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Geometriae Dedicata

, Volume 168, Issue 1, pp 265–289 | Cite as

Discretization of asymptotic line parametrizations using hyperboloid surface patches

  • Emanuel Huhnen-Venedey
  • Thilo Rörig
Original Paper

Abstract

Two-dimensional affine A-nets in \(3\)-space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The defining property of A-nets is planarity of vertex stars, hence elementary quadrilaterals of a generic A-net are skew. The present article deals with the extension of A-nets to differentiable surfaces, by gluing hyperboloid surface patches into the skew quadrilaterals. The obtained surfaces, named “hyperbolic nets”, are a novel, piecewise smooth discretization of surfaces parametrized along asymptotic lines. A simply connected affine A-net can be extended to a hyperbolic net if all quadrilateral strips are “equi-twisted”. The geometric condition of equi-twist implies the combinatorial property, that all inner vertices of the A-net have to be of even degree. If an A-net can be extended to a hyperbolic net, then there exists a 1-parameter family of such extensions. It is briefly explained how the generation of hyperbolic nets can be implemented on a computer. We use a projective model of Plücker line geometry in order to describe A-nets and hyperboloids.

Keywords

Discrete differential geometry Discrete asymptotic line parametrization A-nets Hyperboloids Projective geometry Plücker line geometry 

Mathematics Subject Classification (2000)

51M30 53A05 65D17 

Notes

Acknowledgments

We would like to thank Alexander Bobenko and Wolfgang Schief for fruitful discussions on the subject and comments on previous versions of the manuscript. We further want to thank Charles Gunn for providing us with the pictures for Fig. 11.

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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