Abstract
In this paper, we generalize the problems of finding simple polygons with minimum area, maximum perimeter, and maximum number of vertices, so that they contain a given set of points and their angles are bounded by α + π where α ( 0 ≤ α ≤ π ) is a parameter. We also consider the maximum angle of each possible simple polygon crossing a given set of points, and derive an upper bound for the minimum of these angles. The correspondence between the problems of finding simple polygons with minimum area and maximum number of vertices is investigated from a theoretical perspective. We formulate these three generalized problems as nonlinear programming models, and then present a genetic algorithm to solve them. Finally, the computed solutions are evaluated on several datasets and the results are compared with those from the optimal approach.
Highlights
Polygons are one of the fundamental objects in the field of computational geometry
We define α-minimum area polygonization (MAP), α-MPP, and α-MNP as the problems of computing simple polygons containing a set of points in the plane with minimum area, maximum perimeter, and maximum number of vertex points, respectively, such that all internal angles of the polygons are less than or equal to π + α
We considered the problem of finding optimal simple polygons containing a set of points in the plane
Summary
Polygons are one of the fundamental objects in the field of computational geometry. Simple polygonization is a way to construct all possible simple polygons on a set of points in the plane. We define α-MAP, α-MPP, and α-MNP as the problems of computing simple polygons containing a set of points in the plane with minimum area, maximum perimeter, and maximum number of vertex points, respectively, such that all internal angles of the polygons are less than or equal to π + α. We consider α-MAP, α-MPP and α-MNP as generalizations of computing the convex hull, and formulate them as nonlinear programming models. For a set S of points, an upper bound for θ is obtained, such that θ is the minimum of maximum angles of each simple polygon containing S.
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