Infinitesimally rigid polyhedra. II. Modified spherical frameworks
Infinitesimally rigid polyhedra. II. Modified spherical frameworks
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无限小刚性多面体。
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发表时间：
1988
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通讯作者：
W. Whiteley
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作者：
W. Whiteley
在第一篇论文中，亚历山德罗夫定理进行了研究，并扩大，以表明凸多面体形式静态刚性框架在空间中，当建立与平面刚性面。这第二篇论文研究了这些多面体框架的两个修改：（一）块多面体框架，与一些光盘作为开孔，其他光盘作为spacebridid块，其余的面平面刚性;和（ii）扩展多面体框架，单独添加的酒吧（轴）和选定的边缘删除。归纳方法的发展，以显示静态刚性的特定模式的孔和块和扩展，在一般实现的多面体。该方法是基于证明技术的斯坦尼茨定理，以及一个相关的协调适当实现的3连接的球面多面体。抽样结果表明：（a）一个k-连通块和一个k-连通孔产生静刚度当且仅当该块和孔在顶点意义上是k-连通的;（B）一个4-连通的三角形化球体，加上一个杆，是一个静刚性回路（去掉任何一个杆，留下一个最小的静刚性框架）。结果也被解释为一个描述的二面角在一个三角形的球体将弯曲时，一个酒吧被删除。
In the first paper, Alexandrov's Theorem was studied, and extended, to show that convex polyhedra form statically rigid frameworks in space, when built with plane-rigid faces. This second paper studies two modifications of these polyhedral frameworks: (i) block polyhedral frameworks, with some discs as open holes, other discs as spacbrigid blocks, and the remaining faces plane-rigid; and (ii) extended polyhedral frameworks, with individually added bars (shafts) and selected edges removed. Inductive methods are developed to show the static rigidity of particular patterns of holes and blocks and of extensions, in general realizations of the polyhedron. The methods are based on proof techniques for Steinitz's Theorem, and a related coordinatization of the proper realizations of a 3-connected spherical polyhedron. Sample results show that: (a) a single k-gonal block and a k-gonal hole yield static rigidity if and only if the block and hole are k-connected in a vertex sense; and (b) a 4-connected triangulated sphere, with one added bar, is a statically rigid circuit (removing any one bar leaves a minimal statically rigid framework). The results are also interpreted as a description of which dihedral angles in a triangulated sphere will flex when one bar is removed.