On the convergence of metric and geometric properties of polyhedral surfaces

Hildebrandt, Klaus; Polthier, Konrad; Wardetzky, Max

doi:10.1007/s10711-006-9109-5

Cited by 143 publications

(142 citation statements)

References 35 publications

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“…Hildebrandt et.al's work [23] focuses on the equivalence of convergence of polyhedral meshes under different metrics, such as Hausdorff, normal, area and Laplace-Beltrami. Assuming the Hausdorff convergence and the homeomorphism between the surface and the mesh, all the error estimations are based on the homeomorphism.…”

Section: Comparisons To Previous Theoretical Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Geometric accuracy analysis for discrete surface approximation

Dai

Luo

Jin

et al. 2007

Computer Aided Geometric Design

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Section: Comparisons To Previous Theoretical Resultsmentioning

confidence: 99%

“…These important applications demand the theoretical guarantee for accurate approximation for Riemannian metric and differential operators. It has been shown in [23], Hausdorff convergence and normal field convergence guarantee the convergence of area, Riemannian metric tensor and Laplace-Beltrami operator.…”

Section: Geometric Accuracymentioning

confidence: 97%

Geometric accuracy analysis for discrete surface approximation

Dai

Luo

Jin

et al. 2007

Computer Aided Geometric Design

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“…[23,24] Φ(e) is the signed dihedral angle between the faces, f 1 (e) and f 2 (e), taking a value of π when the faces are coplanar . R(e) is the length of the edge.…”

Section: In-plane Orientational Ordermentioning

confidence: 99%

Modeling Anisotropic Elasticity of Fluid Membranes

Ramakrishnan¹,

Kumar²,

Ipsen³

2011

Macro Theory & Simulations

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The biological membrane, which compartmentalizes the cell and its organelles, exhibit wide variety of macroscopic shapes of varying morphology and topology.s A systematic understanding of the relation of membrane shapes to composition, external field, environmental conditions etc. have important biological relevance. Here we review the triangulated surface model, used in the macroscopic simulation of membranes and the associated Monte Carlo (DTMC) methods. New techniques to calculate surface quantifiers, that will facilitate the study of additional in-plane orientational degrees of freedom, has been introduced. The mere presence of a polar and nematic fields in the ordered phase drives the ground state conformations of the membrane to a cylinder and tetrahedron respectively.

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“…For example, Meyer et al (2002) proposed a cotangent formula for estimating mean curvatures, which is closely related to the formula for Dirichlet energy of Pinkall and Polthier (1993). It was shown that the cotangent formula does not produce converging pointwise mean-curvature estimations except for some special cases, as noted in Borrelli et al (2003), Xu (2004), Hildebrandt et al (2006), Wardetzky (2007). As another example, Langer et al (2007) proposed a tangent-weighted formula for estimating mean-curvature vectors, whose convergence relies on special symmetric patterns of a mesh.…”

Section: Introductionmentioning

confidence: 99%

An analysis and comparison of parameterization-based computation of differential quantities for discrete surfaces

Wang

Clark

Jiao

2009

Computer Aided Geometric Design

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Normals and curvatures are fundamental for geometric modeling and computer-aided design, but their accurate computations on discrete surfaces are challenging. Two types of methods, namely height-function based and parameterization based polynomial fittings, are well founded mathematically and can be proven to deliver convergent results under reasonable assumptions. However, the numerical behaviors of these methods can differ drastically in practice, and no systematic analysis and comparison have been reported previously for these methods. In this paper, we describe a unified framework for these methods based on weighted least squares approximations, and on top of this framework we compare a number of methods in terms of numerical accuracy and stability as well as runtime efficiency and robustness through both theoretical analysis and numerical experiments. Our analysis shows that the choice of parameterization and numerical solver for the least squares problem can have significant impact on the accuracy and stability of polynomial fittings. In addition, we show that the methods based on local orthogonal projection with a safeguard against folding deliver the best combination of simplicity, accuracy, efficiency, and robustness.

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On the convergence of metric and geometric properties of polyhedral surfaces

Cited by 143 publications

References 35 publications

Geometric accuracy analysis for discrete surface approximation

Geometric accuracy analysis for discrete surface approximation

Modeling Anisotropic Elasticity of Fluid Membranes

An analysis and comparison of parameterization-based computation of differential quantities for discrete surfaces

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