Constrained Willmore Surfaces

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


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.


Dr. Albert Chern   +

University: TU Berlin
E-Mail: chern[at]

Prof. Dr. Olga Diamanti   +

Projects: A05
University: TU Graz, Department of Mathematics, NT04012
Address: Kopernikusgasse 24/IV, 8010 Graz, AUSTRIA
Tel: +43 316 8738442
E-Mail: olga.diamanti[at]

Dr. Felix Knöppel   +

Projects: C07
University: TU Berlin
E-Mail: knoeppel[at]

Prof. Dr. Ulrich Pinkall   +

Projects: A05, C07
University: TU Berlin, Institut für Mathematik, MA 822
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31424607
E-Mail: pinkall[at]

Prof. Dr. Peter Schröder   +