Geodesic
What is a hinge mechanism? And what did Kempe prove?
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Classical and Modern Geometry" Dedicated to the 100th Anniversary of the Birth of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 1, Tome 179 (2020), pp. 16-28.

Voir la notice de l'article provenant de la source Math-Net.Ru

For a long time, it was believed that Alfred Kempe proved (1876) a theorem about the possibility of plotting an arbitrary plane algebraic curve by parts using hinge mechanisms. However, at the end of the 20th century, specialists in algebraic geometry, having rediscovered and developed this result in modern language, began to assert that Kempe's reasoning contained significant gaps and errors. In the author's opinion, these charges are unfounded. In this paper, the author tries to substantiate this point of view. Moreover, the notions of a hinge mechanism and its configuration space are analyzed.
Mots-clés : flat hinge mechanism, mechanism configuration space, Kempe theorem. 
@article{INTO_2020_179_a2,
     author = {M. D. Kovalev},
     title = {What is a hinge mechanism? {And} what did {Kempe} prove?},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {16--28},
     publisher = {mathdoc},
     volume = {179},
     year = {2020},
     language = {ru},
     url = {https://geodesic-test.mathdoc.fr/item/INTO_2020_179_a2/}
}
TY  - JOUR
AU  - M. D. Kovalev
TI  - What is a hinge mechanism? And what did Kempe prove?
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2020
SP  - 16
EP  - 28
VL  - 179
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/INTO_2020_179_a2/
LA  - ru
ID  - INTO_2020_179_a2
ER  - 
%0 Journal Article
%A M. D. Kovalev
%T What is a hinge mechanism? And what did Kempe prove?
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2020
%P 16-28
%V 179
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/INTO_2020_179_a2/
%G ru
%F INTO_2020_179_a2
M. D. Kovalev. What is a hinge mechanism? And what did Kempe prove?. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Classical and Modern Geometry" Dedicated to the 100th Anniversary of the Birth of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 1, Tome 179 (2020), pp. 16-28. https://geodesic-test.mathdoc.fr/item/INTO_2020_179_a2/

[1] Artobolevskii I. I., Teoriya mekhanizmov, Nauka, M., 1965 | MR

[2] Gilbert D., Kon-Fossen S., Naglyadnaya geometriya, Nauka, M., 1981

[3] Gorin E. A., “Ob asimptoticheskikh svoistvakh mnogochlenov i algebraicheskikh funktsii ot neskolkikh peremennykh”, Usp. mat. nauk., 16:1 (97) (1961), 91–118 | MR | Zbl

[4] Kovalev M. D., “Geometricheskaya teoriya sharnirnykh ustroistv”, Izv. RAN. Ser. mat., 58:1 (1994), 45–70 | Zbl

[5] Kovalev M. D., “O vosstanovimosti sharnirnikov po vnutrennim napryazheniyam”, Izv. RAN. Ser. mat., 61:4 (1997), 37–66 | MR | Zbl

[6] Kovalev M. D., “Ustoichivost sharnirnikov, sharnirnykh ustroistv i skhem”, Itogi nauki i tekhn. Sovr. mat. prilozh., 68 (1999), 65–86 | Zbl

[7] Kovalev M. D., “Voprosy geometrii sharnirnykh ustroistv i skhem”, Vestn. MGTU. Ser. Mashinostroenie., 2001, no. 4, 33–51

[8] Levitskii N. I., Teoriya mekhanizmov i mashin. Terminologiya, Nauka, M., 1984

[9] Oshemkov A. A., Popelenskii F. Yu., Tuzhilin A. A., Fomenko A. T., Shafarevich A. I., Kurs naglyadnoi geometrii i topologii, LENAND, M., 2015

[10] Abbott T., Generalizations of Kempe’s Universality Theorem, Massachusetts Institute of Technology, Cambridge, Massachusetts, 2008

[11] Asimov L., Roth B., “The rigidity of Graphs. II”, J. Math. Anal. Appl., 68:1 (1979), 171–190 | DOI | MR

[12] Bottema O., Roth B., Theoretical Kinematics, North-Holland, Amsterdam, 1979 | MR | Zbl

[13] Crapo H., “Structural Rigidity”, Struct. Topol., 1979, no. 1, 26–45 | MR | Zbl

[14] Demaine E., O'Rourke J., Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge Univ. Press, 2007 | MR | Zbl

[15] Graver J., Servatius B., Servatius H., Combinatorial Rigidity, Am. Math. Soc., Providence. Rhode Island, 1993 | MR | Zbl

[16] Hunt K. H., Kinematic geometry of mechanisms, Clarendon Press, New York, 1978 | MR | Zbl

[17] Jordan D., Steiner M., “Configuration spaces of mechanical linkages”, Discr. Comput. Geom., 22 (1999), 297-–315 | DOI | MR | Zbl

[18] Kapovich M., Millson J. J., “Universality theorems for configurations of planar linkages”, Topology., 41:6 (2002), 1051–1107 | DOI | MR | Zbl

[19] Kempe A. B., “On a general method of describing plane curves of the $n$th degree by Linkwork”, Proc. London Math. Soc., 7:102 (1876), 213–216 | MR | Zbl

[20] King H. C., Planar linkages and algebraic sets, arXiv: 9807023 [math.AG] | MR

[21] King H. C., Semiconfiguration spaces of planar linkages, arXiv: 9810130 [math.AG]

[22] King H. C., Configuration Spaces of Linkages in $R^n$, arXiv: 9811138 [math.AG]

[23] Kovalev M., “Results and problems of the geometry of hinged devices”, Proc. 12th World Congress in Mechanism and Machine Science, Besançon, France, 2007

[24] Power S., Elementary proofs of Kempe universality, arXiv: 1511.09002v2 [math.MG] | MR