Constrained Willmore Surfaces
ACM Transactions on Graphics · Volume 40 (4) · SIGGRAPH 2021
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.