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.

Publications

2019

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}
}

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}
}

2018

Rational General Solutions of Systems of First-Order Partial Differential Equations

Georg Grasegger, Alberto Lastra, J. Rafael Sendra, Franz Winkler

Journal of Computational and Applied Mathematics 331, pp. 88-103. 2018. ISSN: 0377-0427.
[bib]
@article{RISC5509,
author = {Georg Grasegger and Alberto Lastra and J. Rafael Sendra and Franz Winkler},
title = {{Rational General Solutions of Systems of First-Order Partial Differential Equations}},
language = {english},
journal = {Journal of Computational and Applied Mathematics},
volume = {331},
pages = {88--103},
isbn_issn = {ISSN: 0377-0427},
year = {2018},
refereed = {yes},
length = {16}
}

Graphs with Flexible Labelings

G. Grasegger, J. Legerský, J. Schicho

Discrete & Computational Geometry, pp. 1-20. 2018. 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},
pages = {1--20},
isbn_issn = {1432-0444},
year = {2018},
note = {arXiv:1708.05298},
refereed = {yes},
length = {20},
url = {https://doi.org/10.1007/s00454-018-0026-9}
}

Graphs with Flexible Labelings allowing Injective Realizations

G. Grasegger, J. Legerský, J. Schicho

arXiv. Technical report, 2018. [url]
[bib]
@techreport{RISC5806,
author = {G. Grasegger and J. Legerský and J. Schicho},
title = {{Graphs with Flexible Labelings allowing Injective Realizations}},
language = {english},
year = {2018},
institution = {arXiv},
length = {21},
url = {https://arxiv.org/abs/1811.06709}
}

The Number of Realizations of a Laman Graph

Jose Capco, Matteo Gallet, Georg Grasegger, Christoph Koutschan, Niels Lubbes, Josef Schicho

SIAM Journal on Applied Algebra and Geometry 2(1), pp. 94-125. 2018. 2470-6566. [url]
[bib]
@article{RISC5700,
author = {Jose Capco and Matteo Gallet and Georg Grasegger and Christoph Koutschan and Niels Lubbes and Josef Schicho},
title = {{The Number of Realizations of a Laman Graph}},
language = {english},
journal = {SIAM Journal on Applied Algebra and Geometry},
volume = {2},
number = {1},
pages = {94--125},
isbn_issn = {2470-6566},
year = {2018},
refereed = {yes},
length = {32},
url = {https://doi.org/10.1137/17M1118312}
}

On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs

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

In: ISSAC '18 Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation, C. Arreche (ed.), Proceedings of International Symposium on Symbolic and Algebraic Computation 2018, pp. 55-62. 2018. 978-1-4503-5550-6. [url]
[bib]
@inproceedings{RISC5804,
author = {E. Bartzos and I.Z. Emiris and J. Legerský and E. Tsigaridas},
title = {{On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs}},
booktitle = {{ISSAC '18 Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation}},
language = {english},
pages = {55--62},
isbn_issn = {978-1-4503-5550-6},
year = {2018},
editor = {C. Arreche},
refereed = {yes},
length = {8},
conferencename = {International Symposium on Symbolic and Algebraic Computation 2018},
url = {https://doi.org/10.1145/3208976.3208994}
}

Minimal non-integer alphabets allowing parallel addition

J. Legerský

Acta Polytechnica, Journal of Advanced Engineering 58(5), pp. 285-291. 2018. ISSN 1805-2363. [url]
[bib]
@article{RISC5807,
author = {J. Legerský},
title = {{Minimal non-integer alphabets allowing parallel addition}},
language = {english},
journal = {Acta Polytechnica, Journal of Advanced Engineering},
volume = {58},
number = {5},
pages = {285--291},
isbn_issn = {ISSN 1805-2363},
year = {2018},
refereed = {yes},
length = {7},
url = {https://doi.org/10.14311/AP.2018.58.0285}
}

