Maximal Area Triangles in a Convex Polygon

The widely known linear time algorithm for computing the maximum area triangle in a convex polygon was found incorrect recently by Keikha et. al.(arXiv:1705.11035). We present an alternative algorithm in this paper. Comparing to the only previously known correct solution, ours is much simpler and more efficient. More importantly, our new approach is powerful in solving related problems

Analyzing Difficulties in Problem Solving of the Polygon Area for Elementary Students

Mathematics is part from a human activity; it cannot be separated from daily human life, both in theory and in practice. Problem solving is the most important topic in learning Mathematics. However, problem solving is a complex process that has many components. One of them is the difficulty of students in answering the questions about how to find the area of polygon. Students are hard to find out two or more shapes contained in polygon. The students are hard to find unknown measurements, because they do not understand the questions. Therefore, teachers should give guidance to the students in order to help them to solve problems correctly. The purpose of this study was to analyze students’ difficulties to solve the problems of the polygon area questions for the fourth grade of one state elementary in Padang Sidimpuan City. There were 33 participants consisting of 16 male and 17 female students. The students were given a problem solving test and the questions were related to a wide area of polygon consisting of five questions. The findings of the study indicated that students were hard to find solutions to solve mathematical problems of the polygon area flat shapes in class, suggesting that teachers should strive to develop students’ critical thinking as well as creative thinking so they will have a good problem solving skills

Improved Approximations for Translational Packing of Convex Polygons

Optimal packing of objects in containers is a critical problem in various real-life and industrial applications. This paper investigates the two-dimensional packing of convex polygons without rotations, where only translations are allowed. We study different settings depending on the type of containers used, including minimizing the number of containers or the size of the container based on an objective function. Building on prior research in the field, we develop polynomial-time algorithms with improved approximation guarantees upon the best-known results by Alt, de Berg and Knauer, as well as Aamand, Abrahamsen, Beretta and Kleist, for problems such as Polygon Area Minimization, Polygon Perimeter Minimization, Polygon Strip Packing, and Polygon Bin Packing. Our approach utilizes a sequence of object transformations that allows sorting by height and orientation, thus enhancing the effectiveness of shelf packing algorithms for polygon packing problems. In addition, we present efficient approximation algorithms for special cases of the Polygon Bin Packing problem, progressing toward solving an open question concerning an ?(1)-approximation algorithm for arbitrary polygons

Finding largest small polygons with GloptiPoly

A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices $n$. Many instances are already solved in the literature, namely for all odd $n$, and for $n=4, 6$ and 8. Thus, for even $n\geq 10$, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for $n=10$ and $n=12$. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic

Improved Approximations for Translational Packing of Convex Polygons

Optimal packing of objects in containers is a critical problem in various real-life and industrial applications. This paper investigates the two-dimensional packing of convex polygons without rotations, where only translations are allowed. We study different settings depending on the type of containers used, including minimizing the number of containers or the size of the container based on an objective function. Building on prior research in the field, we develop polynomial-time algorithms with improved approximation guarantees upon the best-known results by Alt, de Berg and Knauer, as well as Aamand, Abrahamsen, Beretta and Kleist, for problems such as Polygon Area Minimization, Polygon Perimeter Minimization, Polygon Strip Packing, and Polygon Bin Packing. Our approach utilizes a sequence of object transformations that allows sorting by height and orientation, thus enhancing the effectiveness of shelf packing algorithms for polygon packing problems. In addition, we present efficient approximation algorithms for special cases of the Polygon Bin Packing problem, progressing toward solving an open question concerning an O(1)-approximation algorithm for arbitrary polygons.Comment: This is the full version of the same-named paper which will be presented at ESA 2023 conferenc

Computing Smallest Convex Intersecting Polygons

Funding Information: Funding Mark de Berg is supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003. Antonis Skarlatos: Part of the work was done during an internship at the Max Planck Institute for Informatics in Saarbrücken, Germany. Publisher Copyright: © 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.A polygon C is an intersecting polygon for a set O of objects in R2 if C intersects each object in O, where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the minimum-area convex intersecting polygon for a given set O of objects. We present an FPTAS for both problems for the case where O is a set of possibly intersecting convex polygons in the plane of total complexity n. Furthermore, we present an exact polynomial-time algorithm for the minimum-perimeter intersecting polygon for the case where O is a set of n possibly intersecting segments in the plane. So far, polynomial-time exact algorithms were only known for the minimum perimeter intersecting polygon of lines or of disjoint segments.Peer reviewe

SmartDissolve User Guide

SmartDissolve is a polygon aggregation tool developed in the frame of the Global Human Settlement Layer (GHSL) project and it is being used to dissolve polygons setting an areal threshold. SmartDissolve is a tool that handles minimum mapping unit, resolution mismatch between layers, or spatial uncertainty problems in GISc. This tool automatically dissolves polygons below a threshold area, updating fields’ values. The toolbox allows to select the ordering of polygon analysis (i.e. from the smallest to the largest area, vice versa, or order of IDs), different dissolve rules (i.e. with smallest, largest, or maximum-border-share adjacent polygon, minimum total perimeter or maximum compactness) and different field updating operations (i.e. sum, mean or text concatenation). The software is available as toolbox for ArcGIS 10.X, a standalone command line tool or a MATLAB function. More details about the algorithm, the method and its actual applications can be found in the paper in the Reference section. This guide provides instructions about installing and using the SmartDissolve toolbox for ArcGIS and the SmartDissolve command line tool on a Windows computer and the SmartDissolve MATLAB function in a MATLAB environment.JRC.E.1-Disaster Risk Managemen