Constrained Willmore Surfaces

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.

This work was supported in part by a National Science Foundation graduate research fellowship, the Kortschak Scholars program at Caltech, the Einstein Foundation Berlin, the Deutsche Forschungs Gemeinschaft (SFB/TRR 109 “Discretization in Geometry and Physics”), and the Taiwanese National Center for Theoretical Sciences (Mathematics Division). Additional support was provided by the Information Science and Technology Initiative at Caltech and SideFX Software.