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

Random packing of regular polygons and star polygons on a flat two-dimensional surface

Random packing of unoriented regular polygons and star polygons on a two-dimensional flat, continuous surface is studied numerically using random sequential adsorption algorithm. Obtained results are analyzed to determine saturated random packing ratio as well as its density autocorrelation function. Additionally, the kinetics of packing growth and available surface function are measured. In general, stars give lower packing ratios than polygons, but, when the number of vertexes is large enough, both shapes approach disks and, therefore, properties of their packing reproduce already known results for disks.Comment: 5 pages, 8 figure

Inscribing or Circumscribing a Histogon to a Convex Polygon

We consider two optimization problems of approximating a convex polygon, one by a largest inscribed histogon and the other by a smallest circumscribed histogon. An axis-aligned histogon is an axis-aligned rectilinear polygon such that every horizontal edge has an integer length. A histogon of orientation ? is a copy of an axis-aligned histogon rotated by ? in counterclockwise direction. The goal is to find a largest inscribed histogon and a smallest circumscribed histogon over all orientations in [0,?). Depending on whether the horizontal width of a histogon is predetermined or not, we consider several different versions of the problem and present exact algorithms. These optimization problems belong to shape analysis, classification, and simplification, and they have applications in various cost-optimization problems

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