Affine rigidity and conics at infinity

Robert Connelly; Steven J. Gortler; Louis Theran

doi:10.1093/imrn/rnx014

Affine rigidity and conics at infinity

Robert Connelly, Steven J. Gortler, Louis Theran

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that if a framework of a graph is neighborhood affine rigid in d-dimensions (or has the stronger property of having an equilibrium stress matrix of rank n — d — 1) then it has an affine flex (an affine, but non Euclidean, transform of space that preserves all of the edge lengths) if and only if the framework is ruled on a single quadric. This strengthens and also simplifies a related result by Alfakih. It also allows us to prove that the property of super stability is invariant with respect to projective transforms and also to the coning and slicing operations. Finally this allows us to unify some previous results on the Strong Arnold Property of matrices.

Original language	English
Pages (from-to)	4084-4102
Number of pages	19
Journal	International Mathematics Research Notices
Volume	2018
Issue number	13
Early online date	26 Feb 2017
DOIs	https://doi.org/10.1093/imrn/rnx014
Publication status	Published - Jul 2018

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Theran_2017_IMRN_infinity_AAM
© 2017, the Author(s). This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at imrn.oxfordjournals.org / https://doi.org/10.1093/imrn/rnx014
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title = "Affine rigidity and conics at infinity",

abstract = "We prove that if a framework of a graph is neighborhood affine rigid in d-dimensions (or has the stronger property of having an equilibrium stress matrix of rank n — d — 1) then it has an affine flex (an affine, but non Euclidean, transform of space that preserves all of the edge lengths) if and only if the framework is ruled on a single quadric. This strengthens and also simplifies a related result by Alfakih. It also allows us to prove that the property of super stability is invariant with respect to projective transforms and also to the coning and slicing operations. Finally this allows us to unify some previous results on the Strong Arnold Property of matrices.",

author = "Robert Connelly and Gortler, {Steven J.} and Louis Theran",

note = "RC is partially supported by NSF grant DMS-1564493. SJG is partially supported by NSF grant DMS-1564473.",

year = "2018",

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doi = "10.1093/imrn/rnx014",

language = "English",

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pages = "4084--4102",

journal = "International Mathematics Research Notices",

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T1 - Affine rigidity and conics at infinity

AU - Connelly, Robert

AU - Gortler, Steven J.

AU - Theran, Louis

N1 - RC is partially supported by NSF grant DMS-1564493. SJG is partially supported by NSF grant DMS-1564473.

PY - 2018/7

Y1 - 2018/7

N2 - We prove that if a framework of a graph is neighborhood affine rigid in d-dimensions (or has the stronger property of having an equilibrium stress matrix of rank n — d — 1) then it has an affine flex (an affine, but non Euclidean, transform of space that preserves all of the edge lengths) if and only if the framework is ruled on a single quadric. This strengthens and also simplifies a related result by Alfakih. It also allows us to prove that the property of super stability is invariant with respect to projective transforms and also to the coning and slicing operations. Finally this allows us to unify some previous results on the Strong Arnold Property of matrices.

AB - We prove that if a framework of a graph is neighborhood affine rigid in d-dimensions (or has the stronger property of having an equilibrium stress matrix of rank n — d — 1) then it has an affine flex (an affine, but non Euclidean, transform of space that preserves all of the edge lengths) if and only if the framework is ruled on a single quadric. This strengthens and also simplifies a related result by Alfakih. It also allows us to prove that the property of super stability is invariant with respect to projective transforms and also to the coning and slicing operations. Finally this allows us to unify some previous results on the Strong Arnold Property of matrices.

UR - https://arxiv.org/abs/1605.07911

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DO - 10.1093/imrn/rnx014

M3 - Article

SN - 1073-7928

VL - 2018

SP - 4084

EP - 4102

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 13

ER -

Affine rigidity and conics at infinity

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Louis Simon Theran

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Affine rigidity and conics at infinity

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Louis Simon Theran

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