Unions of orthogonally convex or orthogonally starshaped polygons
type: Item
instance of: scholarly article
title: Unions of orthogonally convex or orthogonally starshaped polygons
zbMATH Open document ID: 0814.52002
DOI: 10.1007/BF01264043
author: Marilyn Breen
published in: Geometriae Dedicata
publication date: 1995-05-31T00:00:00Z
review text: This paper considers orthogonal polygons obtained as union of finitely many convex polygons whose edges are parallel to the coordinate axes. A simple polygonal path in the plane whose edges are finite and parallel to the coordinate axes is called a staircase path iff the associated vectors alternate in direction. A set \(T\) is said to be an orthogonally convex polygon iff for all \([x,y]\) in \(T\) such that \([x,y]\) is either horizontal or vertical, then \([x,y] \subseteq T\). A set \(T\) is starshaped via staircase paths with respect to a point \(p\) iff \(p\) sees each point of \(T\) via staircase paths. By imposing conditions on finite subsets of a set \(T\) in a linear space, \textit{J. F. Lawrence, W. R. Hare jun.} and \textit{John W. Kenelly} [Proc. Am. Math. Soc. 34, 225-288 (1972; Zbl 0237.52001)] have characterized \(T\) as union of finite number of convex sets. The author in her first theorem obtains a similar result for simply connected orthogonal polygons in the plane. In the theorem it is stated that the necessary and sufficient condition for a simply connected orthogonal polygon in the plane to be a union of \(k\) orthogonally convex sets is that for every finite subset \(F\) of \(T\) there exists a \(k\)- partition \(\{F_ i\}_{1 \leq i \leq k}\) of \(F\) such that every pair of points in \(F_ i\) can be joined by a staircase path in \(T\). In the second theorem the author proves that if a simply connected orthogonal polygon having that for every three points of \(T\), at least two of these see each other via staircase paths in \(T\), the set is a union of three orthogonally convex polygons. In Theorem 3 the necessary and sufficient condition for a decomposition of a simple connected orthogonal polygon in the plane into two orthogonally convex polygons is given by introducing an odd numbered finite point sequence \(\{\nu_ i\}\) where \(\nu_{n + 1} = \nu_ 1\), \(1 \leq i \leq n\) in the set such that at least one consecutive pair of points see each o…
reviewed by: Semin Akdoǧan
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 51A35
zbMATH DE Number: 701142
zbMATH Keywords: orthogonal polygons
zbMATH Keywords: staircase path
zbMATH Keywords: orthogonally convex
zbMATH Keywords: starshaped
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 701142