Infinitesimally rigid polyhedra. I. Statics of frameworks
Infinitesimally rigid polyhedra. I. Statics of frameworks
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无限小刚性多面体。
DOI：
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发表时间：
1984
期刊：
影响因子：
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通讯作者：
W. Whiteley
中科院分区：
文献类型：
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作者：
W. Whiteley
从柯西时代起，数学家们就开始研究凸多面体的运动，这些凸多面体的面是刚性的，而二面角是允许变化的。在20世纪40年代亚历山德罗夫证明，即使有额外的顶点沿着自然边缘，并与任意三角剖分的自然面临这些顶点，这样的多面体是无穷小刚性。本文用框架静刚度的对偶（和等效）概念来描述围绕凸多面体（和其它）建造的杆系和节点框架的行为。引入的静态技术提供了一个新的简化证明亚历山德罗夫定理，以及一个重要的扩展，其特征在于静态属性的框架，更一般的模式上的脸，包括框架与顶点内部的脸。的静态技术，并采用适当的模式，以扩展的任意静态刚性框架周围的任何多面体（非凸，环形等）。该技术也适用于推导张拉整体框架的静态刚度（与电缆和支柱在酒吧的地方），和框架的静态刚度投影等效已知的多面体框架。最后，作为一个练习，给一个额外的角度在3-空间的结果，详细的类似物亚历山德罗夫定理的凸4-多面体建成酒吧和联合框架在4-空间。
From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation of the natural faces on these vertices, such polyhedra are infinitesimally rigid. In this paper the dual (and equivalent) concept of static rigidity for frameworks is used to describe the behavior of bar and joint frameworks built around convex (and other) polyhedra. The static techniques introduced provide a new simplified proof of Alexandrov's theorem, as well as an essential extension which characterizes the static properties of frameworks built with more general patterns on the faces, including frameworks with vertices interior to the faces. The static techniques are presented and employed in a pattern appropriate to the extension of an arbitrary statically rigid framework built around any polyhedron (nonconvex, toroidal, etc.). The techniques are also applied to derive the static rigidity of tensegrity frameworks (with cables and struts in place of bars), and the static rigidity of frameworks projectively equivalent to known polyhedral frameworks. Finally, as an exercise to give an additional perspective to the results in 3-space, detailed analogues of Alexandrov's theorem are presented for convex 4-polytopes built as bar and joint frameworks in 4-space.