Constrained Willmore surfaces.

Yousuf Soliman, Albert Chern, Olga Diamanti, Felix Knöppel, Ulrich Pinkall, Peter Schröder

Research output: Contribution to journalArticlepeer-review


Smooth curves and surfaces can be characterized as minimizers of squared curvature bending energies subject to constraints. In the univariate case with an isometry (length) constraint this leads to classic non-linear splines. For surfaces, isometry is too rigid a constraint and instead one asks for minimizers of the Willmore (squared mean curvature) energy subject to a conformality constraint. We present an efficient algorithm for (conformally) constrained Willmore surfaces using triangle meshes of arbitrary topology with or without boundary. Our conformal class constraint is based on the discrete notion of conformal equivalence of triangle meshes. The resulting non-linear constrained optimization problem can be solved efficiently using the competitive gradient descent method together with appropriate Sobolev metrics. The surfaces can be represented either through point positions or differential coordinates. The latter enable the realization of abstract metric surfaces without an initial immersion. A versatile toolkit for extrinsic conformal geometry processing, suitable for the construction and manipulation of smooth surfaces, results through the inclusion of additional point, area, and volume constraints.

Original languageEnglish
Article number112
Pages (from-to)1-17
JournalACM Transactions on Graphics
Issue number4
Publication statusPublished - 1 Jul 2021


  • conformal geometry
  • differential coordinates
  • discrete differential geometry
  • geometric flows
  • geometric modeling
  • willmore surfaces

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design


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