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    A curvature theory for discrete surfaces based on mesh parallelity

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    Type
    Article
    Authors
    Bobenko, Alexander Ivanovich
    Pottmann, Helmut cc
    Wallner, Johannes
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Applied Mathematics and Computational Science Program
    Visual Computing Center (VCC)
    Date
    2009-12-18
    Preprint Posting Date
    2009-01-29
    Online Publication Date
    2009-12-18
    Print Publication Date
    2010-09
    Permanent link to this record
    http://hdl.handle.net/10754/561482
    
    Metadata
    Show full item record
    Abstract
    We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces' areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discrete isothermic minimal surfaces and surfaces of constant mean curvature. We discuss various types of natural Gauss images, the existence of principal curvatures, constant curvature surfaces, Christoffel duality, Koenigs nets, contact element nets, s-isothermic nets, and interesting special cases such as discrete Delaunay surfaces derived from elliptic billiards. © 2009 Springer-Verlag.
    Citation
    Bobenko, A. I., Pottmann, H., & Wallner, J. (2009). A curvature theory for discrete surfaces based on mesh parallelity. Mathematische Annalen, 348(1), 1–24. doi:10.1007/s00208-009-0467-9
    Sponsors
    This research was supported by grants P19214-N18, S92-06, and S92-09 of the Austrian Science Foundation (FWF), and by the DFG Research Unit "Polyhedral Surfaces".
    Publisher
    Springer Nature
    Journal
    Mathematische Annalen
    DOI
    10.1007/s00208-009-0467-9
    arXiv
    0901.4620
    Additional Links
    http://arxiv.org/abs/arXiv:0901.4620v1
    ae974a485f413a2113503eed53cd6c53
    10.1007/s00208-009-0467-9
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Visual Computing Center (VCC); Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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