A curvature theory for discrete surfaces based on mesh parallelity

Alexander Ivanovich Bobenko; Helmut Pottmann; Johannes Wallner

doi:10.1007/s00208-009-0467-9

A curvature theory for discrete surfaces based on mesh parallelity

Alexander Ivanovich Bobenko, Helmut Pottmann, Johannes Wallner

Research output: Contribution to journal › Article › peer-review

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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.

Original language	English (US)
Pages (from-to)	1-24
Number of pages	24
Journal	Mathematische Annalen
Volume	348
Issue number	1
DOIs	https://doi.org/10.1007/s00208-009-0467-9
State	Published - Dec 18 2009

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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".

ASJC Scopus subject areas

General Mathematics

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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. {\textcopyright} 2009 Springer-Verlag.",

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AU - Pottmann, Helmut

AU - Wallner, Johannes

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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".

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N2 - 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.

AB - 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.

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A curvature theory for discrete surfaces based on mesh parallelity

Abstract

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