论文标题
与多边形的侧面相交
Intersecting the sides of a polygon
论文作者
论文摘要
考虑地图$ s $，该$ s $将平面多边形$ p $发送给新的多边形$ s（p）$，其顶点是$ p $的第二个最接近的侧面的交点。该地图是著名的Pentagram地图的倒数。在本文中，我们研究了地图$ S $的动态。也就是说，我们解决了一个问题，即在$ s $的迭代下是否保持凸的凸。计算机实验表明，这几乎永远不会发生。我们证明，在$ s $的迭代下仍然保持凸的一组多边形的量度为零，而且它是Codimension二的代数亚变量。我们还讨论了削减这种子不同的方程式，以及在五角大州的情况下的几何含义。
Consider the map $S$ which sends a planar polygon $P$ to a new polygon $S(P)$ whose vertices are the intersection points of second nearest sides of $P$. This map is the inverse of the famous pentagram map. In this paper we investigate the dynamics of the map $S$. Namely, we address the question of whether a convex polygon stays convex under iterations of $S$. Computer experiments suggest that this almost never happens. We prove that indeed the set of polygons which remain convex under iterations of $S$ has measure zero, and moreover it is an algebraic subvariety of codimension two. We also discuss the equations cutting out this subvariety, as well as their geometric meaning in the case of pentagons.
