Minimizing the Squared Mean Curvature Integral for Surfaces in Space Forms

Hsu, Lucas; Kusner, Robert B.; Sullivan, John M.

doi:10.1080/10586458.1992.10504258

Cited by 98 publications

(95 citation statements)

References 11 publications

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“…We can now understand why the single structures are found to be so close to minimal surfaces; the minimum of dAH 2 (the so-called Willmore problem [47]) in this case is H = 0, which in turn minimizes f min . Note that this reasoning is not rigorous, since the minimization of the free-energy functional with respect to lattice constant and shape are not independent; the latter step determines A/a 2 , which is taken to be constant in the first step.…”

Section: Hierarchy Of Structuresmentioning

confidence: 99%

Systematic approach to bicontinuous cubic phases in ternary amphiphilic systems

1999

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The Fourier approach and theories of space groups and color symmetries are used to systematically generate and compare bicontinuous cubic structures in the framework of a Ginzburg-Landau model for ternary amphiphilic systems. Both single and double structures are investigated; they correspond to systems with one or two monolayers in a unit cell, respectively. We show how and why single structures can be made to approach triply periodic minimal surfaces very closely, and give improved nodal approximations for G, D, I-WP and P surfaces. We demonstrate that the relative stability of the single structures can be calculated from the geometrical properties of their interfaces only. The single gyroid G turns out to be the most stable bicontinuous cubic phase since it has the smallest porosity. The representations are used to calculate distributions of the Gaussian curvature and 2 H-NMR bandshapes for C(P), C(D), S, C(Y) and F-RD surfaces.

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Section: Hierarchy Of Structuresmentioning

confidence: 99%

Systematic approach to bicontinuous cubic phases in ternary amphiphilic systems

1999

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“…Local descent techniques have been derived for minimizing (1), cf. [7], but they are very dependent on a good initialization.…”

Section: Curvature In Visionmentioning

confidence: 99%

Curvature Regularization for Curves and Surfaces in a Global Optimization Framework

Strandmark

Kahl

2011

Lecture Notes in Computer Science

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Abstract. Length and area regularization are commonplace for inverse problems today. It has however turned out to be much more difficult to incorporate a curvature prior. In this paper we propose several improvements to a recently proposed framework based on global optimization. We identify and solve an issue with extraneous arcs in the original formulation by introducing region consistency constraints. The mesh geometry is analyzed both from a theoretical and experimental viewpoint and hexagonal meshes are shown to be superior. We demonstrate that adaptively generated meshes significantly improve the performance. Our final contribution is that we generalize the framework to handle mean curvature regularization for 3D surface completion and segmentation.

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“…Brakke's Evolver 2 is a general tool for geometric optimization, like minimization of surface area or bending energies [4]. There are several discretizations of the Willmore energy available in the Evolver [15]. In fact, we made use of these before to implement the minimax eversion with p = 2, which is equivalent to Morin's original eversion [14].…”

Section: Symmetric Eversions Driven By Willmore Energymentioning

confidence: 99%

“…An elastic bending energy for surfaces should be quadratic in the principal curvature, and if symmetric can be reduced by the Gauss-Bonnet theorem to the integral of mean curvature squared, W = H 2 dA, known as the Willmore energy [26]. (See [15] for more about the history of this energy, and some early computer experiments minimizing it. )…”

Section: Symmetric Eversions Driven By Willmore Energymentioning

confidence: 99%

Computing Sphere Eversions

Francis

Sullivan

Hartman

1998

Mathematical Visualization

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Abstract. We consider several tools for computing and visualizing sphere eversions. First, we discuss a family of rotationally symmetric eversions driven computationally by minimizing the Willmore bending energy. Next, we describe programs to compute and display the double locus of an immersed surface and to track this along a homotopy. Finally, we consider ways to implement computationally the various eversions originally drawn by hand; this requires interpolation of splined curves in time and space.

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Minimizing the Squared Mean Curvature Integral for Surfaces in Space Forms

Cited by 98 publications

References 11 publications

Systematic approach to bicontinuous cubic phases in ternary amphiphilic systems

Systematic approach to bicontinuous cubic phases in ternary amphiphilic systems

Curvature Regularization for Curves and Surfaces in a Global Optimization Framework

Computing Sphere Eversions

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