Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Stokes, Klara

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Öhman, Lars-Daniel

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

2023 (English)In: EuroCG 2023: Book of abstracts, 2023, article id 41Conference paper, Published paper (Refereed)

Abstract [en]

We present results on rigidity of structures of rigid rods connected in joints: rod configurations. The underlying combinatorial structure of a rod configuration is an incidence structure. Our aim is to find simple ways of determining which rod configurations admit non-trivial motions, using the underlying incidence structure.

Rigidity of graphs in the plane is well understood. Indeed, there is a polynomial time algorithm for deciding whether most realisations of a graph are rigid. One of the results presented here equates rigidity of sufficiently generic rod configurations to rigidity of a related graph. As a consequence, itis possible to determine the rigidity of rod configurations using the previously mentioned polynomial time algorithm. We use this to show that all v3-configurations on up to 15 points and all triangle-free v3-configurations on up to 20 points are rigid in regular position, if such a realisation exists. We also conjecture that the smallest v3-configuration that is flexible in regular position is a previously known 28_{3}-configuration.

National Category

Discrete Mathematics Geometry

Identifiers

urn:nbn:se:umu:diva-215548 (URN)

Conference

The 39th European workshop on computational geometry (EuroCG 2023), Barcelona, Spain, March 29-31, 2023

Available from: 2023-10-22 Created: 2023-10-22 Last updated: 2023-10-23Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Stokes, Klara

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Öhman, Lars-Daniel

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

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

Combinatorial rigidity, Hypergraphs, Incidence geometries, Parallel redrawings, Rod configurations

Knut and Alice Wallenberg Foundation, 2020.0001Knut and Alice Wallenberg Foundation, 2020.0007
Available from: 2024-01-04 Created: 2024-01-04 Last updated: 2024-01-04

Organisations

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics