{"@context":"https:\/\/schema.org\/","@type":"PublicationVolume","@id":"#volume743","volumeNumber":9111,"name":"Dagstuhl Seminar Proceedings, Volume 9111","dateCreated":"2009-06-23","datePublished":"2009-06-23","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume743"},"hasPart":[{"@type":"ScholarlyArticle","@id":"#article2411","name":"09111 Abstracts Collection \u2013 Computational Geometry","abstract":"From March 8 to March 13, 2009, the Dagstuhl Seminar 09111 ``Computational Geometry '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.\r\nDuring the seminar, several participants presented their current\r\nresearch, and ongoing work and open problems were discussed. Abstracts of\r\nthe presentations given during the seminar as well as abstracts of\r\nseminar results and ideas are put together in this paper. The first section\r\ndescribes the seminar topics and goals in general.\r\nLinks to extended abstracts or full papers are provided, if available.","author":[{"@type":"Person","name":"Agarwal, Pankaj Kumar","givenName":"Pankaj Kumar","familyName":"Agarwal"},{"@type":"Person","name":"Alt, Helmut","givenName":"Helmut","familyName":"Alt"},{"@type":"Person","name":"Teillaud, Monique","givenName":"Monique","familyName":"Teillaud"}],"position":1,"pageStart":1,"pageEnd":18,"dateCreated":"2009-06-24","datePublished":"2009-06-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Agarwal, Pankaj Kumar","givenName":"Pankaj Kumar","familyName":"Agarwal"},{"@type":"Person","name":"Alt, Helmut","givenName":"Helmut","familyName":"Alt"},{"@type":"Person","name":"Teillaud, Monique","givenName":"Monique","familyName":"Teillaud"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09111.1","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume743"},{"@type":"ScholarlyArticle","@id":"#article2412","name":"A Pseudopolynomial Algorithm for Alexandrov's Theorem","abstract":"Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of some convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation whose solution is the polyhedron corresponding to a given metric. We describe an algorithm based on this differential equation to compute the polyhedron given the metric, and prove a pseudopolynomial bound on its running time.","keywords":["Folding","metrics","pseudopolynomial","algorithms"],"author":[{"@type":"Person","name":"Kane, Daniel","givenName":"Daniel","familyName":"Kane"},{"@type":"Person","name":"Price, Gregory Nathan","givenName":"Gregory Nathan","familyName":"Price"},{"@type":"Person","name":"Demaine, Erik","givenName":"Erik","familyName":"Demaine"}],"position":2,"pageStart":1,"pageEnd":22,"dateCreated":"2009-06-23","datePublished":"2009-06-23","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kane, Daniel","givenName":"Daniel","familyName":"Kane"},{"@type":"Person","name":"Price, Gregory Nathan","givenName":"Gregory Nathan","familyName":"Price"},{"@type":"Person","name":"Demaine, Erik","givenName":"Erik","familyName":"Demaine"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09111.2","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume743"},{"@type":"ScholarlyArticle","@id":"#article2413","name":"Minimizing Absolute Gaussian Curvature Locally","abstract":"One of the remaining challenges when reconstructing a surface from a\r\n finite sample is recovering non-smooth surface features like sharp\r\n edges. There is practical evidence showing that a two step approach\r\n could be an aid to this problem, namely, first computing a\r\n polyhedral reconstruction isotopic to the sampled surface, and\r\n secondly minimizing the absolute Gaussian curvature of this\r\n reconstruction globally. The first step ensures topological\r\n correctness and the second step improves the geometric accuracy of\r\n the reconstruction in the presence of sharp features without\r\n changing its topology. Unfortunately it is computationally hard to\r\n minimize the absolute Gaussian curvature globally. Hence we study a\r\n local variant of absolute Gaussian curvature minimization problem\r\n which is still meaningful in the context of surface\r\n fairing. Absolute Gaussian curvature like Gaussian curvature is\r\n concentrated at the vertices of a polyhedral surface embedded into\r\n $mathbb{R}^3$. Local optimization tries to move a single vertex in\r\n space such that the absolute Gaussian curvature at this vertex is\r\n minimized. We show that in general it is algebraically hard to find\r\n the optimal position of a vertex. By