{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:07:58Z","timestamp":1776802078460,"version":"3.51.2"},"reference-count":26,"publisher":"American Mathematical Society (AMS)","issue":"303","license":[{"start":{"date-parts":[[2017,4,13]],"date-time":"2017-04-13T00:00:00Z","timestamp":1492041600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P26607"],"award-info":[{"award-number":["P26607"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P26607"],"award-info":[{"award-number":["P26607"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["W1214"],"award-info":[{"award-number":["W1214"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P27229"],"award-info":[{"award-number":["P27229"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P26607"],"award-info":[{"award-number":["P26607"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P26607"],"award-info":[{"award-number":["P26607"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["W1214"],"award-info":[{"award-number":["W1214"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P27229"],"award-info":[{"award-number":["P27229"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P26607"],"award-info":[{"award-number":["P26607"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P26607"],"award-info":[{"award-number":["P26607"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["W1214"],"award-info":[{"award-number":["W1214"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P27229"],"award-info":[{"award-number":["P27229"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>Designing mechanical devices, called linkages, that draw a given plane curve has been a topic that interested engineers and mathematicians for hundreds of years, and recently also computer scientists. Already in 1876, Kempe proposed a procedure for solving the problem in full generality, but his constructions tend to be extremely complicated. We provide a novel algorithm that produces much simpler linkages, but works only for parametric curves. Our approach is to transform the problem into a factorization task over some noncommutative algebra. We show how to compute such a factorization, and how to use it to construct a linkage tracing a given curve.<\/p>","DOI":"10.1090\/mcom\/3120","type":"journal-article","created":{"date-parts":[[2016,3,23]],"date-time":"2016-03-23T14:52:48Z","timestamp":1458744768000},"page":"473-506","source":"Crossref","is-referenced-by-count":28,"title":["Planar linkages following a prescribed motion"],"prefix":"10.1090","volume":"86","author":[{"given":"Matteo","family":"Gallet","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christoph","family":"Koutschan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zijia","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Georg","family":"Regensburger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Josef","family":"Schicho","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nelly","family":"Villamizar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,4,13]]},"reference":[{"key":"1","unstructured":"Timothy G. Abbott, Generalizations of Kempe\u2019s universality theorem, Master\u2019s thesis, Massachusetts Institute of Technology, 2008. Available at \\url{http:\/\/web.mit.edu\/tabbott\/www\/papers\/mthesis.pdf}."},{"key":"2","unstructured":"Wilhelm Blaschke and Hans R. M\u00fcller, Ebene Kinematik, Verlag von R. Oldenbourg, M\u00fcnchen, 1956."},{"key":"3","series-title":"Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03718-8","volume-title":"Real algebraic geometry","volume":"36","author":"Bochnak, Jacek","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/3540646639"},{"key":"4","series-title":"North-Holland Series in Applied Mathematics and Mechanics","isbn-type":"print","volume-title":"Theoretical kinematics","volume":"24","author":"Bottema, O.","year":"1979","ISBN":"https:\/\/id.crossref.org\/isbn\/0444851240"},{"key":"5","doi-asserted-by":"crossref","unstructured":"Robert Connelly and Erik D. Demaine, Geometry and topology of polygonal linkages, In Jacob E. Goodman and Joseph O\u2019Rourke, editors, Handbook of Discrete and Computational Geometry, second edition, Chapman & Hall\/CRC, Boca Raton, FL, 2004, pp. 197\u2013218.","DOI":"10.1201\/9781420035315.ch9"},{"key":"6","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511735172","volume-title":"Geometric folding algorithms","author":"Demaine, Erik D.","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521857574"},{"issue":"5","key":"7","doi-asserted-by":"publisher","first-page":"460","DOI":"10.1007\/BF02948787","article-title":"Automated generation of Kempe linkage and its complexity","volume":"14","author":"Gao, Xiaoshan","year":"1999","journal-title":"J. Comput. Sci. Tech.","ISSN":"https:\/\/id.crossref.org\/issn\/1000-9000","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","unstructured":"Xiao-Shan Gao, Chang-Cai Zhu, Shang-Ching Chou, and Jian-Xin Ge, Automated generation of Kempe linkages for algebraic curves and surfaces, Mechanism and Machine Theory 36 (2001), no. 9, 1019\u20131033.","DOI":"10.1016\/S0094-114X(01)00040-4"},{"key":"9","doi-asserted-by":"crossref","unstructured":"G\u00e1bor Heged\u00fcs, Josef Schicho, and Hans-Peter Schr\u00f6cker, Construction of overconstrained linkages by