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...

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Bibliographic Details
Published inACM transactions on graphics Vol. 40; no. 4; pp. 1 - 17
Main Authors Soliman, Yousuf, Chern, Albert, Diamanti, Olga, Knöppel, Felix, Pinkall, Ulrich, Schröder, Peter
Format Journal Article
LanguageEnglish
Published New York, NY, USA ACM 31.08.2021
Subjects
Computer graphics
Computing methodologies
Discretization
Mathematical analysis
Mathematical optimization
Mathematics of computing
Mesh geometry models
Numerical analysis
Shape analysis
Shape modeling
geometric flows
willmore surfaces
geometric modeling
conformal geometry
discrete differential geometry
differential coordinates
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Summary: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.
ISSN:0730-0301
1557-7368
DOI:10.1145/3450626.3459759