It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.
@article{bwmeta1.element.doi-10_2478_s11533-012-0147-y,
author = {Giorgi Khimshiashvili and Gaiane Panina and Dirk Siersma and Alena Zhukova},
title = {Critical configurations of planar robot arms},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {519-529},
zbl = {1319.51017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0147-y}
}
Giorgi Khimshiashvili; Gaiane Panina; Dirk Siersma; Alena Zhukova. Critical configurations of planar robot arms. Open Mathematics, Tome 11 (2013) pp. 519-529. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0147-y/
[1] Kapovich M., Millson J., On the moduli space of polygons in the Euclidean plane, J. Differential Geom., 1995, 42(2), 430–464 | Zbl 0854.51016
[2] Khimshiashvili G., Cyclic polygons as critical points, Proc. I. Vekua Inst. Appl. Math., 2008, 58, 74–83 | Zbl 1222.52010
[3] Khimshiashvili G., Panina G., Siersma D., Zhukova A., Extremal Configurations of Polygonal Linkages, Oberwolfach Preprints, 24, Mathematisches Forschungsinstitut, Oberwolfach, 2011, available at http://www.mfo.de/scientificprogramme/publications/owp/2011/OWP2011_24.pdf | Zbl 1319.51017
[4] Khimshiashvili G., Siersma D., Cyclic configurations of planar multiply penduli, preprint available at http://users.ictp.it/~pub_off/preprints-sources/2009/IC2009047P.pdf | Zbl 06269927
[5] Panina G., Khimshiashvili G., Cyclic polygons are critical points of area, J. Math. Sci. (N.Y.), 2009, 158(6), 899–903 http://dx.doi.org/10.1007/s10958-009-9417-z[Crossref] | Zbl 1193.52015
[6] Panina G., Zhukova A., Morse index of a cyclic polygon, Cent. Eur. J. Math., 2011, 9(2), 364–377 http://dx.doi.org/10.2478/s11533-011-0011-5[Crossref][WoS] | Zbl 1242.52018
[7] Takens F., The minimal number of critical points of a function on a compact manifold and the Lusternik-Schnirelman cathegory, Invent. Math., 1968, 6, 197–244 http://dx.doi.org/10.1007/BF01404825[Crossref] | Zbl 0198.56603