Mobility of geometric constraint systems with extrusion symmetry
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909261866467328 |
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| author | Owen, John Schulze, Bernd |
| author_facet | Owen, John Schulze, Bernd |
| contents | If we take a (bar-joint) framework, prepare an identical copy of this framework, translate it by some vector $τ$, and finally join corresponding points of the two copies, then we obtain a framework with `extrusion' symmetry in the direction of $τ$. This process may be repeated $t$ times to obtain a framework whose underlying graph has $\mathbb{Z}_2^t$ as a subgroup of its automorphism group and which has `$t$-fold extrusion' symmetry. We show that while $t$-fold extrusion symmetry is not a point-group symmetry, the rigidity matrix of a framework with $t$-fold extrusion symmetry can still be transformed into a block-decomposed form in the analogous way as for point-group symmetric frameworks. This allows us to use Fowler-Guest-type character counts to analyse the mobility of such frameworks. We show that this entire theory also extends to the more general point-hyperplane frameworks with $t$-fold extrusion symmetry. Moreover, we show that under suitable regularity conditions the infinitesimal flexes we detect with our symmetry-adapted counts extend to finite (continuous) motions. Finally, we establish an algorithm that checks for finite motions via linearly displacing framework points along velocity vectors of infinitesimal motions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_12740 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mobility of geometric constraint systems with extrusion symmetry Owen, John Schulze, Bernd Metric Geometry Combinatorics 52C25, 70B99, 20C35 If we take a (bar-joint) framework, prepare an identical copy of this framework, translate it by some vector $τ$, and finally join corresponding points of the two copies, then we obtain a framework with `extrusion' symmetry in the direction of $τ$. This process may be repeated $t$ times to obtain a framework whose underlying graph has $\mathbb{Z}_2^t$ as a subgroup of its automorphism group and which has `$t$-fold extrusion' symmetry. We show that while $t$-fold extrusion symmetry is not a point-group symmetry, the rigidity matrix of a framework with $t$-fold extrusion symmetry can still be transformed into a block-decomposed form in the analogous way as for point-group symmetric frameworks. This allows us to use Fowler-Guest-type character counts to analyse the mobility of such frameworks. We show that this entire theory also extends to the more general point-hyperplane frameworks with $t$-fold extrusion symmetry. Moreover, we show that under suitable regularity conditions the infinitesimal flexes we detect with our symmetry-adapted counts extend to finite (continuous) motions. Finally, we establish an algorithm that checks for finite motions via linearly displacing framework points along velocity vectors of infinitesimal motions. |
| title | Mobility of geometric constraint systems with extrusion symmetry |
| topic | Metric Geometry Combinatorics 52C25, 70B99, 20C35 |
| url | https://arxiv.org/abs/2304.12740 |