Locked and Unlocked Chains of Planar Shapes
We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to c...
| Main Authors: | , , , , , , , |
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| Format: | Article |
| Language: | en_US |
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Spring New York
2011
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| Online Access: | http://hdl.handle.net/1721.1/62017 https://orcid.org/0000-0003-3803-5703 |
| _version_ | 1811084448683261952 |
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| author | Connelly, Robert Demaine, Erik D. Demaine, Martin L. Fekete, Sandor P. Langerman, Stefan Mitchell, Joseph S. B. Ribo, Ares Rote, Gunter |
| author2 | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory |
| author_facet | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory Connelly, Robert Demaine, Erik D. Demaine, Martin L. Fekete, Sandor P. Langerman, Stefan Mitchell, Joseph S. B. Ribo, Ares Rote, Gunter |
| author_sort | Connelly, Robert |
| collection | MIT |
| description | We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle <90° admit locked chains, which is precisely the threshold beyond which the slender property no longer holds. |
| first_indexed | 2024-09-23T12:50:53Z |
| format | Article |
| id | mit-1721.1/62017 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:50:53Z |
| publishDate | 2011 |
| publisher | Spring New York |
| record_format | dspace |
| spelling | mit-1721.1/620172022-10-01T11:28:41Z Locked and Unlocked Chains of Planar Shapes Connelly, Robert Demaine, Erik D. Demaine, Martin L. Fekete, Sandor P. Langerman, Stefan Mitchell, Joseph S. B. Ribo, Ares Rote, Gunter Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Demaine, Erik D. Demaine, Erik D. Demaine, Martin L. We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle <90° admit locked chains, which is precisely the threshold beyond which the slender property no longer holds. 2011-04-01T18:34:56Z 2011-04-01T18:34:56Z 2010-05 2007-05 Article http://purl.org/eprint/type/JournalArticle 0179-5376 1432-0444 http://hdl.handle.net/1721.1/62017 Connelly, Robert et al. “Locked and Unlocked Chains of Planar Shapes.” Discrete & Computational Geometry 44.2 (2010): 439-462-462. https://orcid.org/0000-0003-3803-5703 en_US http://dx.doi.org/10.1007/s00454-010-9262-3 Discrete and Computational Geometry Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Spring New York MIT web domain |
| spellingShingle | Connelly, Robert Demaine, Erik D. Demaine, Martin L. Fekete, Sandor P. Langerman, Stefan Mitchell, Joseph S. B. Ribo, Ares Rote, Gunter Locked and Unlocked Chains of Planar Shapes |
| title | Locked and Unlocked Chains of Planar Shapes |
| title_full | Locked and Unlocked Chains of Planar Shapes |
| title_fullStr | Locked and Unlocked Chains of Planar Shapes |
| title_full_unstemmed | Locked and Unlocked Chains of Planar Shapes |
| title_short | Locked and Unlocked Chains of Planar Shapes |
| title_sort | locked and unlocked chains of planar shapes |
| url | http://hdl.handle.net/1721.1/62017 https://orcid.org/0000-0003-3803-5703 |
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