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

arXiv. Technical report, 2018. [url]
[bib]
@techreport{RISC5810,
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},
year = {2018},
institution = {arXiv},
length = {22},
url = {https://arxiv.org/abs/1811.12800}
}

On sets defining few ordinary circles

A. Lin, M. Makhul, H. N. Mojarrad, J. Schicho, K. Swanepoel, F. de Zeeuw

Discrete and Computational Geometry 59, pp. 59-87. 2018. 0179-5376.
[bib]
@article{RISC5535,
author = {A. Lin and M. Makhul and H. N. Mojarrad and J. Schicho and K. Swanepoel and F. de Zeeuw},
title = {{On sets defining few ordinary circles}},
language = {english},
journal = {Discrete and Computational Geometry},
volume = {59},
pages = {59--87},
isbn_issn = {0179-5376},
year = {2018},
refereed = {yes},
length = {29}
}

Kinematic generation of Darboux cyclides

N. Lubbes, J. Schicho

Comp. Aided Geom. Des. 64, pp. -. 2018. 0167-8396.
[bib]
@article{RISC5876,
author = {N. Lubbes and J. Schicho},
title = {{Kinematic generation of Darboux cyclides}},
language = {english},
journal = {Comp. Aided Geom. Des.},
volume = {64},
pages = {--},
isbn_issn = {0167-8396},
year = {2018},
refereed = {yes},
length = {0}
}

Kempe's Universality Theorem for Rational Space Curves

Z. Li, J. Schicho, H.-P. Schröcker

Found. Comp. Math. 18, pp. 509-536. 2018. 1615-3375.
[bib]
@article{RISC5879,
author = {Z. Li and J. Schicho and H.-P. Schröcker},
title = {{Kempe's Universality Theorem for Rational Space Curves}},
language = {english},
journal = {Found. Comp. Math.},
volume = {18},
pages = {509--536},
isbn_issn = {1615-3375},
year = {2018},
refereed = {yes},
length = {28}
}

2017

Planar linkages following a prescribed motion

M. Gallet, C. Koutschan, Z. Li, G. Regensburger, J. Schicho, N. Villamizar

Math. Comp. 87, pp. 473-506. 2017. ISSN: 1088-6842. [url]
[bib]
@article{RISC5410,
author = {M. Gallet and C. Koutschan and Z. Li and G. Regensburger and J. Schicho and N. Villamizar},
title = {{Planar linkages following a prescribed motion}},
language = {english},
journal = {Math. Comp.},
volume = {87},
pages = {473--506},
isbn_issn = {ISSN: 1088-6842},
year = {2017},
refereed = {yes},
length = {34},
url = {http://www.ams.org/journals/mcom/2017-86-303/S0025-5718-2016-03120-8/}
}

The number of realizations of a Laman graph

Jose Capco, Georg Grasegger, Matteo Gallet, Christoph Koutschan, Niels Lubbes, Josef Schicho

Research Institute for Symbolic Computation (RISC/JKU). Technical report, 2017. [url] [pdf]
[bib]
@techreport{RISC5418,
author = {Jose Capco and Georg Grasegger and Matteo Gallet and Christoph Koutschan and Niels Lubbes and Josef Schicho},
title = {{The number of realizations of a Laman graph}},
language = {english},
abstract = {Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Using ideas from algebraic and tropical geometry, we provide a recursion formula for the number of complex solutions of such systems. },
year = {2017},
institution = {Research Institute for Symbolic Computation (RISC/JKU)},
length = {42},
url = {http://www.koutschan.de/data/laman/}
}

Computing the number of realizations of a Laman graph

Jose Capco, Georg Grasegger, Matteo Gallet, Christoph Koutschan, Niels Lubbes, Josef Schicho