algebraically hard we mean that\r\n in general an optimal solution is not constructible, i.e., there\r\n exist no finite sequence of expressions starting with rational numbers, \r\nwhere each expression is either the sum, difference,\r\n product, quotient or $k$'th root of preceding expressions and the\r\n last expressions give the coordinates of an optimal solution. Hence\r\n the only option left is to approximate the optimal position. We\r\n provide an approximation scheme for the minimum possible value of\r\n the absolute Gaussian curvature at a vertex.","keywords":["Absolute Gaussian curvature","surface reconstruction","mesh smoothing"],"author":[{"@type":"Person","name":"Giesen, Joachim","givenName":"Joachim","familyName":"Giesen"},{"@type":"Person","name":"Madhusudan, Manjunath","givenName":"Manjunath","familyName":"Madhusudan"}],"position":3,"pageStart":1,"pageEnd":16,"dateCreated":"2009-06-23","datePublished":"2009-06-23","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Giesen, Joachim","givenName":"Joachim","familyName":"Giesen"},{"@type":"Person","name":"Madhusudan, Manjunath","givenName":"Manjunath","familyName":"Madhusudan"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09111.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume743"},{"@type":"ScholarlyArticle","@id":"#article2414","name":"Open Problem Session","abstract":"This is a scribing of the open problems posed at the Tuesday evening open problem session. Posers of problems provided input after the session.","keywords":["Open problems","computational geometry"],"author":{"@type":"Person","name":"Mitchell, Joseph S.","givenName":"Joseph S.","familyName":"Mitchell"},"position":4,"pageStart":1,"pageEnd":3,"dateCreated":"2009-06-23","datePublished":"2009-06-23","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Mitchell, Joseph S.","givenName":"Joseph S.","familyName":"Mitchell"},"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09111.4","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume743"},{"@type":"ScholarlyArticle","@id":"#article2415","name":"Shortest Path Problems on a Polyhedral Surface","abstract":"We develop algorithms to compute edge sequences, Voronoi diagrams, shortest\r\npath maps, the Fr\u00e9chet distance, and the diameter for a polyhedral surface. Distances on the surface are measured either by the length of a Euclidean shortest path or by link distance. Our main result is a linear-factor speedup for computing all shortest path edge sequences on a convex polyhedral surface.","keywords":["Shortest paths","edge sequences"],"author":[{"@type":"Person","name":"Wenk, Carola","givenName":"Carola","familyName":"Wenk"},{"@type":"Person","name":"Cook, Atlas F.","givenName":"Atlas F.","familyName":"Cook"}],"position":5,"pageStart":1,"pageEnd":30,"dateCreated":"2009-06-23","datePublished":"2009-06-23","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Wenk, Carola","givenName":"Carola","familyName":"Wenk"},{"@type":"Person","name":"Cook, Atlas F.","givenName":"Atlas F.","familyName":"Cook"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09111.5","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume743"},{"@type":"ScholarlyArticle","@id":"#article2416","name":"Two Applications of Point Matching","abstract":"The two following problems can be solved by a reduction\r\nto a minimum-weight bipartite matching problem (or a related\r\nnetwork flow problem):\r\n\r\na) Floodlight illumination:\r\nWe are given $n$ infinite wedges (sectors, spotlights) that can cover\r\nthe whole plane when placed at the origin.\r\nThey are to be assigned to $n$ given locations\r\n(in arbitrary order, but without rotation)\r\nsuch that they still cover the whole plane.\r\n(This extends results of Bose et al. from 1997.)\r\n\r\nb) Convex partition:\r\nPartition a convex $m$-gon into $m$ convex parts, each part\r\ncontaining one of the edges and a given number of points from a given\r\npoint set. (Garcia and Tejel 1995, Aurenhammer 2008)","keywords":["Bipartite matching","least-squares"],"author":{"@type":"Person","name":"Rote, G\u00fcnter","givenName":"G\u00fcnter","familyName":"Rote"},"position":6,"pageStart":1,"pageEnd":3,"dateCreated":"2009-06-24","datePublished":"2009-06-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Rote, G\u00fcnter","givenName":"G\u00fcnter","familyName":"Rote"},"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.09111.6","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume743"}]}