{"title":"Fast triangulation of the plane with respect to simple polygons","authors":"Stefan Hertel, Kurt Mehlhorn","doi":"10.1016/S0019-9958(85)80044-9","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>P</em><sub>1</sub>,…, <em>P<sub>k</sub></em> be pairwise non-intersecting simple polygons with a total of <em>n</em> vertices and <em>s</em> start vertices. A start vertex, in general, is a vertex both of which neighbors have larger <em>x</em> coordinate. We present an algorithm for triangulating <em>P</em><sub>1</sub>,…, <em>P<sub>k</sub></em> in time <em>O</em>(<em>n</em> + <em>s</em> log <em>s</em>). <em>s</em> may be viewed as a measure of non-convexity. In particular, <em>s</em> is always bounded by the number of concave angles + 1, and is usually much smaller. We also describe two new applications of triangulation. Given a triangulation of the plane with respect to a set of <em>k</em> pairwise non-intersecting simple polygons, then the intersection of this set with a convex polygon <em>Q</em> can be computed in time linear with respect to the combined number of vertices of the <em>k</em> + 1 polygons. Such a result had only be known for two <em>convex polygons</em>. The other application improves the bound on the number of convex parts into which a polygon can be decomposed.</p></div>","PeriodicalId":38164,"journal":{"name":"信息与控制","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1985-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0019-9958(85)80044-9","citationCount":"16","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"信息与控制","FirstCategoryId":"1093","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019995885800449","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 16
Abstract
Let P1,…, Pk be pairwise non-intersecting simple polygons with a total of n vertices and s start vertices. A start vertex, in general, is a vertex both of which neighbors have larger x coordinate. We present an algorithm for triangulating P1,…, Pk in time O(n + s log s). s may be viewed as a measure of non-convexity. In particular, s is always bounded by the number of concave angles + 1, and is usually much smaller. We also describe two new applications of triangulation. Given a triangulation of the plane with respect to a set of k pairwise non-intersecting simple polygons, then the intersection of this set with a convex polygon Q can be computed in time linear with respect to the combined number of vertices of the k + 1 polygons. Such a result had only be known for two convex polygons. The other application improves the bound on the number of convex parts into which a polygon can be decomposed.