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Discrete & Computational Geometry

, Volume 60, Issue 4, pp 831–858 | Cite as

Infinitesimal Conformal Deformations of Triangulated Surfaces in Space

  • Wai Yeung LamEmail author
  • Ulrich Pinkall
Article

Abstract

We study infinitesimal conformal deformations of a triangulated surface in Euclidean space and investigate the change in its extrinsic geometry. A deformation of vertices is conformal if it preserves length cross-ratios. On one hand, conformal deformations generalize deformations preserving edge lengths. On the other hand, there is a one-to-one correspondence between infinitesimal conformal deformations in space and infinitesimal isometric deformations of the stereographic image on the sphere. The space of infinitesimal conformal deformations can be parametrized in terms of the change in dihedral angles, which is closely related to the Schläfli formula.

Keywords

Discrete conformality Infinitesimal rigidity Schläfli formula 

Mathematics Subject Classification

52C26 53A05 53A30 52C25 

Notes

Acknowledgements

This research was supported by the DFG Collaborative Research Centre SFB/TRR 109 Discretization in Geometry and Dynamics.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBrown UniversityProvidenceUSA
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

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