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    Visual Smoothness of Polyhedral Surfaces

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    Type
    Article
    Authors
    PELLIS, DAVIDE
    Kilian, Martin
    DELLINGER, FELIX
    Wallner, Johannes
    Pottmann, Helmut cc
    KAUST Department
    Applied Mathematics and Computational Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Office of the VP
    Visual Computing Center (VCC)
    Date
    2019-07-12
    Online Publication Date
    2019-07-12
    Print Publication Date
    2019-07-12
    Permanent link to this record
    http://hdl.handle.net/10754/653104
    
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    Abstract
    Representing smooth geometric shapes by polyhedral meshes can be quite difficult in situations where the variation of edges and face normals is prominently visible. Especially problematic are saddle-shaped areas of the mesh, where typical vertices with six incident edges are ill suited to emulate the more symmetric smooth situation. The importance of a faithful discrete representation is apparent for certain special applications like free form architecture, but is also relevant for simulation and geometric computing. In this paper we discuss what exactly is meant by a good representation of saddle points, and how this requirement is stronger than a good approximation of a surface plus its normals. We characterize good saddles in terms of the normal pyramid in a vertex. We show several ways to design meshes whose normals enjoy small variation (implying good saddle points). For this purpose we define a discrete energy of polyhedral surfaces, which is related to a certain total absolute curvature of smooth surfaces. We discuss the minimizers of both functionals and in particular show that the discrete energy is minimal not for triangle meshes, but for principal quad meshes. We demonstrate our procedures for optimization and interactive design by means of meshes intended for architectural design.
    Citation
    Pellis, D., Kilian, M., Dellinger, F., Wallner, J., & Pottmann, H. (2019). Visual smoothness of polyhedral surfaces. ACM Transactions on Graphics, 38(4), 1–11. doi:10.1145/3306346.3322975
    Sponsors
    This work was supported by the SFB-Transregio programme Discretization in geometry and dynamics, through grant I2978 of the Austrian Science Fund. The authors are grateful to Felix Günther for useful discussions, and to the anonymous referees for extensive reviews.
    Publisher
    Association for Computing Machinery (ACM)
    DOI
    10.1145/3306346.3322975
    ae974a485f413a2113503eed53cd6c53
    10.1145/3306346.3322975
    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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