Abstract
AbstractFor a polygon $$x=(x_j)_{j\in \mathbb {Z}}$$
x
=
(
x
j
)
j
∈
Z
in $$\mathbb {R}^n$$
R
n
we consider the midpoints polygon $$(M(x))_j=\left( x_j+x_{j+1}\right) /2.$$
(
M
(
x
)
)
j
=
x
j
+
x
j
+
1
/
2
.
We call a polygon a soliton of the midpoints mapping M if its midpoints polygon is the image of the polygon under an invertible affine map. We show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on $$\mathbb {R}^n.$$
R
n
.
These smooth curves are also characterized as solutions of the differential equation $$\dot{c}(t)=Bc (t)+d$$
c
˙
(
t
)
=
B
c
(
t
)
+
d
for a matrix B and a vector d. For $$n=2$$
n
=
2
these curves are curves of constant generalized-affine curvature $$k_{ga}=k_{ga}(B)$$
k
ga
=
k
ga
(
B
)
depending on B parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.
Publisher
Springer Science and Business Media LLC
Reference11 articles.
1. Berlekamp, E.R., Gilbert, E.N., Sinden, F.W.: A polygon problem. Am. Math. Mon. 72, 233–241 (1965)
2. Blaschke, W.: Vorlesungen über Differentialgeometrie I. 2. Auflage (Grundlehren der math. Wiss. in Einzeldarstellungen, Bd. I). Verlag J. Springer, Berlin (1924)
3. Calabi, E., Olver, P.J., Tannenbaum, A.: Affine geometry, curve flows, and invariant numerical approximations. Adv. Math. 124, 154–196 (1996)
4. Darboux, G.: Sur un problème de géométrie élémentaire. Bull. Sci. Math. Astron. 2e série 2, 298–304 (1878)
5. Glickenstein, D., Liang, J.: Asymptotic behaviour of$$\beta $$-polygon flows. J. Geom. Anal. 28, 2902–2952 (2018)