A technique for solving the polygon inclusion problems

TL;DR

This paper introduces the Rotate-and-Kill technique, providing efficient $O(n)$ algorithms for various polygon inclusion and circumscribing problems, improving upon previous methods in simplicity and computational complexity.

Contribution

The paper presents a new Rotate-and-Kill technique that achieves linear-time algorithms for key polygon problems, surpassing prior $O(n ext{log} n)$ or higher complexity solutions.

Findings

Linear-time algorithms for maximum area triangle in convex polygons

Linear-time algorithms for minimum area and perimeter triangles enclosing polygons

Improved simplicity and efficiency over previous algorithms

Abstract

We propose a technique called Rotate-and-Kill for solving the polygon inclusion and circumscribing problems. By applying this technique, we obtain $O(n)$ time algorithms for computing (1) the maximum area triangle in a given $n$-sided convex polygon $P$, (2) the minimum area triangle enclosing $P$, (3) the minimum area triangle enclosing $P$ touching edge-to-edge, i.e. the minimum area triangle that is the intersection of three half-planes out of the $n$ half-planes defining $P$, and (4) the minimum perimeter triangle enclosing $P$ touching edge-to-edge. Our algorithm for computing the maximum area triangle is simpler than the alternatives given in [Chandran and Mount, IJCGA'92] and [Kallus, arXiv'17]. Our algorithms for computing the minimum area or perimeter triangle enclosing $P$ touching edge-to-edge improve the $O(n\log n)$ or $O(n\log^2n)$ time algorithms given in [Boyce \emph{et al.}, STOC'82], [Aggarwal \emph{et al.}, Algorithmica'87], [Aggarwal and J. Park., FOCS'88], [Aggarwal \emph{et al.}, DCG'94], and [Schieber, SODA'95].