An upper bound for min-max angle of polygons

TL;DR

This paper establishes an upper bound on the maximum angle of polygons passing through a set of points, providing a construction algorithm and applications in robotics path planning.

Contribution

It introduces a new upper bound for the min-max angle of polygons through point sets and offers an algorithm to construct such polygons, improving previous bounds.

Findings

Proved the upper bound $ heta \\leq 2\\\pi - \\\frac{2\\\pi}{r.m}$.

Developed an algorithm to construct polygons meeting the angular bound.

Proved the bound is tight for the case when there is only one inner point.

Abstract

Let $S$ be a set of $n$ points in the plane, $\wp(S)$ be the set of all simple polygons crossing $S$, $\gamma_P$ be the maximum angle of polygon $P \in \wp(S)$ and $\theta =min_{P\in\wp(S)} \gamma_P$. In this paper, we prove that $\theta\leq 2\pi-\frac{2\pi}{r.m}$ where $m$ and $r$ are the number of edges and inner points of the convex hull of $S$, respectively. We also propose an algorithm to construct a polygon with the said upper bound on its angles. Constructing a simple polygon with angular constraint on a given set of points in the plane can be used for path planning in robotics. Moreover, we improve our upper bound on $\theta$ and prove that this is tight for $r=1$.