论文标题
关于多边形的投影演变
On projective evolutes of polygons
论文作者
论文摘要
曲线的发展是其正常的信封。在本说明中，我们考虑了这种结构的一个自然离散的类似物：我们定义了射影平面中多边形侧面的投射垂直等分线，并研究了将多边形发送到其侧面的射击垂直双分配器形成的新的多边形的地图。我们认为该地图作用于投影多边形的模量空间。我们分析了五角大州的案例；在这种情况下，模量空间是二维的。地图的第二个迭代具有一个积分不可分割的，其水平曲线是立方曲线，并且在这些水平曲线上的转换与地图$ x \ mapsto -4x $ mod 1。我们还介绍了在己糖的情况下进行实验研究的结果。
The evolute of a curve is the envelope of its normals. In this note we consider a projectively natural discrete analog of this construction: we define projective perpendicular bisectors of the sides of a polygon in the projective plane, and study the map that sends a polygon to the new polygon formed by the projective perpendicular bisectors of its sides. We consider this map acting on the moduli space of projective polygons. We analyze the case of pentagons; the moduli space is 2-dimensional in this case. The second iteration of the map has one integral whose level curves are cubic curves, and the transformation on these level curves is conjugated to the map $x\mapsto -4x$ mod 1. We also present the results of an experimental study in the case of hexagons.
