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dc.contributor.authorGünther, Felix
dc.contributor.authorJiang, Caigui
dc.contributor.authorPottmann, Helmut
dc.date.accessioned2017-12-28T07:32:15Z
dc.date.available2017-12-28T07:32:15Z
dc.date.issued2020-01-29
dc.date.submitted2017-03-20
dc.identifier.citationGünther, F., Jiang, C., & Pottmann, H. (2020). Smooth polyhedral surfaces. Advances in Mathematics, 363, 107004. doi:10.1016/j.aim.2020.107004
dc.identifier.doi10.1016/j.aim.2020.107004
dc.identifier.urihttp://hdl.handle.net/10754/626545
dc.description.abstractPolyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the aim of our work is to find and to discuss suitable assessments of smoothness of polyhedral surfaces that only take the geometry of the polyhedral surface itself into account. Motivated by analogies to classical differential geometry, we propose a theory of smoothness of polyhedral surfaces including suitable notions of normal vectors, tangent planes, asymptotic directions, and parabolic curves that are invariant under projective transformations. It is remarkable that seemingly mild conditions significantly limit the shapes of faces of a smooth polyhedral surface. Besides being of theoretical interest, we believe that smoothness of polyhedral surfaces is of interest in the architectural context, where vertices and edges of polyhedral surfaces are highly visible.
dc.description.sponsorshipThe authors are grateful to Thomas Banchoff for fruitful discussions concerning the Gauss image of a vertex star and to Günter Rote for pointing out the connection to [7]. This research was initiated during the first author's stay at the Erwin Schrödinger International Institute for Mathematical Physics in Vienna and continued during his stays at the Institut des Hautes Études Scientifiques in Bures-sur-Yvette and the Max Planck Institute for Mathematics in Bonn. The first author thanks the institutes for their hospitality and the European Post-Doctoral Institute for Mathematical Sciences for the opportunity to visit the afore mentioned research institutes. The first and last author are grateful for support by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics” and corresponding FWF grants I 706-N26 and I 2978-N35. The first author was also partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).
dc.publisherElsevier BV
dc.relation.urlhttps://linkinghub.elsevier.com/retrieve/pii/S0001870820300293
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics, [[Volume], [Issue], (2020-01-29)] DOI: 10.1016/j.aim.2020.107004 . © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
dc.titleSmooth polyhedral surfaces
dc.typeArticle
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentVisual Computing Center (VCC)
dc.identifier.journalAdvances in Mathematics
dc.rights.embargodate2022-01-29
dc.eprint.versionPost-print
dc.contributor.institutionErwin Schrödinger International Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Vienna, Austria
dc.contributor.institutionMax Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
dc.contributor.institutionInstitut für Mathematik, MA 8-3, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
dc.contributor.institutionMax Planck Institute for Informatics, Campus E1 4, 66123 Saarbrücken, Germany
dc.contributor.institutionCenter for Geometry and Computational Design, Technische Universität Wien, Wiedner Hauptstraße 8/104, 1040 Vienna, Austria
dc.identifier.arxividarXiv:1703.05318
kaust.personJiang, Caigui
kaust.personPottmann, Helmut
dc.date.accepted2020-01-06
dc.versionv1
refterms.dateFOA2018-06-14T09:29:44Z


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