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Published 2005 | public
Book Section - Chapter

Discrete Willmore Flow

Abstract

The Willmore energy of a surface, ∫(H^2 - K) dA, as a function of mean and Gaussian curvature, captures the deviation of a surface from (local) sphericity. As such this energy and its associated gradient flow play an important role in digital geometry processing, geometric modeling, and physical simulation. In this paper we consider a discrete Willmore energy and its flow. In contrast to traditional approaches it is not based on a finite element discretization, but rather on an ab initio discrete formulation which preserves the Möbius symmetries of the underlying continuous theory in the discrete setting. We derive the relevant gradient expressions including a linearization (approximation of the Hessian), which are required for non-linear numerical solvers. As examples we demonstrate the utility of our approach for surface restoration, n-sided hole filling, and non-shrinking surface smoothing.

Additional Information

© 2005 ACM. This work was supported in part by NSF (DMS-0220905, DMS-0138458, ACI-0219979), DFG (Research Center MATHEON "Mathematics for Key Technologies," Berlin), DOE (W-7405-ENG-48/B341492), nVidia, the Center for Integrated Multiscale Modeling and Simulation, Alias, and Pixar. Special thanks to Kevin Bauer, Oscar Bruno, Mathieu Desbrun, Ilja Friedel, Cici Koenig, Nathan Litke, and Fabio Rossi.

Additional details

Created:
August 22, 2023
Modified:
October 20, 2023