Abstract
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.
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This work was supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.
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Hildebrandt, K., Polthier, K. & Wardetzky, M. On the convergence of metric and geometric properties of polyhedral surfaces. Geom Dedicata 123, 89–112 (2006). https://doi.org/10.1007/s10711-006-9109-5
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DOI: https://doi.org/10.1007/s10711-006-9109-5