On curves and polygons with the equiangular chord property
On curves and polygons with the equiangular chord property
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关于具有等角弦性质的曲线和多边形
DOI：
10.2140/pjm.2015.274.305
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发表时间：
2013
影响因子：
0.6
通讯作者：
Yuwen Wang
中科院分区：
文献类型：
--
作者：
Tarik Aougab;X. Sun;S. Tabachnikov;Yuwen Wang
设C是球面S2、双曲平面H2或欧氏平面E2上的光滑凸曲线，具有以下性质：存在C的参数化x（t）;y（t），使得对于每个t，连接x（t）和y（t）的弦与C的夹角在两端.假设C不是一个圆，则E。古特金完全刻画了在欧几里得情形下存在这样一条曲线的角度。我们研究这个问题的无穷小版本的上下文中的其他两个常曲率的几何形状，特别是我们提供了一个完整的表征的角度，存在一个非平凡的无穷小变形的圆通过这样的曲线与相应的角度。我们还考虑了一个离散版本的欧氏多边形的这个属性，在这种情况下，我们给出了一个完整的描述所有非平凡的解决方案。
Let C be a smooth, convex curve on either the sphere S 2 , the hyperbolic plane H 2 or the Euclidean plane E 2 , with the following property: there exists , and parameterizations x(t);y(t) of C such that for each t, the angle between the chord connecting x(t) to y(t) and C is at both ends. Assuming that C is not a circle, E. Gutkin completely charac- terized the angles for which such a curve exists in the Euclidean case. We study the infinitesimal version of this problem in the context of the other two constant curvature geometries, and in particular we provide a complete characterization of the angles for which there exists a non-trivial infinitesimal deformation of a circle through such curves with corresponding angle . We also consider a discrete version of this property for Euclidean polygons, and in this case we give a complete description of all non-trivial solutions.