We consider a one-parametric family of conformal mappings from the upper half-plane onto the family of the polygons obtained by an angle-preserving shift of one or more vertices of some initial polygon. We obtain some differential equation for the family of mappings and some system of ordinary second-order differential equations for the preimages of the vertices of the family of polygons with Cauchy initial-value conditions.

中文翻译：

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从半平面到多边形族的一参数共形映射族

我们考虑从上半平面到多边形族的一参数共形映射族，该保角映射是通过某个初始多边形的一个或多个顶点的保角移动获得的。我们获得了带有映射初值条件的多边形族的顶点的原图像的一些微分方程和一些普通的二阶微分方程系统。