University of KashanMathematics Interdisciplinary Research2538-36398320230901An Upper Bound for Min-Max Angle of Polygons24726011399110.22052/mir.2023.246534.1363ENSaeed AsaeediDepartment of Computer Science, University of Kashan, Kashan, I. R. Iran0000-0002-2980-3013Farzad DidehvarDepartment of Mathematics and computer Science, Amirkabir university of Technology, Tehran, I. R. IranAli MohadesDepartment of Mathematics and computer Science, Amirkabir university of Technology, Tehran, I. R. Iran0000-0002-6118-2245Journal Article20220628Let $S$ be a set of $n$ points in the plane, $\nabla(S)$ the set of all simple polygons crossing $S$, $\gamma_P$ the maximum angle of polygon $P \in \nabla(S)$ and $\theta =min_{P\in\nabla(S)} \gamma_P$. 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 upper bound on its angles. Constructing a simple polygon with the 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 $r=1$.https://mir.kashanu.ac.ir/article_113991_5f60217ec36459aeb0d0cbd4eb6fb175.pdf