Differential Geometry of Surfaces

Carmo, Manfredo P. do

doi:10.1007/978-3-642-57951-6_5

Cited by 832 publications

(1,324 citation statements)

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“…It is the basis for developing continuum elastic potentials from the atomistic description of the system (Cousins, 1978;Ericksen, 1984;Zanzotto, 1996), without other phenomenological input, and has proven very e ective in space-ÿlling crystals. It states that the crystal vectors deÿned by two nuclei deform according to the local deformation gradient.…”

Section: Introductionmentioning

confidence: 99%

An atomistic-based finite deformation membrane for single layer crystalline films

Arroyo

Belytschko

2002

Journal of the Mechanics and Physics of Solids

320

241

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A general methodology to develop hyper-elastic membrane models applicable to crystalline ÿlms one-atom thick is presented. In this method, an extension of the Born rule based on the exponential map is proposed. The exponential map accounts for the fact that the lattice vectors of the crystal lie along the chords of the curved membrane, and consequently a tangent map like the standard Born rule is inadequate. In order to obtain practical methods, the exponential map is locally approximated. The e ectiveness of our approach is demonstrated by numerical studies of carbon nanotubes. Deformed conÿgurations as well as equilibrium energies of atomistic simulations are compared with those provided by the continuum membrane resulting from this method discretized by ÿnite elements.

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Section: Introductionmentioning

confidence: 99%

An atomistic-based finite deformation membrane for single layer crystalline films

Arroyo

Belytschko

2002

Journal of the Mechanics and Physics of Solids

320

241

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“…When g = 0, (2.3a) is the minimal surface equation [70,170,34,76], whose solution is identified as the minimizer of the surface area…”

Section: 1mentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

142

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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“…For simplicity, we use the Gaussian curvature curv(i) at point i as geometric descriptor, which is invariant to isometric transformation [14], as well as the texture value tex(i) at point i as photometric descriptor if texture information is available. Then, the singleton potential θ a for a correspondence a = (i, j) is defined as follows:…”

Section: Non-rigid 3d Surface Matchingmentioning

confidence: 99%

“…In practice we often want to eliminate such an ambiguity, for which we can define another type of third-order terms based on the Gaussian map of the surface. The Gaussian map is defined as the mapping of the normal at each point on the surface to the unit sphere [14]. Due to the fact that two triplets have the same orientation of the Gaussian maps if and only if the determinant of their normals have the same sign, we can define the below higher-order term as a penalty for extrinsic orientation inconsistency:…”

Section: Non-rigid 3d Surface Matchingmentioning

confidence: 99%

Modeling Shapes with Higher-Order Graphs: Methodology and Applications

Wang

Zeng

Samaras

et al. 2013

Shape Perception in Human and Computer Vision

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Extrinsic factors such as object pose and camera parameters are a main source of shape variability and pose an obstacle to efficiently solving shape matching and inference. Most existing methods address the influence of extrinsic factors by decomposing the transformation of the source shape (model) into two parts: one corresponding to the extrinsic factors and the other accounting for intra-class variability and noise, which are solved in a successive or alternating manner. In this chapter, we consider a methodology to circumvent the influence of extrinsic factors by exploiting shape properties that are invariant to them. Based on higher-order graph-based models, we implement such a methodology to address various important vision problems, such as non-rigid 3D surface matching and knowledge-based 3D segmentation, in a one-shot optimization scheme. Experimental results demonstrate the superior performance and potential of this type of approach.

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Differential Geometry of Surfaces

Cited by 832 publications

References 0 publications

An atomistic-based finite deformation membrane for single layer crystalline films

An atomistic-based finite deformation membrane for single layer crystalline films

A review of numerical methods for nonlinear partial differential equations

Modeling Shapes with Higher-Order Graphs: Methodology and Applications

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