{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,2,8]],"date-time":"2023-02-08T09:27:17Z","timestamp":1675848437640},"reference-count":11,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Comput. Geom. Appl."],"published-print":{"date-parts":[[2007,6]]},"abstract":"<jats:p> Let S be a set of points in the plane. What is the minimum possible dilation of all plane graphs that contain S? Even for a set S as simple as five points evenly placed on the circle, this question seems hard to answer; it is not even clear if there exists a lower bound &gt; 1. In this paper we provide the first upper and lower bounds for the embedding problem. <\/jats:p><jats:p> 1. Each finite point set can be embedded into the vertex set of a finite triangulation of dilation \u2264 1.1247. <\/jats:p><jats:p> 2. Each embedding of a closed convex curve has dilation \u2265 1.00157. <\/jats:p><jats:p> 3. Let P be the plane graph that results from intersecting n infinite families of equidistant, parallel lines in general position. Then the vertex set of P has dilation [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0218195907002318","type":"journal-article","created":{"date-parts":[[2007,6,22]],"date-time":"2007-06-22T02:42:56Z","timestamp":1182480176000},"page":"201-230","source":"Crossref","is-referenced-by-count":6,"title":["EMBEDDING POINT SETS INTO PLANE GRAPHS OF SMALL DILATION"],"prefix":"10.1142","volume":"17","author":[{"given":"ANNETTE","family":"EBBERS-BAUMANN","sequence":"first","affiliation":[{"name":"Institute of Computer Science I, University of Bonn, D-53117 Bonn, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ANSGAR","family":"GR\u00dcNE","sequence":"additional","affiliation":[{"name":"Institute of Computer Science I, University of Bonn, D-53117 Bonn, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ROLF","family":"KLEIN","sequence":"additional","affiliation":[{"name":"Institute of Computer Science I, University of Bonn, D-53117 Bonn, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MAREK","family":"KARPINSKI","sequence":"additional","affiliation":[{"name":"Institute of Computer Science V, University of Bonn, D-53117 Bonn, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"CHRISTIAN","family":"KNAUER","sequence":"additional","affiliation":[{"name":"Institute of Computer Science, FU Berlin, D-14195 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ANDRZEJ","family":"LINGAS","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Lund University, 22100 Lund, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","author":"Agarwal P.","journal-title":"Discrete Comput. 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