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For a polygon $x=(x_j)_{j\in \mathbb{Z}}$ in $\mathbb{R}^n$ we consider themidpoints polygon $(M(x))_j=\left(x_j+x_{j+1}\right)/2\,.$ We call a polygon asoliton of the midpoints mapping $M$ if its midpoints polygon is the image ofthe polygon under an invertible affine map. We show that a large class of thesepolygons lie on an orbit of a one-parameter subgroup of the affine group actingon $\mathbb{R}^n.$ These smooth curves are also characterized as solutions ofthe differential equation $\dot{c}(t)=Bc (t)+d$ for a matrix $B$ and a vector$d.$ For $n=2$ these curves are curves of constant generalized-affine curvature$k_{ga}=k_{ga}(B)$ depending on $B$ parametrized by generalized-affine arclength unless they are parametrizations of a parabola, an ellipse, or ahyperbola.

Solitons of the midpoint mapping and affine curvature