- NixonAssurDirectionLength
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- http://ctw2020.iasi.cnr.it/nixon_353098/
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- https://link.springer.com/chapter/10.1007/978-3-030-63072-0_11
Final published version

Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Chapter (peer-reviewed) › peer-review

Published

**Assur decompositions of direction-length frameworks.** / Nixon, Anthony.

Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Chapter (peer-reviewed) › peer-review

Nixon, A 2020, Assur decompositions of direction-length frameworks. in C Gentile, G Stecca & P Ventura (eds), *Graphs and Combinatorial Optimization : from Theory to Applications : CTW2020 Proceedings.* AIRO Springer, Springer, Cham, pp. 131-143, 18th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, 14/09/20. https://doi.org/10.1007/978-3-030-63072-0_11

Nixon, A. (2020). Assur decompositions of direction-length frameworks. In C. Gentile, G. Stecca, & P. Ventura (Eds.), *Graphs and Combinatorial Optimization : from Theory to Applications : CTW2020 Proceedings *(pp. 131-143). (AIRO Springer). Springer. https://doi.org/10.1007/978-3-030-63072-0_11

Nixon A. Assur decompositions of direction-length frameworks. In Gentile C, Stecca G, Ventura P, editors, Graphs and Combinatorial Optimization : from Theory to Applications : CTW2020 Proceedings. Cham: Springer. 2020. p. 131-143. (AIRO Springer). doi: 10.1007/978-3-030-63072-0_11

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title = "Assur decompositions of direction-length frameworks",

abstract = "A bar-joint framework is a realisation of a graph consisting of stiff barslinked by universal joints. The framework is rigid if the only bar-length preserving continuous motions of the joints arise from isometries. A rigid framework is isostatic if deleting any single edge results in a flexible framework. Generically, rigidity depends only on the graph and we say an Assur graph is a pinned isostatic graph with no proper pinned isostatic subgraphs. Any pinned isostatic graph can be decomposed into Assur components which may be of use for mechanical engineers in decomposing mechanisms for simpler analysis and synthesis. A direction-length framework is a generalisation of bar-joint framework where some distance constraints are replaced by direction constraints. We initiate a theory of Assur graphs and Assur decompositions for direction-length frameworks using graph orientations and spanning trees and then analyse choices of pinning set.",

keywords = "Assur decomposition, Assur graph, Bar-joint framework, Direction-length framework, Pinned framework, Rigid graph",

author = "Anthony Nixon",

year = "2020",

month = nov,

day = "9",

doi = "10.1007/978-3-030-63072-0_11",

language = "English",

isbn = "9783030630713",

series = "AIRO Springer",

publisher = "Springer",

pages = "131--143",

editor = "Claudio Gentile and Giuseppe Stecca and Paolo Ventura",

booktitle = "Graphs and Combinatorial Optimization : from Theory to Applications",

note = "18th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, CTW 2020 ; Conference date: 14-09-2020 Through 16-09-2020",

url = "http://ctw2020.iasi.cnr.it/",

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AB - A bar-joint framework is a realisation of a graph consisting of stiff barslinked by universal joints. The framework is rigid if the only bar-length preserving continuous motions of the joints arise from isometries. A rigid framework is isostatic if deleting any single edge results in a flexible framework. Generically, rigidity depends only on the graph and we say an Assur graph is a pinned isostatic graph with no proper pinned isostatic subgraphs. Any pinned isostatic graph can be decomposed into Assur components which may be of use for mechanical engineers in decomposing mechanisms for simpler analysis and synthesis. A direction-length framework is a generalisation of bar-joint framework where some distance constraints are replaced by direction constraints. We initiate a theory of Assur graphs and Assur decompositions for direction-length frameworks using graph orientations and spanning trees and then analyse choices of pinning set.

KW - Assur decomposition

KW - Assur graph

KW - Bar-joint framework

KW - Direction-length framework

KW - Pinned framework

KW - Rigid graph

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