorienting the triangle or require its orientation to be fixed. We present exact algorithms for all versions of the problem. In the case with reorientations for convex polygons with $n$ vertices, we also present $(1-\varepsilon)$-approximation algorithms

A Geometric Structure Associated with the Convex Polygon

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.Comment: 50 pages, 39 figures, 1 beautiful structur