factorization of rational motions, In Jadran Lenar\u010di\u010d and Manfred Husty, editors, Latest Advances in Robot Kinematics, Springer Netherlands, 2012, pp. 213\u2013220.","DOI":"10.1007\/978-94-007-4620-6_27"},{"key":"10","doi-asserted-by":"crossref","unstructured":"G\u00e1bor Heged\u00fcs, Josef Schicho, and Hans-Peter Schr\u00f6cker, Factorization of Rational Curves in the Study Quadric, Mechanism and Machine Theory 69 (2013), 142\u2013152.","DOI":"10.1016\/j.mechmachtheory.2013.05.010"},{"key":"11","doi-asserted-by":"crossref","unstructured":"Manfred L. Husty and Hans-Peter Schr\u00f6cker, Kinematics and algebraic geometry, In J. Michael McCarthy, editor, 21st Century Kinematics, Springer London, 2013, pp. 85\u2013123.","DOI":"10.1007\/978-1-4471-4510-3_4"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1007\/PL00009462","article-title":"Configuration spaces of mechanical linkages","volume":"22","author":"Jordan, D.","year":"1999","journal-title":"Discrete Comput. Geom.","ISSN":"https:\/\/id.crossref.org\/issn\/0179-5376","issn-type":"print"},{"issue":"6","key":"13","doi-asserted-by":"publisher","first-page":"1051","DOI":"10.1016\/S0040-9383(01)00034-9","article-title":"Universality theorems for configuration spaces of planar linkages","volume":"41","author":"Kapovich, Michael","year":"2002","journal-title":"Topology","ISSN":"https:\/\/id.crossref.org\/issn\/0040-9383","issn-type":"print"},{"key":"14","doi-asserted-by":"crossref","unstructured":"Alfred B. Kempe, On a general method of describing plane curves of the nth degree by linkwork, Proc. London Math. Soc. 213 (1876), S1-7(1).","DOI":"10.1112\/plms\/s1-7.1.213"},{"issue":"1","key":"15","first-page":"33","article-title":"Planar linkages and algebraic sets","volume":"23","author":"King, Henry C.","year":"1999","journal-title":"Turkish J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1300-0098","issn-type":"print"},{"key":"16","unstructured":"Alexander Kobel, Automated generation of Kempe linkages for algebraic curves in a dynamic geometry system, Bachelor\u2019s thesis, University of Saarbr\u00fccken, 2008. Available at \\url{https:\/\/people.mpi-inf.mpg.de\/ akobel\/publications\/Kobel08-kempe-linkages.pdf}."},{"key":"17","unstructured":"Christoph Koutschan, Mathematica package PlanarLinkages.m and electronic supplementary material for the paper Planar linkages following a prescribed motion, this issue, 2016. Available at \\url{http:\/\/www.koutschan.de\/data\/link\/}."},{"key":"18","first-page":"Art. B42q, 67","article-title":"Advanced determinant calculus","volume":"42","author":"Krattenthaler, C.","year":"1999","journal-title":"S\\'{e}m. Lothar. Combin."},{"issue":"2","key":"19","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/PL00012583","article-title":"Jacobians of symmetric polynomials","volume":"6","author":"Lascoux, Alain","year":"2002","journal-title":"Ann. Comb.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-0006","issn-type":"print"},{"key":"20","volume-title":"Le\\c{c}ons sur les Constructions G\\'{e}om\\'{e}triques","author":"Lebesgue, Henri","year":"1950"},{"key":"21","unstructured":"Zijia Li, Josef Schicho, and Hans-Peter Schr\u00f6cker, Factorization of motion polynomials, CoRR, abs\/1502.07600, 2015. Available at \\url{http:\/\/arxiv.org\/abs\/1502.07600}."},{"key":"22","unstructured":"Joseph Malkevitch, Linkages: From fingers to robot arms, AMS Feature Column, September 2002. Available at \\url{https:\/\/www.ams.org\/samplings\/feature-column\/fcarc-linkages1}."},{"key":"23","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511975028","volume-title":"How to fold it","author":"O\u2019Rourke, Joseph","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521145473"},{"issue":"6","key":"24","doi-asserted-by":"publisher","first-page":"1105","DOI":"10.1112\/blms\/bdp087","article-title":"Continuous curves from infinite Kempe linkages","volume":"41","author":"Owen, John","year":"2009","journal-title":"Bull. Lond. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"key":"25","doi-asserted-by":"crossref","unstructured":"Mark Plecnik, J. Michael McCarthy, and Charles W. Wampler, Kinematic Synthesis of a Watt I Six-Bar Linkage for Body Guidance, In Advances in Robot Kinematics, Springer International Publishing, 2014, pp. 317\u2013325.","DOI":"10.1007\/978-3-319-06698-1_33"},{"key":"26","series-title":"Monographs in Computer Science","isbn-type":"print","volume-title":"Geometric fundamentals of robotics","author":"Selig, J. M.","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/0387208747","edition":"2"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2017-86-303\/S0025-5718-2016-03120-8\/S0025-5718-2016-03120-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-303\/S0025-5718-2016-03120-8\/S0025-5718-2016-03120-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:00:05Z","timestamp":1776798005000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-303\/S0025-5718-2016-03120-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,4,13]]},"references-count":26,"journal-issue":{"issue":"303","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["S0025-5718-2016-03120-8"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3120","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2016,4,13]]}}}