Maxwell-Laman counts for bar-joint frameworks in normed spaces. (English) Zbl 1318.52018
Summary: The rigidity matrix is a fundamental tool for studying the infinitesimal rigidity properties of Euclidean bar-joint frameworks. In this paper, we generalise this tool and introduce a rigidity matrix for bar-joint frameworks in arbitrary finite dimensional real normed vector spaces. Using this new matrix, we derive necessary Maxwell-Laman-type counting conditions for a well-positioned bar-joint framework in a real normed vector space to be infinitesimally rigid. Moreover, we derive symmetry-extended counting conditions for a bar-joint framework with a non-trivial symmetry group to be isostatic (i.e., minimally infinitesimally rigid). These conditions imply very simply stated restrictions on the number of those structural components that are fixed by the various symmetry operations of the framework. Finally, we offer some observations and conjectures regarding combinatorial characterisations of symmetric, isostatic bar-joint frameworks in \((\mathbb{R}^2, \parallel \cdot \parallel_{\mathcal{P}})\), where the unit ball \(\mathcal{P}\) is a quadrilateral.
MSC:
| 52C25 | Rigidity and flexibility of structures (aspects of discrete geometry) |
| 20C35 | Applications of group representations to physics and other areas of science |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 51B20 | Minkowski geometries in nonlinear incidence geometry |
| 52A21 | Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) |
Keywords:
rigidity matrix; bar-joint framework; infinitesimal rigidity; Minkowski geometry; symmetric frameworkReferences:
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