Geometriae Dedicata

, Volume 123, Issue 1, pp 89–112

On the convergence of metric and geometric properties of polyhedral surfaces


  • Klaus Hildebrandt
    • Department of MathematicsFreie Universität Berlin
  • Konrad Polthier
    • Department of MathematicsFreie Universität Berlin
    • Department of MathematicsFreie Universität Berlin
Original Paper

DOI: 10.1007/s10711-006-9109-5

Cite this article as:
Hildebrandt, K., Polthier, K. & Wardetzky, M. Geom Dedicata (2006) 123: 89. doi:10.1007/s10711-006-9109-5


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 geometryPolyhedral surfacesMinimal surfacesNumerical analysis

Mathematics Subject Classification (2000)


Copyright information

© Springer Science + Business Media B.V. 2007