Geometriae Dedicata

, Volume 123, Issue 1, pp 89–112 | Cite as

On the convergence of metric and geometric properties of polyhedral surfaces

  • Klaus Hildebrandt
  • Konrad Polthier
  • Max Wardetzky
Original Paper


We provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in Euclidean 3-space. Under the assumption of convergence of surfaces in Hausdorff distance, we show that convergence of the following properties are equivalent: surface normals, surface area, metric tensors, and Laplace–Beltrami operators. Additionally, we derive convergence of minimizing geodesics, mean curvature vectors, and solutions to the Dirichlet problem.


Discrete differential geometry Polyhedral surfaces Minimal surfaces Numerical analysis 

Mathematics Subject Classification (2000)



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Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  • Klaus Hildebrandt
    • 1
  • Konrad Polthier
    • 1
  • Max Wardetzky
    • 1
  1. 1.Department of MathematicsFreie Universität BerlinBerlinGermany

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