Geodesic
Isometric deformations of planar quadrilaterals with constant index
Sbornik. Mathematics, Tome 205 (2014) no. 5, pp. 663-683 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider isometric deformations (motions) of polygons (so-called carpenter's rule problem) in the case of self-intersecting polygons with the additional condition that the index of the polygon is preserved by the motion. We provide general information about isometric deformations of planar polygons and give a complete solution of the carpenter's problem for quadrilaterals. Bibliography: 17 titles.
Keywords: planar polygons, linkages, carpenter's rule problem. 
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E. S. Zaputryaeva. Isometric deformations of planar quadrilaterals with constant index. Sbornik. Mathematics, Tome 205 (2014) no. 5, pp. 663-683. http://geodesic.mathdoc.fr/item/SM_2014_205_5_a3/

[1] R. Connelly, E. D. Demaine, G. Rote, “Straightening polygonal arcs and convexifying polygonal cycles”, Discrete Comput. Geom., 30:2 (2003), 205–239 | DOI | MR | Zbl

[2] I. Streinu, “A combinatorial approach to planar non-colliding robot arm motion planning”, 41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA, 2000), IEEE Comput. Soc. Press, Los Alamitos, CA, 2000, 443–453 | DOI | MR | Zbl

[3] W. J. Lenhart, S. H. Whitesides, “Reconfiguring closed polygonal chains in Euclidian $d$-space”, Discrete Comput. Geom., 13:1 (1995), 123–140 | DOI | MR | Zbl

[4] Y. Benoist, D. Hulin, “Itération de pliages de quadrilatères”, Invent. Math., 157:1 (2004), 147–194 | DOI | MR | Zbl

[5] K. Charter, T. Rogers, “The dynamics of quadrilateral folding”, Experiment. Math., 2:3 (1993), 209–222 | DOI | MR | Zbl

[6] G. Darboux, “De l'emploi des fonctions elliptiques dans la théorie du quadrilatère plan”, C. R. Acad. Sci. Paris, 88 (1879), 1183–1185, 1252–1255

[7] G. Darboux, “De l'emploi des fonctions elliptiques dans la théorie du quadrilatère plan”, Bull. Sci. Math. Astron., 3 (1879), 109–128

[8] F. Pécaut, Quadrilatères articulés, Journées nationales APM, 1997, 24 pp. http://www.math.unm.edu/~vageli/papers/FLEX/quadri_articule.pdf

[9] G. Toussaint, Simple proofs of a geometric property of four-bar linkages, Technical Report No. SOCS-01.4, School of Computer Science, McGill University, 2001, 15 pp. http://cgm.cs.mcgill.ca/~godfried/publications/four-bar.pdf

[10] H. Maehara, “Distance graphs in euclidean space”, Ryukyu Math. J., 5 (1992), 33–51 | MR | Zbl

[11] D. Leborgne, “Systèmes articulés et Géométrie au 19ème siècle”, Gazette des mathématiciens, Soc. Math. France, 14 (1980), 55–72 | Zbl

[12] I. Kh. Sabitov, “Chemu ravna summa uglov mnogougolnika?”, Kvant, 3 (2001), 7–12

[13] I. Kh. Sabitov, “Around the proof of the Legendre–Cauchy lemma on convex polygons”, Siberian Math. J., 45:4 (2004), 740–762 | DOI | MR | Zbl

[14] B. Jaggi, Configuration spaces of point sets with distance constrains, Ph. D. thesis, Univ. of Bern, 1992

[15] I. Kh. Sabitov, “Local theory of the bendings of surfaces”, Encyclopaedia Math. Sci., 48 (1992), 179–250 | DOI | MR | Zbl

[16] D. Zvonkine, “Configuration spaces of hinge constructions”, Russian J. Math. Phys., 5:2 (1997), 247–266 | MR | Zbl

[17] M. Kapovich, J. Millson, “On the moduli spaces of polygons in the Euclidean plane”, J. Differential Geom., 42:1 (1995), 133–164 | MR | Zbl