Constrained willmore surfaces

Abstract

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 ecient 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 eciently using the competitive gradient descent method together with appropriate Sobolev metrics. The surfaces can be represented either through point positions or dierential 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.

Mem3DG: Modeling membrane mechanochemical dynamics in 3D using discrete differential geometry

“…The length-cross-ratio, λ ij = l il l jk / l ki l jl is a metric of discrete conformality on triangulated mesh, where the indices is illustrated in Fig. 1 A and B ( 148 ). Regularization forces require the input of a reference value for geometric measurements, , , and , which can be derived from a well-conditioned reference mesh (usually the initial input mesh for the simulation).…”

Modeling Membrane Dynamics in 3D using Discrete Differential Geometry

“…The length-cross-ratio, λ ij = l il l jk /l ki l jl is a metric of discrete conformality on triangulated mesh, where the indices is illustrated in Fig. 1A-B (139 ). Regularization forces require the input of a reference value for geometric measurements, l, Ā, and λ, which can be derived from a well-conditioned reference mesh (usually the initial input mesh for the simulation).…”

Mem3DG: Modeling Membrane Mechanochemical Dynamics in 3D using Discrete Differential Geometry

“…The length-cross-ratio, λ ij = l il l jk / l ki l jl is a metric of discrete conformality on triangulated mesh, where the indices is illustrated in Fig. 1A-B ( 139 ). Regularization forces require the input of a reference value for geometric measurements, and , which can be derived from a well-conditioned reference mesh (usually the initial input mesh for the simulation).…”