Let $S$ be a set of $n$ points in the plane, $\wp(S)$ be the set of all simple polygons crossing $S$, $\gamma_P$ be the maximum angle of polygon $P \in \wp(S)$ and $\theta =min_{P\in\wp(S)} \gamma_P$. In this paper, we prove that $\theta\leq 2\pi-\frac{2\pi}{r.m}$ where $m$ and $r$ are the number of edges and inner points of the convex hull of $S$, respectively. We also propose an algorithm to construct a polygon with the said upper bound on its angles. Constructing a simple polygon with angular constraint on a given set of points in the plane can be used for path planning in robotics. Moreover, we improve our upper bound on $\theta$ and prove that this is tight for $r=1$.

翻译：让$S$成为平面上一套美元点数, $\wp(S) $(S) 是所有简单的多边形跨越$S$, $\gamma_P$(P$) 是多边形$P\ wp(S)$和$\theta=min ⁇ P\in\wp(S)}\gamma_P$。 在本文中, 我们证明$\theta\leq 2\pi-\frac{2\pi ⁇ r.m} 是所有简单的多边形跨越$S$, $和$$( $) 的集合体的边缘和内端点数, $( $) 。 我们还提议了一种算法, 用来在角上方建造一个多边形, 上面绑定的圆形。 构造一个对飞机上指定一组点有角限制的简单多边形, 可用于机器人的路径规划。 此外, 我们改进$\\\\\\\\\\\r\r\r\r. m} 并证明这很紧。