Geodesic
Proof of convergence in the problem of rectification
Sbornik. Mathematics, Tome 23 (1974) no. 1, pp. 69-83 
G. A. Gal'perin. Proof of convergence in the problem of rectification. Sbornik. Mathematics, Tome 23 (1974) no. 1, pp. 69-83. http://geodesic.mathdoc.fr/item/SM_1974_23_1_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The behavior of the vertices $A_1(t),\dots,A_n(t)$ of a polygonal line $\mathbf A(t)$ situated in $k$-dimensional Euclidean space is considered as $t\to\infty$ (each point $A_i(t\pm1)$, $1, is a linear combination of the points $A_{i-1}(t)$, $A_i(t)$ and $A_{i+1}(t)$; the points $A_1(t+1)$ and $A_n(t+1)$ are linear combinations of $A_1(t)$ and $A_2(t)$, and $A_{n-1}(t)$ and $A_n(t)$, respectively). It is proved that for any initial position $\mathbf A(0)$ the polygonal lines $\mathbf A(t)$ converge to one of two possible limits, namely a stationary or quasistationary polygonal line. Figures: 1. Bibliography: 2 titles.

[1] A. M. Leontovich, I. I. Pyatetskii-Shapiro, O. N. Stavskaya, “Nekotorye matematicheskie zadachi, svyazannye s formoobrazovaniem”, Avtomatika i telemekhanika, 1970, no. 4, 94–107 | MR | Zbl

[2] O. N. Stavskaya, “Issledovanie skhodimosti v zadache spryamleniya”, Matem. sb., 88 (130) (1972), 118–136