An upper bound for min-max angle of polygons

07/28/2018

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by   Saeed Asaeedi, et al.

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Let S be a set of points in the plane, CH be the convex hull of S, (S) be the set of all simple polygons crossing S, γ_P be the maximum angle of polygon P ∈(S) and θ =min_P∈(S)γ_P. In this paper, we prove that θ≤ 2π-2π/r.m such that m and r are the number of edges and internal points of CH, respectively. We also introduce an innovative polynomial time 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 is highly applicable to the fields of robotics, path planning, image processing, GIS, etc. Moreover, we improve our upper bound on θ and prove that this is tight.