On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces
On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces
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多面体曲面无穷小与有限变形的可积性
DOI：
10.1007/978-3-7643-8621-4_4
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发表时间：
2008
影响因子：
--
通讯作者：
T. Hoffmann
中科院分区：
文献类型：
--
作者：
W. Schief;A. Bobenko;T. Hoffmann
证明了多面体曲面的等距变形与离散可积系统之间存在密切联系。特别地，采用Sauer的运动学方法证明了由平面四边形(离散共轭网)组成的离散曲面的二阶无穷小等距变形是由控制共轭网有限等距变形的Bianci经典方程的可积离散形式的解决定的。此外，还证明了离散共轭网的有限等距变形完全封装在一种特殊的受约束的非线性σ模型的标准可积离散格式中。从而以自然的方式检索离散VOSS表面的可变形性。
It is established that there exists an intimate connection between isometric deformations of polyhedral surfaces and discrete integrable systems. In particular, Sauer’s kinematic approach is adopted to show that second-order infinitesimal isometric deformations of discrete surfaces composed of planar quadrilaterals (discrete conjugate nets) are determined by the solutions of an integrable discrete version of Bianchi’s classical equation governing finite isometric deformations of conjugate nets. Moreover, it is demonstrated that finite isometric deformations of discrete conjugate nets are completely encapsulated in the standard integrable discretization of a particular nonlinear σ-model subject to a constraint. The deformability of discrete Voss surfaces is thereby retrieved in a natural manner.