论文标题
中点映射和仿射曲率的孤子
Solitons of the midpoint mapping and affine curvature
论文作者
论文摘要
对于polyon $ x =（x_j）_ {j \ in \ mathbb {z}} $ in $ \ mathbb {r}^n $，我们考虑中点polygon $（m（x）_ j = \ left（x_j+x_+x_+x_+x_+1}}它的中点多边形是在可逆仿射图下多边形的图像。我们表明，其中一大批这些多边形位于仿期组的一个参数亚组的轨道上，该组作用于$ \ mathbb {r}^n。$这些平滑曲线也被特征为微分方程$ \ dot $ \ dot {c}（c}（c}（c}（c}（t）= bc（t）= bc（t）+d $ n Contrandion $ n = $ n = $ n = cluves $ n = vector $ d。概括 - 抗曲率$ k_ {ga} = k_ {ga}（b）$，具体取决于$ b $由概括性 - 折叠弧长参数化，除非它们是抛物线，椭圆形或多金的参数化。
For a polygon $x=(x_j)_{j\in \mathbb{Z}}$ in $\mathbb{R}^n$ we consider the midpoints polygon $(M(x))_j=\left(x_j+x_{j+1}\right)/2\,.$ 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 $\mathbb{R}^n.$ These smooth curves are also characterized as solutions of the 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 arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.
