Abstract
Polyhedral 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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 107004 |
| Journal | Advances in Mathematics |
| Volume | 363 |
| DOIs | |
| State | Published - Jan 29 2020 |
Bibliographical note
KAUST Repository Item: Exported on 2020-10-01Acknowledgements: The 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).