Curvature formulas for implicit curves and surfaces

Goldman, Ron

doi:10.1016/j.cagd.2005.06.005

Cited by 360 publications

(192 citation statements)

References 5 publications

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“…At every gridpoint within the narrow band, the mean, K M , and Gaussian, K G , curvatures are calculated from the smoothed order parameter according to [38]:…”

Section: Surface Curvaturesmentioning

confidence: 99%

Phase wettability and microstructural evolution in solid oxide fuel cell anode materials

et al. 2014

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Recent experimental and theoretical findings suggest that high-temperature solid oxide fuel cells (SOFCs) often suffer from performance degradation due to coarsening of the metallic-phase particles within the anode. In this study, we explore the feasibility of improving the microstructural stability of SOFC anode materials by tuning the contact angle between the metallic phase and electrolyte particles. To this end, a continuum diffuse-interface model is employed to capture the coarsening behavior of the metallic phase and simulate a range of equilibrium contact angles. The evolution of performance-critical, microstructural features is presented for varying degrees of phase wettability. It is found that both the density of electrochemically active triple-phase regions and contiguity of the electron-conducting phase display undesirable minima near the contact angle of conventional SOFC materials. Our results suggest that tailoring the interfacial properties of the constituent phases could lead to a significant increase in the performance and lifetime of SOFCs.

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“…At every gridpoint within the narrow band, the mean, K M , and Gaussian, K G , curvatures are calculated from the smoothed order parameter according to [38]:…”

Section: Surface Curvaturesmentioning

confidence: 99%

Phase wettability and microstructural evolution in solid oxide fuel cell anode materials

et al. 2014

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“…Expressions for the mean and Gaussian curvature of implicit surfaces are given in some differential geometry textbooks [191,192]. The following considerations are based on GOLDMAN, where also rigorous proofs are provided [193]. For an implicit surface defined by f ( r , ρ iso ) = 0, the mean and Gaussian curvature at any point r of the surface are…”

Section: Differential Geometry Of Implicit Surfacesmentioning

confidence: 99%

“…(6.10) and (6.11), respectively. Therefore, H is invariant under rescaling, while K changes its sign for λ < 0 [193]. Due to Schwarz's theorem, Hess( f ) and Hess * ( f ) are symmetric.…”

mentioning

confidence: 99%

Stalk structures in lipid bilayer fusion studied by x-ray diffraction

Aeffner¹

2012

Göttingen Series in X-Ray Physics

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“…In this implicit case, Goldman [7] provides a formula for Gaussian curvature. In particular, the scalar ðvÞ can be expressed as…”

Section: Linear Precisionmentioning

confidence: 99%

Barycentric coordinates for convex sets

et al. 2006

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In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulation valid for arbitrary convex polytopes has been proposed to extend this interpolation in higher dimensions. Moreover, there has been no attempt to extend these functions into the continuous domain, where barycentric coordinates are related to Green's functions and construct functions that satisfy a boundary value problem. First, we review the properties and construction of barycentric coordinates in the discrete domain for convex polytopes. Next, we show how these concepts extend into the continuous domain to yield barycentric coordinates for continuous functions. We then provide a proof that our functions satisfy all the desirable properties of barycentric coordinates in arbitrary dimensions. Finally, we provide an example of constructing such barycentric functions over regions bounded by parametric curves and show how they can be used to perform freeform deformations.

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Curvature formulas for implicit curves and surfaces

Cited by 360 publications

References 5 publications

Phase wettability and microstructural evolution in solid oxide fuel cell anode materials

Phase wettability and microstructural evolution in solid oxide fuel cell anode materials

Stalk structures in lipid bilayer fusion studied by x-ray diffraction

Barycentric coordinates for convex sets

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