In this paper, we generalize the problems of finding simple polygons with the minimum area, maximum perimeter and maximum number of vertices so that they contain a given set of points and their angles are bounded by $\alpha+\pi$ where $\alpha$ ($0\leq\alpha\leq \pi$) 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 the minimum area and maximum number of vertices is investigated from a theoretical perspective. We formulate the 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.
The traveling salesman problem (TSP) is a well-known and most widely studied problem in combinatorial optimization. The Dubins TSP is a variant of TSP for curvature-constrained vehicles. In this paper, another variant of TSP, called angle-constrained TSP, is considered for vehicles that can either move straight or turn, i.e., no voluntary curve is possible. The vehicles are only allowed to make on-the-spot turns and straight motions. We impose turn angle constraint on vehicles. Hence, some additional arbitrary points, called stop points, are added to reach an unreachable target. Here, computing the minimum number of stop points walking on the optimal TSP path is presented as the solution of angle-constrained TSP. In addition, we consider two other problems: computing an angle-constrained path on the vertices with the minimum number of stop points and computing an angle-constrained path with the minimum perimeter. In both problems, there is no need to stay on the optimal TSP path. Here, some algorithms are developed to solve these three problems and are examined on some datasets. These algorithms can be used for the path planning of robots with turn angle constraints.
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.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.