In: Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017), Vadim Lozin (ed.), Proceedings of The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)61, pp. 207-213. 2017. ISSN 1571-0653. [url]
[bib]
@inproceedings{RISC5478,
author = {Jose Capco and Georg Grasegger and Matteo Gallet and Christoph Koutschan and Niels Lubbes and Josef Schicho},
title = {{Computing the number of realizations of a Laman graph}},
booktitle = {{Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017)}},
language = {english},
abstract = {Laman graphs model planar frameworks which are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. In a recent paper we provide a recursion formula for this number of realizations using ideas from algebraic and tropical geometry. Here, we present a concise summary of this result focusing on the main ideas and the combinatorial point of view.},
volume = {61},
pages = {207--213},
isbn_issn = {ISSN 1571-0653},
year = {2017},
editor = {Vadim Lozin},
refereed = {yes},
keywords = {Laman graph; minimally rigid graph; tropical geometry; euclidean embedding; graph realization},
length = {7},
conferencename = {The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)},
url = {http://www.koutschan.de/data/laman/}
}

Mobile Icosapods

M. Gallet, G. Nawratil, J. Schicho, J. Selig

Adv. Appl. Math. 88, pp. 1-25. 2017. ISSN: 0196-8858. [url]
[bib]
@article{RISC5411,
author = {M. Gallet and G. Nawratil and J. Schicho and J. Selig},
title = {{Mobile Icosapods}},
language = {english},
journal = {Adv. Appl. Math.},
volume = {88},
pages = {1--25},
isbn_issn = {ISSN: 0196-8858},
year = {2017},
refereed = {yes},
length = {25},
url = {http://dx.doi.org/10.1016/j.aam.2016.12.002}
}

New developments in geometry. Theory and applications

U. Hertrich-Jeromin, B. Jüttler, J. Schicho

Comp. Aided Geom. Des. 47, pp. 1-2. 2017. ISSN: 0167-8396. [url]
[bib]
@article{RISC5413,
author = {U. Hertrich-Jeromin and B. Jüttler and J. Schicho},
title = {{New developments in geometry. Theory and applications}},
language = {english},
journal = {Comp. Aided Geom. Des.},
volume = {47},
pages = {1--2},
isbn_issn = {ISSN: 0167-8396},
year = {2017},
refereed = {yes},
length = {2},
url = {http://dx.doi.org/10.1016/j.cagd.2016.08.002}
}

Liaison linkages

M. Gallet, G. Nawratil, J. Schicho

J. Symb. Comp. 79, pp. 65-98. 2017. ISSN: 0747-7171. [url]
[bib]
@article{RISC5414,
author = {M. Gallet and G. Nawratil and J. Schicho},
title = {{Liaison linkages}},
language = {english},
journal = {J. Symb. Comp.},
volume = {79},
pages = {65--98},
isbn_issn = {ISSN: 0747-7171},
year = {2017},
refereed = {yes},
length = {34},
url = {http://dx.doi.org/10.1016/j.jsc.2016.08.006}
}

2016

Algebraic Geometry methods in Kinematics: Mobile Pods

Matteo Gallet

Johannes Kepler University. PhD Thesis. October 2016.
[bib]
@phdthesis{RISC5393,
author = {Matteo Gallet},
title = {{Algebraic Geometry methods in Kinematics: Mobile Pods}},
language = {English},
year = {2016},
month = {October},
translation = {0},
school = {Johannes Kepler University},
length = {82}
}

The rational motion of minimal dual quaternion degree with prescribed trajectory

Zijia Li, , J. Schicho, Hans-Peter Schröcker

Comp. Aided Geom. Des. 41, pp. 1-9. 2016. ISSN: 0167-8396. [url]
[bib]
@article{RISC5408,
author = {Zijia Li and and J. Schicho and Hans-Peter Schröcker},
title = {{The rational motion of minimal dual quaternion degree with prescribed trajectory}},
language = {english},
journal = {Comp. Aided Geom. Des.},
volume = {41},
pages = {1--9},
isbn_issn = {ISSN: 0167-8396},
year = {2016},
refereed = {yes},
length = {9},
url = {http://dx.doi.org/10.1016/j.cagd.2015.10.002}
}

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