Discrete Differential Geometry. Integrability as Consistency

Bobenko, Alexander I.

doi:10.1007/978-3-540-40357-9_4

Cited by 58 publications

(120 citation statements)

References 30 publications

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“…Chebyshev nets give a natural discretization of surfaces with constant negative curvature that respects both the intrinsic and the extrinsic geometry of the surface. This idea has already been used for constructing discrete isometric immersions (K-surfaces) for such surfaces [29]. This can be interpreted as a t → 0 limit for the variational problem (2.1).…”

Section: Discussionmentioning

confidence: 99%

“…The coloring of the disks corresponds to contours of arclength data and indicate how the different geodesic circles cut from these surfaces would appear. These surfaces are in fact not true Amsler surfaces but are discrete Amsler surfaces creating using the algorithm presented in [29].…”

Section: Maximum Radius Of Periodic Amsler Surfacesmentioning

confidence: 99%

“…As evidenced by the sphere, C-nets in general are only local parameterizations of a surface and a global C-net exists if the integral of the magnitude of the Gaussian curvature over a quadrilateral is less than 2π [28]. C-nets are useful parameterizations in that they lead to a natural way to discretize a surface [29] and projections of C-nets onto paper retain the three dimensional properties of the surface [30]. Remarkably, for surfaces with constant negative curvature, every sufficiently smooth isometric immersion generates a C-net on the surface obtained by taking the asymptotic curves as the coordinate curves and conversely every C-net on a surface of constant negative curvature generates an immersion whose asymptotic curves are the coordinate lines [14].…”

Section: Introductionmentioning

confidence: 99%

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Shape selection in non-Euclidean plates

Gemmer

Venkataramani

2011

Physica D: Nonlinear Phenomena

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We investigate isometric immersions of disks with constant negative curvature into R 3 , and the minimizers for the bending energy, i.e. the L 2 norm of the principal curvatures over the class of W 2,2 isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H 2 into R 3 . In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the L ∞ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W 2,2 , and numerically have lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grow approximately exponentially with the radius of the disc. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates.

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Section: Discussionmentioning

confidence: 99%

Section: Maximum Radius Of Periodic Amsler Surfacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Shape selection in non-Euclidean plates

Gemmer

Venkataramani

2011

Physica D: Nonlinear Phenomena

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“…Likewise, the quadrilateral and circular lattices are described by integrable difference equations. The key idea of the geometric approach [24,[29][30][31][32][33][34][35][36]] to integrability of discrete classical systems is to utilize various consistency conditions [37] arising from geometric relations between elements of the lattice. It is quite remarkable that these conditions ultimately reduce to certain incidence theorems of elementary geometry.…”

Section: Introductionmentioning

confidence: 99%

Quantum geometry of three-dimensional lattices

Bazhanov¹,

Mangazeev²,

Sergeev³

2008

J. Stat. Mech.

100

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We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable "ultra-local" Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation). These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasi-classical limit.

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“…дискретных интегрируемых систем, см. [15]). Дискретные интегрируе-мые системы и коммутационные соотношения Лакса выписаны и изучались в [12,16].…”

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