Symbolic Methods in Kinematics

Designing mechanical devices, called linkages, that perform a prescribed motion has been a topic that interested engineers and mathematicians for hundreds of years.

We apply techniques from algebraic geometry and symbolic computation to various problems in this area, for example the classification of closed 6R linkages, the study of pentapods and hexapods, or the construction of planar linkages. In this context, the concept of motion polynomial and the theory of bonds have been developed, which are powerful tools to answer questions arising in kinematics.

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Ongoing Projects

Publications

2021

[Legersky]

On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2

E. Bartzos, I.Z. Emiris, J. Legerský, E. Tsigaridas

Journal of Symbolic Computation 102, pp. 189-208. 2021. ISSN 0747-7171. [url]
[bib]
@article{RISC5992,
author = {E. Bartzos and I.Z. Emiris and J. Legerský and E. Tsigaridas},
title = {{On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2}},
language = {english},
journal = {Journal of Symbolic Computation},
volume = {102},
pages = {189--208},
isbn_issn = {ISSN 0747-7171},
year = {2021},
refereed = {yes},
length = {20},
url = {https://doi.org/10.1016/j.jsc.2019.10.015}
}

2020

[Capco]

Robots, computer algebra and 8 connected components

J. Capco, M. Safey el Din, J. Schicho

In: Proceedings of ISSAC 2020, I. Emiris, L. Zhi (ed.), pp. 62-69. 2020. 978-1-4503-7100-1. [url]
[bib]
@inproceedings{RISC6247,
author = {J. Capco and M. Safey el Din and J. Schicho},
title = {{Robots, computer algebra and 8 connected components}},
booktitle = {{Proceedings of ISSAC 2020}},
language = {english},
pages = {62--69},
isbn_issn = {978-1-4503-7100-1},
year = {2020},
editor = {I. Emiris and L. Zhi},
refereed = {yes},
length = {8},
url = {https://arxiv.org/abs/2008.13392}
}
[Grasegger]

Graphs with Flexible Labelings allowing Injective Realizations

G. Grasegger, J. Legerský, J. Schicho

Discrete Mathematics 343(6), pp. Art. 111713-. 2020. ISSN 0012-365X. [url]
[bib]
@article{RISC6012,
author = {G. Grasegger and J. Legerský and J. Schicho},
title = {{Graphs with Flexible Labelings allowing Injective Realizations}},
language = {english},
journal = {Discrete Mathematics},
volume = {343},
number = {6},
pages = {Art. 111713--},
isbn_issn = {ISSN 0012-365X},
year = {2020},
refereed = {yes},
length = {14},
url = {https://doi.org/10.1016/j.disc.2019.111713}
}
[Grasegger]

Flexible placements of graphs with rotational symmetry

S.Dewar, G. Grasegger, J. Legerský

arXiv. Technical report, 2020. [url]
[bib]
@techreport{RISC6091,
author = {S.Dewar and G. Grasegger and J. Legerský},
title = {{Flexible placements of graphs with rotational symmetry}},
language = {english},
year = {2020},
institution = {arXiv},
length = {9},
url = {https://arxiv.org/abs/2003.09328}
}
[Grasegger]

Combinatorics of Bricard’s octahedra

M. Gallet, G. Grasegger, J. Legerský, J. Schicho

arXiv. Technical report, 2020. [url]
[bib]
@techreport{RISC6095,
author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},
title = {{Combinatorics of Bricard’s octahedra}},
language = {english},
year = {2020},
institution = {arXiv},
length = {40},
url = {https://arxiv.org/pdf/2004.01236.pdf}
}
[Grasegger]

Computing Animations of Linkages with Rotational Symmetry (Media Exposition)

Sean Dewar, Georg Grasegger, Jan Legerský

In: 36th International Symposium on Computational Geometry (SoCG 2020), Sergio Cabello and Danny Z. Chen (ed.), Leibniz International Proceedings in Informatics (LIPIcs) 164, pp. 77:1-77:4. 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl, Germany, ISBN 978-3-95977-143-6. [url]
[bib]
@inproceedings{RISC6128,
author = {Sean Dewar and Georg Grasegger and Jan Legerský},
title = {{Computing Animations of Linkages with Rotational Symmetry (Media Exposition)}},
booktitle = {{36th International Symposium on Computational Geometry (SoCG 2020)}},
language = {english},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
volume = {164},
pages = {77:1--77:4},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum für Informatik},
address = {Dagstuhl, Germany},
isbn_issn = {ISBN 978-3-95977-143-6},
year = {2020},
editor = {Sergio Cabello and Danny Z. Chen},
refereed = {no},
length = {4},
url = {https://doi.org/10.4230/LIPIcs.SoCG.2020.77}
}
[Grasegger]

