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Coincident-point rigidity in normed planes

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Coincident-point rigidity in normed planes. / Dewar, Sean; Hewetson, John; Nixon, Anthony.
In: Ars Mathematica Contemporanea, Vol. 24, No. 1, 07.09.2023.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Dewar, S, Hewetson, J & Nixon, A 2023, 'Coincident-point rigidity in normed planes', Ars Mathematica Contemporanea, vol. 24, no. 1. https://doi.org/10.26493/1855-3974.2826.3dc

APA

Dewar, S., Hewetson, J., & Nixon, A. (2023). Coincident-point rigidity in normed planes. Ars Mathematica Contemporanea, 24(1). https://doi.org/10.26493/1855-3974.2826.3dc

Vancouver

Dewar S, Hewetson J, Nixon A. Coincident-point rigidity in normed planes. Ars Mathematica Contemporanea. 2023 Sept 7;24(1). Epub 2023 Mar 31. doi: 10.26493/1855-3974.2826.3dc

Author

Dewar, Sean ; Hewetson, John ; Nixon, Anthony. / Coincident-point rigidity in normed planes. In: Ars Mathematica Contemporanea. 2023 ; Vol. 24, No. 1.

Bibtex

@article{4e9fa5f07e374e109d3d99bdec56e0c6,
title = "Coincident-point rigidity in normed planes",
abstract = "A bar-joint framework (G,p) is the combination of a graph G and a map p assigning positions, in some space, to the vertices of G. The framework is rigid if every edge-length-preserving continuous motion of the vertices arises from an isometry of the space. We will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated vertices are mapped to the same position. This non-genericity assumption leads us to a count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that independence in this matroid is equivalent to independence as a suitably regular bar-joint framework in a normed plane with two coincident points; this characterises when a regular normed plane coincident-point framework is rigid and allows us to deduce a delete-contract characterisation. We then apply this result to show that an important construction operation (generalised vertex splitting) preserves the stronger property of global rigidity in normed planes and use this to construct rich families of globally rigid graphs when the normed plane is analytic.",
keywords = "Bar-joint framework, global rigidity, non-Euclidean framework, count matroid, recursive construction, normed spaces, analytic norm",
author = "Sean Dewar and John Hewetson and Anthony Nixon",
year = "2023",
month = sep,
day = "7",
doi = "10.26493/1855-3974.2826.3dc",
language = "English",
volume = "24",
journal = "Ars Mathematica Contemporanea",
issn = "1855-3966",
publisher = "DMFA Slovenije",
number = "1",

}

RIS

TY - JOUR

T1 - Coincident-point rigidity in normed planes

AU - Dewar, Sean

AU - Hewetson, John

AU - Nixon, Anthony

PY - 2023/9/7

Y1 - 2023/9/7

N2 - A bar-joint framework (G,p) is the combination of a graph G and a map p assigning positions, in some space, to the vertices of G. The framework is rigid if every edge-length-preserving continuous motion of the vertices arises from an isometry of the space. We will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated vertices are mapped to the same position. This non-genericity assumption leads us to a count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that independence in this matroid is equivalent to independence as a suitably regular bar-joint framework in a normed plane with two coincident points; this characterises when a regular normed plane coincident-point framework is rigid and allows us to deduce a delete-contract characterisation. We then apply this result to show that an important construction operation (generalised vertex splitting) preserves the stronger property of global rigidity in normed planes and use this to construct rich families of globally rigid graphs when the normed plane is analytic.

AB - A bar-joint framework (G,p) is the combination of a graph G and a map p assigning positions, in some space, to the vertices of G. The framework is rigid if every edge-length-preserving continuous motion of the vertices arises from an isometry of the space. We will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated vertices are mapped to the same position. This non-genericity assumption leads us to a count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that independence in this matroid is equivalent to independence as a suitably regular bar-joint framework in a normed plane with two coincident points; this characterises when a regular normed plane coincident-point framework is rigid and allows us to deduce a delete-contract characterisation. We then apply this result to show that an important construction operation (generalised vertex splitting) preserves the stronger property of global rigidity in normed planes and use this to construct rich families of globally rigid graphs when the normed plane is analytic.

KW - Bar-joint framework

KW - global rigidity

KW - non-Euclidean framework

KW - count matroid

KW - recursive construction

KW - normed spaces

KW - analytic norm

U2 - 10.26493/1855-3974.2826.3dc

DO - 10.26493/1855-3974.2826.3dc

M3 - Journal article

VL - 24

JO - Ars Mathematica Contemporanea

JF - Ars Mathematica Contemporanea

SN - 1855-3966

IS - 1

ER -