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When is a planar rod configuration infinitesimally rigid?
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0002-7040-4006
2023 (English)In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444Article in journal (Refereed) Epub ahead of print
Abstract [en]

We investigate the rigidity properties of rod configurations. Rod configurations are realizations of rank two incidence geometries as points (joints) and straight lines (rods) in the Euclidean plane, such that the lines move as rigid bodies, connected at the points. Note that not all incidence geometries have such realizations. We show that under the assumptions that the rod configuration exists and is sufficiently generic, its infinitesimal rigidity is equivalent to the infinitesimal rigidity of generic frameworks of the graph defined by replacing each rod by a cone over its point set. To put this into context, the molecular conjecture states that the infinitesimal rigidity of rod configurations realizing 2-regular hypergraphs is determined by the rigidity of generic body and hinge frameworks realizing the same hypergraph. This conjecture was proven by Jackson and Jordán in the plane, and by Katoh and Tanigawa in arbitrary dimension. Whiteley proved a version of the molecular conjecture for hypergraphs of arbitrary degree that have realizations as independent body and joint frameworks. Our result extends his result to hypergraphs that do not necessarily have realizations as independent body and joint frameworks, under the assumptions listed above.

Place, publisher, year, edition, pages
Springer Nature, 2023.
Keywords [en]
Combinatorial rigidity, Hypergraphs, Incidence geometries, Parallel redrawings, Rod configurations
National Category
Discrete Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-218895DOI: 10.1007/s00454-023-00617-7Scopus ID: 2-s2.0-85180169240OAI: oai:DiVA.org:umu-218895DiVA, id: diva2:1824064
Funder
Knut and Alice Wallenberg Foundation, 2020.0001Knut and Alice Wallenberg Foundation, 2020.0007Available from: 2024-01-04 Created: 2024-01-04 Last updated: 2024-01-04

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Lundqvist, SigneStokes, KlaraÖhman, Lars-Daniel

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