FlexRiLoG - A SageMath Package for Motions of Graphs

G. Grasegger, J. Legerský

In: Mathematical Software – ICMS 2020, Bigatti A., Carette J., Davenport J., Joswig M., de Wolff T. (ed.), Proceedings of ICMS 2020, Lecture Notes in Computer Science 12097, pp. 442-450. 2020. Springer, Cham, ISBN 978-3-030-52199-8. [url]
[bib]
@inproceedings{RISC6182,
author = {G. Grasegger and J. Legerský},
title = {{FlexRiLoG - A SageMath Package for Motions of Graphs}},
booktitle = {{ Mathematical Software – ICMS 2020}},
language = {english},
series = {Lecture Notes in Computer Science},
volume = {12097},
pages = {442--450},
publisher = {Springer, Cham},
isbn_issn = {ISBN 978-3-030-52199-8},
year = {2020},
editor = {Bigatti A. and Carette J. and Davenport J. and Joswig M. and de Wolff T.},
refereed = {no},
length = {9},
conferencename = {ICMS 2020},
url = {https://doi.org/10.1007/978-3-030-52200-1_44}
}
[Grasegger]

Bracing frameworks consisting of parallelograms

G. Grasegger, J. Legerský

arXiv. Technical report, 2020. [url]
[bib]
@techreport{RISC6203,
author = {G. Grasegger and J. Legerský},
title = {{Bracing frameworks consisting of parallelograms}},
language = {english},
year = {2020},
institution = {arXiv},
length = {20},
url = {https://arxiv.org/abs/2008.11521}
}
[Grasegger]

Zero-sum cycles in flexible polyhedra

M. Gallet, G. Grasegger, J. Legerský, J. Schicho

arXiv. Technical report, 2020. [url]
[bib]
@techreport{RISC6211,
author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},
title = {{Zero-sum cycles in flexible polyhedra}},
language = {english},
year = {2020},
institution = {arXiv},
length = {16},
url = {https://arxiv.org/abs/2009.14041}
}
[Grasegger]

On the Classification of Motions of Paradoxically Movable Graphs

G. Grasegger, J. Legerský, J. Schicho

Journal of Computational Geometry 11(1), pp. 548-575. 2020. ISSN: 1920-180X. [url]
[bib]
@article{RISC6239,
author = {G. Grasegger and J. Legerský and J. Schicho},
title = {{On the Classification of Motions of Paradoxically Movable Graphs}},
language = {english},
journal = {Journal of Computational Geometry},
volume = {11},
number = {1},
pages = {548--575},
isbn_issn = {ISSN: 1920-180X},
year = {2020},
refereed = {yes},
length = {27},
url = {https://doi.org/10.20382/jocg.v11i1a22}
}
[Grasegger]

Counting realizations of {L}aman graphs on the sphere

M. Gallet, G. Grasegger, J. Schicho

Electronic Journal of Combinatorics 27(2), pp. 1-18. 2020. 1077-8926 . [url]
[bib]
@article{RISC6243,
author = {M. Gallet and G. Grasegger and J. Schicho},
title = {{Counting realizations of {L}aman graphs on the sphere}},
language = {english},
journal = {Electronic Journal of Combinatorics},
volume = {27},
number = {2},
pages = {1--18},
isbn_issn = {1077-8926 },
year = {2020},
refereed = {yes},
length = {18},
url = {https://arxiv.org/abs/1903.01145}
}
[Li]

Invertible Paradoxic Loop Structures for Transformable Design

Li, Z. and Nawratil, G. and Rist, F. and Hensel, M.

Computer Graphics Forum, pp. 261-275. 2020. 1467-8659. [url]
[bib]
@article{RISC6233,
author = {Li and Z. and Nawratil and G. and Rist and F. and Hensel and M.},
title = {{Invertible Paradoxic Loop Structures for Transformable Design}},
language = {english},
journal = {Computer Graphics Forum},
pages = {261--275},
isbn_issn = {1467-8659},
year = {2020},
refereed = {yes},
length = {15},
url = {http://doi.org/10.1111/cgf.13928}
}
[Schicho]

Probabilities of incidence between lines and a plane curve over finite field

M. Gallet, M. Makhul, J. Schicho

Finite Fields and Their Applications 61, pp. 1-22. 2020. 1071-5797.
[bib]
@article{RISC6073,
author = {M. Gallet and M. Makhul and J. Schicho},
title = {{Probabilities of incidence between lines and a plane curve over finite field}},
language = {english},
journal = {Finite Fields and Their Applications},
volume = {61},
pages = {1--22},
isbn_issn = {1071-5797},
year = {2020},
refereed = {yes},
length = {22}
}
[Schicho]

