Geometry of Point-Hyperplane and Spherical Frameworks
Geometry of Point-Hyperplane and Spherical Frameworks
复制标题
点超平面和球面框架的几何形状
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Y. Eftekhari
中科院分区：
文献类型：
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作者：
Y. Eftekhari
本文证明了En中点超平面框架的无穷小刚度等价于Sn中杆节点框架的无穷小刚度，其中节点集（对应于超平面）位于Sn中的超平面上。这是通过比较欧几里德点超平面框架的刚度矩阵和球框架的刚度矩阵来完成的。这个结果清楚地表明了射影空间和欧氏空间中的一阶刚性是如何全局连通的。这个几何意义上的结果是论文的核心。这一结果导致的一阶刚度的点超平面框架在En与一组关节在En的超平面的杆关节框架的等效。这个结果和它的一些重要结果也在合著的论文[17]中给出。我们还研究了En中点超平面框架的刚性，并刻画了它们的刚性。接下来，我们强调点线框架和滑块机制在平面上的关系。点-线框架被用来模拟各种类型的滑块机构。作为平面内铰接杆-关节框架的类似结果的直接结果，直接导出了平面内铰接滑块框架刚度的组合表征。使用fixednormal点线框架，我们模拟了第二种类型的滑块系统中的滑块方向不改变。此外，引入了第三种类型的滑块机构，其中滑块可以仅围绕固定点旋转但不平移。该滑块机制使用具有旋转线的点线框架来定义（不允许线的平移运动）。这些框架的通用刚性的组合表征在[17]中合著。然后，我们介绍了点超平面张拉整体框架在En。我们调查的刚度和无穷小刚度的这些框架在En使用张拉整体框架在Sn。我们描述了这些不同类型的刚度点超平面张拉整体框架在En和显示如何这些类型的刚度
In this thesis we show that the infinitesimal rigidity of point-hyperplane frameworks in En is equivalent to the infinitesimal rigidity of bar-joint frameworks in Sn with a set of joints (corresponding to the hyperplanes) located on a hyperplane in Sn. This is done by comparing the rigidity matrix of Euclidean point-hyperplane frameworks and the rigidity matrix of spherical frameworks. This result clearly shows how the first-order rigidity in projective spaces and Euclidean spaces are globally connected. This geometrically significant result is central to the thesis. This result leads to the equivalence of the first-order rigidity of point-hyperplane frameworks in En with that of bar-joint frameworks with a set of joints in a hyperplane in En. This result and some of its important consequences are also presented in the coauthored paper [17]. We also study the rigidity of point-hyperplane frameworks in En and characterize their rigidity. We next highlight the relationship between point-line frameworks and slider mechanisms in the plane. Point-line frameworks are used to model various types of slider mechanisms. A combinatorial characterization of the rigidity of pinnedslider frameworks in the plane is derived directly as an immediate consequence of the analogous result for pinned bar-joint frameworks in the plane. Using fixednormal point-line frameworks, we model a second type of slider system in which the slider directions do not change. Also, a third type of slider mechanism is introduced in which the sliders may only rotate around a fixed point but do not translate. This slider mechanism is defined using point-line frameworks with rotatory lines (no translational motion of the lines is allowed). A combinatorial characterization of the generic rigidity of these frameworks is coauthored in [17]. Then we introduce point-hyperplane tensegrity frameworks in En. We investigate the rigidity and the infinitesimal rigidity of these frameworks in En using tensegrity frameworks in Sn. We characterize these different types of rigidity for point-hyperplane tensegrity frameworks in En and show how these types of rigidity
点超平面框架、滑块关节和刚度保持变换
DOI：
10.1016/j.jctb.2018.07.008
发表时间：
2019
期刊：
Journal of Combinatorial Theory, Series B
影响因子：
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作者：
Eftekhari Y
通讯作者：
Eftekhari Y