## Planar linkages following a prescribed motion

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- by Matteo Gallet, Christoph Koutschan, Zijia Li, Georg Regensburger, Josef Schicho and Nelly Villamizar PDF
- Math. Comp.
**86**(2017), 473-506 Request permission

## Abstract:

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.## References

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## Additional Information

**Matteo Gallet**- Affiliation: Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria
- MR Author ID: 1094242
- Email: matteo.gallet@ricam.oeaw.ac.at
**Christoph Koutschan**- Affiliation: Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria
- Email: christoph.koutschan@ricam.oeaw.ac.at
**Zijia Li**- Affiliation: Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria
- MR Author ID: 978116
- Email: zijia.li@ricam.oeaw.ac.at
**Georg Regensburger**- Email: georg.regensburger@ricam.oeaw.ac.at
**Josef Schicho**- Affiliation: Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenberger Straße 69, 4040 Linz, Austria
- MR Author ID: 332588
- Email: josef.schicho@ricam.oeaw.ac.at
**Nelly Villamizar**- Email: nelly.villamizar@ricam.oeaw.ac.at
- Received by editor(s): February 19, 2015
- Received by editor(s) in revised form: June 13, 2015
- Published electronically: April 13, 2016
- Additional Notes: The first, second, and third authors were supported by the Austrian Science Fund (FWF): W1214.

The first author was also supported by the Austrian Science Fund (FWF): P26607 - “Algebraic Methods in Kinematics: Motion Factorisation and Bond Theory”

The fourth author was supported by the Austrian Science Fund (FWF): P27229. - © Copyright 2016 American Mathematical Society
- Journal: Math. Comp.
**86**(2017), 473-506 - MSC (2010): Primary 70B15, 68W30, 70G55, 20G20, 16Z05, 14P05, 12Y05
- DOI: https://doi.org/10.1090/mcom/3120
- MathSciNet review: 3557808