On the multihomogeneous Bezout bound on the number of embeddings of minimally rigid graphs

V. Bartzos, I. Emiris, J. Schicho

Applicable Algebra in Engineering, Communication and Computing 31, pp. 325-357. 2020. 0938-1279. [url]
[bib]
@article{RISC6245,
author = {V. Bartzos and I. Emiris and J. Schicho},
title = {{On the multihomogeneous Bezout bound on the number of embeddings of minimally rigid graphs}},
language = {english},
journal = {Applicable Algebra in Engineering, Communication and Computing},
volume = {31},
pages = {325--357},
isbn_issn = {0938-1279},
year = {2020},
refereed = {yes},
length = {32},
url = {https://arxiv.org/abs/2005.14485}
}
[Schicho]

Counting Projections of Rational Curves

M. Gallet, J. Schicho

Israel Journal of Mathematics 240, pp. 503-538. 2020. 0021-2172 . [url]
[bib]
@article{RISC6246,
author = {M. Gallet and J. Schicho},
title = {{Counting Projections of Rational Curves}},
language = {english},
journal = {Israel Journal of Mathematics},
volume = {240},
pages = {503--538},
isbn_issn = {0021-2172 },
year = {2020},
refereed = {yes},
length = {36},
url = {https://arxiv.org/abs/1707.05264}
}

2019

[Capco]

Sum of Squares over Rationals

J. Capco, C. Scheiderer

RISC. Technical report, 2019. [url] [pdf]
[bib]
@techreport{RISC5884,
author = {J. Capco and C. Scheiderer},
title = {{Sum of Squares over Rationals}},
language = {english},
abstract = {Recently it has been shown that a multivariate (homogeneous) polynomialwith rational coefficients that can be written as a sum of squares offorms with real coefficients, is not necessarily a sum of squares offorms with rational coefficients. Essentially, only one constructionfor such forms is known, namely taking the $K/\Q$-norm of a sufficientlygeneral form with coefficients in a number field $K$. Whether thisconstruction yields a form with the desired properties depends onGalois-theoretic properties of $K$ that are not yet well understood.We construct new families of examples, and we shed new light on somewell-known open questions.},
year = {2019},
institution = {RISC},
length = {0},
url = {https://www3.risc.jku.at/~jcapco/public_files/ss18/sosq.html}
}
[Grasegger]

Graphs with Flexible Labelings

G. Grasegger, J. Legerský, J. Schicho

Discrete & Computational Geometry 62(2), pp. 461-480. 2019. 1432-0444. arXiv:1708.05298. [url]
[bib]
@article{RISC5803,
author = {G. Grasegger and J. Legerský and J. Schicho},
title = {{Graphs with Flexible Labelings}},
language = {english},
journal = {Discrete & Computational Geometry},
volume = {62},
number = {2},
pages = {461--480},
isbn_issn = {1432-0444},
year = {2019},
note = {arXiv:1708.05298},
refereed = {yes},
length = {20},
url = {https://doi.org/10.1007/s00454-018-0026-9}
}
[Grasegger]

On the existence of paradoxical motions of generically rigid graphs on the sphere

M. Gallet, G. Grasegger, J. Legerský, J. Schicho

arXiv. Technical report, 2019. [url]
[bib]
@techreport{RISC5977,
author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},
title = {{On the existence of paradoxical motions of generically rigid graphs on the sphere}},
language = {english},
year = {2019},
institution = {arXiv},
length = {40},
url = {https://arxiv.org/abs/1908.00467}
}
[Legersky]

Flexible and Rigid Labelings of Graphs

Jan Legerský

Research Institute for Symbolic Computation, Johannes Kepler University Linz. PhD Thesis. 2019. [url] [pdf]
[bib]
@phdthesis{RISC5941,
author = {Jan Legerský},
title = {{Flexible and Rigid Labelings of Graphs}},
language = {english},
year = {2019},
translation = {0},
school = {Research Institute for Symbolic Computation, Johannes Kepler University Linz},
length = {108},
url = {https://jan.legersky.cz/project/movablegraphs/}
}
[Schicho]

Projective and affine symmetries and equivalences of rational and polynomial surfaces

M. Hauer, B. Jüttler, J. Schicho

J. Comp. Appl. Math. 349, pp. 424-437. 2019. 0377-0427.
[bib]
@article{RISC5875,
author = {M. Hauer and B. Jüttler and J. Schicho},
title = {{Projective and affine symmetries and equivalences of rational and polynomial surfaces}},
language = {english},
journal = {J. Comp. Appl. Math.},
volume = {349},
pages = {424--437},
isbn_issn = {0377-0427},
year = {2019},
refereed = {yes},
length = {14}
}

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