Discrete quadratic curvature energies

Max Wardetzky, Miklós Bergou, David Harmon, Denis Zorin, Eitan Grinspun

Research output: Contribution to journalArticlepeer-review

108 Citations (SciVal)


We present a family of discrete isometric bending models (IBMs) for triangulated surfaces in 3-space. These models are derived from an axiomatic treatment of discrete Laplace operators, using these operators to obtain linear models for discrete mean curvature from which bending energies are assembled. Under the assumption of isometric surface deformations we show that these energies are quadratic in surface positions. The corresponding linear energy gradients and constant energy Hessians constitute an efficient model for computing bending forces and their derivatives, enabling fast time-integration of cloth dynamics with a two- to three-fold net speedup over existing nonlinear methods, and near-interactive rates for Willmore smoothing of large meshes.

Original languageEnglish
Pages (from-to)499-518
Number of pages20
JournalComputer Aided Geometric Design
Issue number8-9
Publication statusPublished - Nov 2007
Externally publishedYes


  • Bending energy
  • Cloth simulation
  • Discrete Laplace operator
  • Discrete mean curvature
  • Non-conforming finite elements
  • Thin plates
  • Willmore flow


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